Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces
aa r X i v : . [ m a t h . M G ] F e b Discrete groups of packed, non-positively curved,Gromov hyperbolic metric spaces
Nicola Cavallucci, Andrea Sambusetti
Abstract.
We prove a quantitative version of the classical Tits’ alternative for discretegroups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicomb-ing. Some geometric consequences, as uniform estimates on systole, diastole, algebraicentropy and critical exponent of the groups, will be presented. Finally we will study thebehaviour of these group actions under limits, providing new examples of compact classesof metric spaces.
Contents δ -hyperbolic spaces . . . . . . . . . 22 δ -hyperbolic GCB-spaces 25 ℓ ≤ ε . . . . . . . . . . . . . . . . . . . . 335.2 Proof of Theorem 1.1, case ℓ > ε . . . . . . . . . . . . . . . . . . . . 375.2.1 Ping-pong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2.2 Proof of Theorem 1.1 (ii), when d ( α, β ) ≤ ε . . . . . . . . . . . . 415.2.3 Proof of Theorem 1.1(ii), when ε < d ( α, β ) ≤ δ. . . . . . . . . 445.2.4 Proof of Theorem 1.1 (ii), when d ( α, β ) > δ . . . . . . . . . . . . 45 Introduction
The aim of this paper is to extend the results of [BCGS17] to non-cocompact actions of discrete groups of Gromov hyperbolic spaces. In fact, while non-cocompact actions of groups Γ on general metric spaces X are consideredto some extent in [BCGS17] (especially in Chapter 6), the underlying as-sumption in that work is that the group Γ admits also a cocompact actionor a “well-behaved” action on some Gromov δ -hyperbolic space X , namelywith some prescribed lower bound | g | ≥ ℓ for the minimal or asymptoticdisplacement of every g ∈ Γ on X (such groups are called ( δ , ℓ ) -thick bythe authors).On the other hand, we will be interested here to general discrete group ac-tions on Gromov-hyperbolic metric spaces X satisfying a packing condition :that is, an upper bound of the cardinality of any r -separated set inside anyball of radius r . This condition bounds the metric complexity of the space X at some fixed scale , and should be thought as a macroscopic, weak replace-ment of a lower bound on the curvature: for instance, in the case of Rie-mannian manifolds, it is strictly weaker than a lower Ricci curvature bound.It also seems, a priori, a much stronger condition than the bounded entropy assumption adopted in [BCGS17] (see Lemma 6.2); but, actually, most ofthe work in [BCGS17] is proving that a group Γ acting cocompactly on a δ -hyperbolic space X with bounded entropy does satisfies an inequality à-la Bishop-Gromov, hence a uniform packing condition (see Theorem 5.1 of[BCGS17], and Sec.2 therein for a complete comparison of the packing as-sumptions with curvature or entropy bounds).We will restrict our attention to actions of groups on Gromov-hyperbolicspaces which are complete and possess a convex geodesic bicombing (see Sec. 2.1 for precise definitions and properties). The notion of met-ric space with a convex geodesic bicombing is one of the weakest forms ofnon-positive curvature, and includes many geometric classes like CAT (0) andBusemann convex metric spaces, Banach spaces, injective metric spaces, etc.For instance, it is known that any Gromov hyperbolic group acts geometri-cally on a proper metric space supporting a convex geodesic bicombing (cp.[Des15]), while the existence of a geometric action on a CAT(0)-space forthese groups is a well-known open problem. Moreover this class of spaces isclosed under limit operations (as ultralimits), while Busemann convex spacesare not; a property that makes it preferable to work in this setting.We will require an additional property to our classes of metric spaces:the geodesically completeness of the bicombing, which is equivalent to thestandard geodesically completeness assumption in case of CAT (0) or Buse-mann convex spaces. This is an assumption which is usually required, even incase of CAT (0) -spaces, in order to control much better the local and asymp-totic geometry (see the foundational works of B. Kleiner [Kle99] and of A.Lytchak and K. Nagano [Nag18], [LN19], [LN20]). For instance in [CS20] weproved that, for complete and geodesically complete CAT (0) -spaces, a pack-ing condition at some scale r yields explicit control of the packing conditionat any scale. With similar techniques the same is true for spaces supportinga geodesically complete, convex, geodesic bicombing (see Proposition 2.5).2 couple ( X, σ ) , where X is a complete metric space and σ a geodesicallycomplete, convex geodesic bicombing, will be called a GCB-space for short.The set up of δ -hyperbolic GCB-spaces satisfying a uniform P -packing con-dition at some fixed scale r establishes a solid framework where many toolsare available, and which is large enough to contain many interesting exam-ples besides Riemannian manifolds (see Sec.2.2 and [CS20] for examples).First of all, due to the propagation of the packing property at scales smallerthan the hyperbolicity constant, it is possible to obtain precise estimates ofthe ε -Margulis’ domain of an isometry on such a space X , for values of ε smaller than δ , a tool which will be extensively used in the paper.Secondly the P -packing condition at some fixed scale r on X implies, byBreuillard-Green-Tao’s work, an analogue for metric spaces of the celebrated Margulis Lemma for Riemannian manifolds: there exists a constant ε , onlydepending on the packing constants P , r , such that for every discrete groupof isometries Γ of X the ε -almost stabilizer Γ ε ( x ) of any point x is virtuallynilpotent (cp. [BGT11], Corollary 11.17); that is, the elements of Γ whichdisplace x less than ε generate a virtually nilpotent group.Finally, this class is compact for the Gromov-Hausdorff topology, for fixed δ , P and r , a fact which we will use to investigate limits of group actions onthese spaces in Section 6.4.The first basic tool we needed to develop for our analysis, is a quanti-tative form of the Tits alternative in the framework of Gromov-hyperbolic,GCB-spaces. Recall that the classical Tits alternative, proved by J. Tits[Tit72], says that any finitely generated linear group Γ over a commutativefield K either is virtually solvable or contains a non-abelian free subgroup.This result has been extended to other classes of groups during the years, forinstance (restricting ourselves to non-positively curved spaces, and withoutintending to be exhaustive) to discrete, non-elementary groups of isometriesof Gromov-hyperbolic spaces, of proper CAT(0)-spaces of rank one, of finitedimensional CAT (0) -cube complexes etc., [Gro87], [Ham09], [SW05], [Osi16].A weaker form of the alternative, generally easier to establish, asks for theexistence of free semigroups in Γ instead of free subgroups, provided that Γ isnot virtually solvable . The original result has also been improved by quanti-fying the depth in Γ of the free subgroup with respect to some fixed generat-ing set S of Γ ; that is the minimal N such that S N contains a free subgroupor sub-semigroup. The first forms of quantification of the Tits Alternative forGromov hyperbolic groups were proved by T. Delzant [Del96], by M. Koubi[Kou98] (for Gromov hyperbolic groups with torsion) and by G. Arzant-sheva and I. Lysenok [AL06] (for subgroups of a given hyperbolic group),for a constant N depending however on the group Γ under consideration. Notice that, in this weaker form, the Tits Alternative is no longer a dichotomy forlinear groups, since it is well known that there exist solvable groups of GL ( n, R ) whichalso contain free semigroups (and actually, any finitely generated solvable group which isnot virtually nilpotent contains a free semigroup on two generators [Ros74]). It remains adichotomy for those classes of groups for which “virtually solvable” implies sub-exponentialgrowth, e.g. hyperbolic groups, groups acting geometrically on CAT(0)-spaces etc. N ( C ) such thatfor any finite, symmetric subset S of isometries of a δ -hyperbolic space X ,either the group G generated by S is elementary or S N contains two ele-ments a, b which generate a free semigroup, provided that X satisfies a pack-ing condition at scale δ with constant C (or also simply if the countingmeasure of G satisfies a doubling condition at some scale r with constant C ).For discrete isometry groups of pinched, negatively curved manifolds, S. Dey,M. Kapovich and B. Liu [DKL18] recently improved [BCGS17], proving aquantitative, true Tits Alternative: there exists N = N ( k, d ) such that forany couple of isometries a, b of a complete, simply connected, d -dimensionalRiemannian manifold X with negative sectional curvature − k ≤ K X ≤ − ,generating a discrete non-elementary group, with a not elliptic, one can findan isometry w , which can be written as a word w = w ( a, b ) on { a, b } oflength less than N , such that { a N , w } generate a non-abelian free subgroup.Noticeably, the authors not only find a true free subgroup with quantifica-tion, but they also specify that one of the generators of the free group canbe prescribed a-priori, provided it is chosen non-elliptic: a property that wewill call specification property .The first result of the paper is a generalization of Dey-Kapovich-Liu’sresult in the framework of GCB Gromov-hyperbolic spaces. While Gromov-hyperbolicity is the most natural metric replacement for the classical hypoth-esis of negative curvature, the packing condition at some scale is the correct(weak) metric replacement of the lower curvature bound. In this setting weprove the following: Theorem 1.1 (Quantitative free subgroup theorem with specification) . Let P , r and δ be fixed positive constants. Then, there exists an integer N ( P , r , δ ) , only depending on P , r , δ , satisfying the following properties.Let ( X, σ ) be any δ -hyperbolic, GCB -space which is P -packed at scale r :(i) for any couple of σ -isometries S = { a, b } of X , where a is non elliptic,such that the group h a, b i is discrete and non-elementary, there existsa word w ( a, b ) in a, b of length ≤ N such that one of the semigroups h a N , w ( a, b ) i + , h a − N , w ( a, b ) i + is free;(ii) for any couple of σ -isometries S = { a, b } of X such that the group h a, b i is discrete, non-elementary and torsion-free, there exists a word w ( a, b ) in a, b of length ≤ N such that the group h a N , w ( a, b ) i is free. Here by σ -isometries we mean the natural isometries of ( X, σ ) , that is thosepreserving the geodesic bicombing. In case of Busemann convex or CAT (0) -spaces, this condition is trivially satisfied by all the isometries of the space.4he first part of the theorem precises Proposition 5.18 in [BCGS17] andTheorem 5.11 of [BF18], showing the specification property, under the ad-ditional hypothesis of a convex bicombing. The difficulty here is that thereis no a-priori bounded power a N such that ℓ ( a N ) is greater than a spec-ified constant (for instance, when a is parabolic this is false for every N ).To avoid replacing a with a bounded word in { a, b } (as in [BCGS17], [BF18])we follow the strategy of [DKL18], which however requires a bit of convexity.For the second part of Theorem 1.1, it should be remarked that the torsion-less assumption is actually necessary , as shown by some examples of groupsacting on simplicial trees (with bounded valency, hence packed) produced inProposition 12.2 of [BF18]. Finally, notice that in our setting elementary isthe same as virtually nilpotent , because of the bounded packing assumption(see Section 3.4 for a proof of this fact; notice however that, without thepacking assumption, there exist elementary isometry groups of negativelycurved manifolds which are even free non-abelian, cp.[Bow93]).The motivation for us behind the quantification and the specificationproperty in the Tits Alternative is geometric. For instance, the specifica-tion property will be used to bound from below the systole of the actionin terms of the upper nilradius (see below Corollary 1.2 for the definition)thus yielding another version of the classical Margulis’ Lemma in our context.This will allow us to extend some classical convergence results of Riemannianmanifolds with curvature bounded below to our setting, see further below.Another application of the quantification of free subgroups in Γ is an ana-logue of the classical thick-thin decomposition for Kleinian groups or isome-try groups of pinched, negatively curved Riemannian manifolds (see [Thu97],[Bow95]) for discrete groups Γ acting on any packed, Gromov-hyperbolicGCB-space, a topic which will be developed by the authors elsewhere [CS].Some applications we will describe do not need the specification propertyor the quantitative Tits Alternative, but the bounded packing condition re-mains almost everywhere essential, in particular for Breuillard-Green-Tao’sgeneralized Margulis’ Lemma.We want to stress that the proof we give of the above theorem heavilydraws from techniques developed in [DKL18] and [BCGS17]. However someingredients are new (for instance the estimate of the distance between dif-ferent levels of the Margulis domains, cp. Propositions 4.5, 4.6 and 4.7, theuse of convexity for controlling the packing at arbitrarily small scales, etc),as well as all the applications to non-cocompact groups, which we are nowgoing to discuss.A first, direct consequence of a quantitative free group or semigrouptheorem is a uniform estimate from below of the algebraic entropy of thegroups under consideration and of the entropy of the spaces they act on. The authors are not able to understand the proof in [DKL18] without the torsionlessassumption, which seems to be used at page 14, below Lemma 4.5. X that is P -packed at scale r , the (covering)entropy of X is defined asEnt ( X ) = lim sup R → + ∞ log Pack ( B ( x, R ) , r ) R where Pack ( B ( x, R ) , r ) denotes the packing number of the ball of radius R centered at x ∈ X , i.e. the maximal cardinality of a r -net inside B ( x, R ) (the definition does not depend on the point x , by the triangular inequality).For a discussion on this notion of entropy and its properties see [Cav21b].For instance when X is CAT (0) then Ent ( X ) equals the volume entropyof the natural measure of X (as introduced in [LN19], see also [CS20]).Recall that in case of non-positively curved Riemannian manifolds the nat-ural measure is exactly the Riemannian volume.On the other hand, given any group Γ acting discretely on X , one also definesthe entropy of the action as the exponential growth rateEnt (Γ , X ) = lim sup R → + ∞ log card ( B ( x, R ) ∩ Γ x ) R .
When X is the Cayley graph of Γ with respect to some finite generating set S ,we will also write Ent (Γ , S ) . The algebraic entropy of a finitely generatedgroup, denoted EntAlg (Γ) , is accordingly defined as the infimum, over allpossible finite generating sets S , of the exponential growth rates Ent (Γ , S ) .We record here two estimates for the entropy of the groups and the spacesunder consideration, which stem directly from the quantification of existenceof a free semigroup (already proved in [BF18], Theorem 13.9, or in [BCGS17],Proposition 5.18(i)), combined with the packing assumption: Corollary 1.2.
Let ( X, σ ) be a δ -hyperbolic GCB -space, P -packed at scale r > , admitting a non-elementary, discrete group Γ of σ -isometries. Then:(i) EntAlg (Γ) ≥ C ,(ii) Ent ( X ) ≥ Ent (Γ , X ) ≥ C · nilrad (Γ , X ) − ,where C = C ( P , r , δ ) > is a constant depending only on P , r and δ . The invariant nilrad (Γ , X ) appearing here is the nilradius of the action of Γ on X , defined as the infimum over all x ∈ X of the largest radius r suchthat the r -almost stabilizer Γ r ( x ) of x is virtually nilpotent. By definition,nilrad (Γ , X ) is always bounded from below by Breuillard-Green-Tao’s gen-eralized Margulis constant ε , which can be expressed in terms of P , r .On the other hand, the nilradius can be arbitrarily large if the orbits of Γ are very sparse in X . However, if Γ is non-elementary (which in our casemeans non-virtually nilpotent) it is always a finite number.A second geometric consequence of Theorem 1.1 is a lower bound ofthe systole of the action of Γ on X , that is the smallest non-trivial dis-placement of points of X under the action of the group (for a non posi-tively curved manifold X and a torsionless group of isometries Γ , this isexactly twice the injectivity radius of the quotient manifold ¯ X = Γ \ X ).6amely, let X ε ⊆ X be the subset of points which are displaced lessthan the generalized Margulis constant ε by some nontrivial element of Γ :this is classically called the ε -thin subset of X . We define the upper nil-radius of Γ , as opposite to the nilradius, as the supremum over x ∈ X ε of the largest r such that Γ r ( x ) is virtually nilpotent. In other words theupper nilradius, denoted nilrad + (Γ , X ) , measures how far we need to travelfrom any x of X ε to find two points g x, g x of the orbit such that thesubgroup h g , g i is non-elementary. A natural bound of the upper nilra-dius is given, for cocompact actions, by the diameter of the quotient Γ \ X .However the upper nilradius can well be finite even for non-cocompact ac-tions, for instance when Γ is a quasiconvex-cocompact group, or a sub-group of infinite index of a cocompact group of X (see Examples 6.4, 6.5).The specification property in the quantitative Tits Alternative yields a lowerbound of the systole of the action in terms of the geometric parameters P , r , δ and of an upper bound of the upper nilradius: Corollary 1.3.
Let ( X, σ ) be a δ -hyperbolic, GCB -space that is P -packedat scale r > . Then, for any torsionless, non-elementary, discrete group of σ -isometries Γ of X it holds: sys (Γ , X ) ≥ min (cid:26) ε , H e − H · nilrad + (Γ ,X ) (cid:27) where H = H ( P , r , δ ) is a constant depending only on P , r , δ and where ε = ε ( P , r ) is the generalized Margulis constant introduced before. If Γ has torsion, a similar estimate holds for the free systole , cp. Section 1.1and formula (26).In the restricted framework of nonpositively curved manifolds, this givesa new group-theoretic estimate of the length of the shortest closed geodesicin ¯ X = Γ \ X , without any lower curvature bound assumption.Remark that without any bound of the upper nilradius of Γ there is no hopeof estimating sys (Γ , X ) from below in terms of δ and the packing constants.This is clear for groups acting with parabolics (cp. Example 6.3), but italso fails for groups without parabolics. It is enough to consider compacthyperbolic manifolds ¯ X = Γ \ X possessing very small periodic geodesics γ of length ε , much smaller than the Margulis constant: by the classicaltheory of Kleinian groups, γ has a very long tubular neighbourhood and,consequently, Γ has arbitrarily large upper nilradius (this picture generalizesto actions of discrete groups on convex, packed, Gromov-hyperbolic spaces,see §6 in [BCGS] and [CS]).Even without any a-priori bound of the upper nilradius of the action of Γ ,one can always find a point x where the minimal displacement is boundedbelow by a universal function of the geometric parameters P , r , δ of X .We call diastole of Γ acting on X , denoted dias (Γ , X ) , the supremum over all x ∈ X of the minimal displacement of x under all non trivial elements of thegroup. The next result generalizes one of the classical versions of the MargulisLemma on Riemannian manifolds with pinched, negatively curvature:7 orollary 1.4. Let ( X, σ ) be a δ -hyperbolic, GCB -space that is P -packedat scale r > . Then, for any torsionless, discrete, non-elementary group of σ -isometries Γ of X we have: dias (Γ , X ) = sup x ∈ X inf g ∈ Γ ∗ d ( x, gx ) ≥ ε (where ε = ε ( P , r ) is the generalized Margulis constant). A version of this estimate for groups Γ with torsion is proved in Section 6.3.Notice that the estimate, which holds also for cocompact groups, does notdepend on the diameter (in contrast with Proposition 5.25 of [BCGS17];also notice that our groups do not belong to any of the classes consideredin [BCGS17], as they do not have a-priori a cocompact action on a convex,Gromov-hyperbolic space or an action on a Gromov-hyperbolic space withasymptotic displacement uniformly bounded below).In Section 6.4 we use these estimates to investigate limits of group actionson our class of spaces. Given a sequence of faithful, isometric group actions Γ n y ( X n , x n ) on general, pointed metric spaces and a non-principal ultrafil-ter ω , one can define a limit group Γ ω by taking limits of any possible admissi-ble sequence ( g n ) , for g n ∈ Γ n for every n ; then, the limit group (which heav-ily depends on the choice of base points x n in X n ) naturally acts by isome-tries on the ultralimit space ( X ω , x ω ) . When restricted to the class of closedisometry groups of proper metric spaces, this convergence, is easily seento be equivalent to the pointed, equivariant Gromov-Hausdorff convergence of some subsequence Γ n k y ( X n k , x n k ) , as defined by Fukaya (see [Fuk86],[FY92] and [Cav21a] for comparison with other notions of convergence ofgroups).Since the class of δ -hyperbolic, GCB-spaces is closed under ultralimits, itis natural to ask for the properties of the limit actions on these spaces.According to the behaviour of the distance of the base points x n from thethin subsets of the X n ’s, the type of the limit g ω of a sequence of hyperbolic orparabolic isometries ( g n ) can change (see Proposition 6.9 and Example 6.10),elliptic elements can appear and the limit action may be non-discrete.The following result resumes the possibilities for the limit action: Theorem 1.5 (Extract from Theorems 6.11& 6.13 and Corollary 6.12) . Let ( X n , x n , σ n ) be δ -hyperbolic, GCB -spaces, P -packed at scale r > ,and let Γ n be torsion-free, discrete σ n -isometry groups of the spaces X n .Let ω be a non-principal ultrafilter and let Γ ω be the ultralimit group of σ ω -isometries of the δ -hyperbolic, GCB -space ( X ω , σ ω ) . Then:(i) Γ ω is either discrete and torsion-free, or elementary;(ii) Γ ω is not elementary if and only if there exist admissible sequences ( g k ) , ( h k ) such that the group h g k , h k i is not elementary for ω -a.e. ( k ) .Moreover if the groups Γ n are non-elementary one can always choose thebase points x n in X n so that the limit group is discrete and torsion-free.Finally when Γ ω is discrete and torsion-free then the ultralimit of the quo-tients Γ n \ ( X n , x n ) is isometric to the quotient space Γ ω \ X ω . Γ ω can be discrete, torsion-free and elementary. But, if it is not discrete,then it is necessarily elementary. We will see that the existence of points x n which make the limit action discrete is a direct consequence of the diastolicestimate given in Corollary 1.4. The condition of non-elementarity of the Γ n is necessary, essentially to rule out actions by Z on the real line with smallerand smaller systole.We conclude with some compactness results which are consequence ofTheorem 1.5 and of the closure of our class under ultralimits. For any givenparameters P , r , δ and D > consider the classGCB c ( P , r , δ ; D ) of compact quotients of δ -hyperbolic, GCB-spaces ( X, σ ) which are P -packedat scale r , by some torsionless, non-elementary, discrete group Γ of σ -isometries, with codiameter diam (Γ \ X ) ≤ D . This class is compact withrespect to the Gromov-Hausdorff topology, and contains only finitely manyhomotopy types, as a consequence of results proved in [BG20] and [BCGS].Dropping the assumption on the diameter, thus allowing non-compact spacesin our class, we cannot hope for any finiteness result. However in section 6.4we will prove: Theorem 1.6.
The class of abelian (resp. m -step nilpotent, m -step solvable)coverings of spaces in the class GCB c ( P , r, δ ; D ) is closed under ultralimits,hence compact w.r. to pointed equivariant Gromov-Hausdorff convergence.Namely, for any sequence ( ˆ X n , ˆ x n ) → ( ¯ X n , ¯ x n ) of normal coverings of pointedspaces belonging to GCB c ( P , r, δ ; D ) whose group of deck transformations Γ n is abelian (resp. m -step nilpotent, m -step solvable), there exists a pointedspace ( ¯ X, ¯ x ) in GCB c ( P , r, δ ; D ) and a normal covering ( ˆ X, ˆ x ) → ( ¯ X, ¯ x ) with abelian (resp. m -step nilpotent, m -step solvable) group of deck trans-formations Γ , such that the actions Γ n y ( ˆ X n , ˆ x n ) tend to Γ y ( ˆ X, ˆ x ) inpointed, equivariant Gromov-Hausdorff convergence. Recall that a group is m -step nilpotent (resp. m -step solvable) if its lowercentral series (resp. its derived series) has length at most m .The next theorem generalizes the above result and extends the compactnessof the class GCB c ( P , r, δ ; D ) , by replacing the diameter with the uppernilradius. Let GCB ( P , r , δ ; ∆) be the class of quotients of δ -hyperbolic,GCB-spaces ( X, σ ) which are P -packed at scale r , by a torsionless, non-elementary, discrete group of σ -isometries with upper nilradius ≤ ∆ . Then: Theorem 1.7.
The class
GCB ( P , r , δ, ∆) is closed under ultralimits andcompact with respect to pointed, Gromov-Hausdorff convergence. Remark that, as the condition of being CAT (0) is stable under ultralimits,then the compactness Theorems 1.6 & 1.7 also hold in this restricted class.
Acknowledgments.
The authors are grateful to G. Besson, G. Courtois and S.Gallot for many helpful discussions. In [BG20] and [BCGS] the authors only consider compact quotients of convex , δ -hyperbolic spaces, under the weaker assumption of bounded entropy; however, the com-pactness of the class GCB ( P , r , δ ; D ) follows from the same arguments and the con-tractibility of convex, geodesically bicombed spaces. .1 Notation Given a subset A of a metric space X and a real number r ≥ , we denote by B ( A, r ) and B ( A, r ) the open and closed r -neighbourhood of A , respectively.In particular if A = { x } , where x ∈ X is a point, then B ( x, r ) and B ( x, r ) denote the open and closed ball of radius r and center x .A geodesic segment is a curve γ : [ a, b ] → X such that d ( γ ( t ) , γ ( t ′ )) = | t − t ′ | for all t, t ′ ∈ [ a, b ] ; by abuse of notation, a geodesic joining x to y is denoted by [ x, y ] even if it is not unique. A geodesic ray is an isometric embedding from [0 , + ∞ ) to X , while a geodesic line (or, simply, a geodesic ) is an isometricembedding from R to X . The space X is called geodesic if for all x, y ∈ X there exists a geodesic segment joining x to y .If X is any proper metric space we denote by Isom ( X ) its group of isometries,endowed with the uniform convergence on compact subsets of X . A subgroup Γ of Isom ( X ) is called discrete if the following (equivalent) conditions (see[BCGS17]) hold:(a) Γ is discrete as a subspace of Isom ( X ) ;(b) ∀ x ∈ X and R ≥ the set Σ R ( x ) = { g ∈ Γ | gx ∈ B ( x, R ) } is finite.We will denote by Γ ∗ the subset of nontrivial elements of Γ , while Γ ⋄ ⊂ Γ will denote the subset of elements with finite order. Moreover, if S is asymmetric, finite generating set for Γ we will denote by S n the ball of radius n of Γ with respect to the word metric associated to S .We register here also, for easy reference, a number of invariants associatedto the action of Γ on X we will be interested in throughout the paper.For every r > and every point x ∈ X the r -almost stabilizer of x in Γ isthe subgroup Γ r ( x ) = h Σ r ( x ) i . The r -thin subset of X (with respect to the action of Γ ) is the subset X r = { x ∈ X | ∃ g ∈ Γ ∗ s.t. d ( x, gx ) < r } and the free r -thin subset X ⋄ r is obtained by replacing Γ ∗ in the definitionabove with Γ \ Γ ⋄ . Some numerical invariants associated to the action of Γ we will be interested in are:• the minimal displacement of g ∈ Γ , defined as ℓ ( g ) = inf x ∈ X d ( x, gx ) ;• the asymptotic displacement of g ∈ Γ , defined as k g k = lim n → + ∞ d ( x,g n x ) n ;• the minimal displacement of Γ at x , defined as sys (Γ , x )=inf g ∈ Γ ∗ d ( x, gx ) ;• the minimal free displacement of Γ at x, defined assys ⋄ (Γ , x ) = inf g ∈ Γ \ Γ ⋄ d ( x, gx ); • the nilpotence radius of Γ at x , defined asnilrad (Γ , x ) = sup { r ≥ s.t. Γ r ( x ) is virtually nilpotent } ; systole and the free systole , defined respectively as :sys (Γ , X ) = inf x ∈ X sys (Γ , x ) , sys ⋄ (Γ , X ) = inf x ∈ X sys ⋄ (Γ , x ) • the diastole and the free diastole , defined as :dias (Γ , X ) = sup x ∈ X sys (Γ , x ) , dias ⋄ (Γ , X ) = sup x ∈ X sys ⋄ (Γ , x ) • the nilradius : nilrad (Γ , X ) = inf x ∈ X nilrad (Γ , x ) .Moreover, for convex, packed metric spaces we can define an upper analogueof the nilradius (like the diastole for the systole):• the upper nilradius : nilrad + (Γ , X ) = sup x ∈ X ε nilrad (Γ , x ) where the constant ε appearing in the definition is the generalized Margulisconstant , which will be introduced in see Section 2.3. In the first part of this section we introduce the notion of geodesically com-plete, convex geodesic bicombings. In the second part we define the packingcondition on a metric space and we see how it interacts with the convexity ofa geodesic bicombing. As a consequence we can apply a result of [ BGT11 ] toobtain a uniform estimate of the nilradius of groups acting on packed met-ric spaces, introducing Breuillard-Green-Tao’s generalized Margulis constant ,which plays a key-role in the proof of the Free Subgroup Theorem and itsapplications. A geodesic bicombing on a metric space X is a map σ : X × X × [0 , → X with the property that for all ( x, y ) ∈ X × X the restriction σ xy : t σ ( x, y, t ) is a geodesic from x to y , i.e. d ( σ xy ( t ) , σ xy ( t ′ )) = | t − t ′ | d ( x, y ) for all t, t ′ ∈ [0 , and σ xy (0) = x, σ xy (1) = y . A geodesic bicombing is:• convex if the map t d ( σ xy ( t ) , σ x ′ y ′ ( t )) is convex on [0 , for all x, y, x ′ , y ′ ∈ X ;• consistent if for all x, y ∈ X , for all ≤ s ≤ t ≤ and for all λ ∈ [0 , it holds σ pq ( λ ) = σ xy ((1 − λ ) s + λt ) , where p := σ xy ( s ) and q := σ xy ( t ) ;• reversible if σ xy ( t ) = σ yx (1 − t ) for all t ∈ [0 , .For instance, all convex spaces in the sense of Busemann (hence, all CAT (0) -space) have a unique convex, consistent, reversible geodesic bicombing.11iven a geodesic bicombing σ we say that a geodesic (segment, ray, line) γ is a σ -geodesic (segment, ray, line) if for all x, y ∈ γ we have that σ xy is aparametrization of the subsegment of γ between x and y . We say that ageodesic bicombing is locally geodesically complete if every σ -geodesic seg-ment is contained in a σ -geodesic segment whose endpoints are both differentfrom x and y . We say that it is geodesically complete if every σ -geodesic seg-ment is contained in a σ -geodesic line. A convex metric space in the sense ofBusemann is (locally) geodesically complete if and only if its unique convex,consistent, reversible geodesic bicombing is (locally) geodesically complete.We report here some trivial facts: Lemma 2.1.
Let σ be a convex, consistent, reversible geodesic bicombing ofa metric space X . Then:(i) the map σ is continuous;(ii) X is geodesic and contractible;(iii) if X is complete and σ is locally geodesically complete then σ is geode-sically complete.Proof. Assertion (ii) is a consequence of (i), and (i) follows directly from theconvexity condition. Assertion (iii) is standard.A geodesically complete, convex, consistent, reversible geodesic bicomb-ing will be called GCB for short. A couple ( X, σ ) , where σ is a GCB on acomplete metric space X will be called a GCB-space.
The interest in thisweak notion of non-positive curvature is given by its stability under limits(while the Busemann convexity condition is not stable in general):
Lemma 2.2.
The class of pointed
GCB -spaces is closed under ultralimits.Proof.
Let ( X n , x n , σ n ) be pointed GCB-spaces and let ω be a non-principalultrafilter. We define the map σ ω : X ω × X ω × [0 , → X ω by σ ω ( ω - lim y n , ω - lim z n , t ) = ω - lim σ n ( y n , z n , t ) . It is easy to see it is well defined and it satisfies all the required properties.A subset C of a GCB-space ( X, σ ) is σ -convex if for all x, y ∈ C the σ -geodesic segment [ x, y ] is contained in C . We say that a map f : X → R is σ -convex (resp. strictly σ -convex) if it is convex (resp. strictly convex)when restricted to every σ -geodesic. Lemma 2.3.
Let ( X, σ ) be a proper GCB -space and let C ⊂ X be a σ -convexsubset. Then:(i) for all x ∈ X the distance map d x from x is σ -convex;(ii) the distance map d C from C is σ -convex;(iii) if C is closed then ∀ x ∈ X there exists a unique projection of x on C .Proof. Claim (i) follows directly by the definition of σ -convexity.12o show (ii), let x , x ∈ X and for all ε > pick points y , y ∈ C suchthat d ( x , y ) ≤ d C ( x ) + ε and d ( x , y ) ≤ d C ( x ) + ε . For all t ∈ [0 , wedenote by x t the point σ x x ( t ) and by y t the point σ y y ( t ) . Observe that y t ∈ C by σ -convexity of C , so d C ( x t ) ≤ d ( x t , y t ) ≤ td ( x , y )+(1 − t ) d ( x , y ) ≤ td C ( x )+(1 − t ) d C ( x )+2 ε. By the arbitrariness of ε we get (i). Finally, the existence of the projectionfollows from the fact that C is closed and X is proper; moreover the map x d ( x, x ) is strictly σ -convex, which implies uniqueness.The radius of a bounded subset Y ⊆ X , denoted by r Y , is the infimum ofthe positive numbers r such that Y ⊆ B ( x, r ) for some x ∈ X . The followingfact is well-known for CAT (0) -spaces and will be used later to characterizeelliptic isometries: Lemma 2.4.
Let ( X, σ ) be a proper GCB -space. Then for any boundedsubset Y of X there exists a unique point x ∈ X such that Y ⊆ B ( x, r Y ) .Such a point is called the center of Y .Proof. The existence of such a point is easy: just take a sequence of points x n almost realizing the infimum in the definition of r Y , i.e. B ( x n , r Y + n ) ⊇ Y .Since Y is bounded, then the sequence x n is bounded. We may suppose,up to taking a subsequence, that the sequence x n converges to a point x ∞ .We then have d ( x ∞ , y ) = lim n → + ∞ d ( x n , y ) ≤ r Y for any y ∈ Y .Assume now that there exist two points x = x ′ satisfying the thesis, thatis Y ⊆ B ( x, r Y ) ∩ B ( x ′ , r Y ) . We take the midpoint m between x and x ′ along the σ -geodesic connecting them and we claim that there exists ε > such that B ( m, r Y − ε ) ⊇ B ( x, r Y ) ∩ B ( x ′ , r Y ) . Otherwise there exist asequence of points x n ∈ B ( x, r Y ) ∩ B ( x ′ , r Y ) with x n / ∈ B ( m, r Y − n ) , thatis d ( x n , x ) ≤ r Y , d ( x n , x ′ ) ≤ r Y and d ( x n , m ) > r Y − n . Then again we mayassume that the x n ’s converge to a point x ∞ satisfying d ( x ∞ , x ) ≤ r Y , d ( x ∞ , x ′ ) ≤ r Y , d ( x ∞ , m ) ≥ r Y . By Lemma 2.3.(i) the distance from x ∞ to any point of the σ -geodesic [ x, x ′ ] is constant; this is impossible as the projection of x ∞ on [ x, x ′ ] is unique byLemma 2.3.(iii). We have therefore proved that there exists ε > such that B ( m, r Y − ε ) ⊇ B ( x, r Y ) ∩ B ( x ′ , r Y ) . which contradicts with the definition of r Y . Then x = x ′ .Let ( X, σ ) be a GCB-space, x ∈ X and < r ≤ R . The contraction map ϕ Rr : B ( x, R ) → B ( x, r ) (1)is the map sending any y ∈ B ( x, R ) to σ xy ( r/R ) . This map is rR -Lipschitz,by the convexity of σ . Since σ is geodesically complete, ϕ Rr is also surjective:given y ∈ B ( x, r ) we can extend the geodesic segment σ xy to a σ -geodesicsegment σ xy ′ with d ( x, y ′ ) = Rr d ( x, y ) , so y ′ ∈ B ( x, R ) and ϕ Rr ( y ′ ) = y .13 .2 The packing condition and some examples Recall that a subset S of a metric space X is called r -separated if for anytwo different s, s ′ ∈ S it holds d ( s, s ′ ) > r , while S is r -dense if for any x ∈ X there exists s ∈ S such that d ( x, s ) ≤ r . Accordingly, given a metricspace X we define the packing and the covering function of a subset Y ⊆ X at scale r > , respectively as:Pack ( Y, r ) = cardinality of a maximal r -separated subset of Y, Cov ( Y, r ) = cardinality of a minimal r -dense subset of Y. These numbers are related by the following, classical inequalitiesPack ( Y, r ) ≤ Cov ( Y, r ) ≤ Pack ( Y, r ) . (2)Then for every < r ≤ R we define the packing function of X asPack ( R, r ) = sup x ∈ X Pack ( B ( x, R ) , r ) . We say X is P -packed at scale r if Pack ( B ( x, r ) , r ) ≤ P ∀ x ∈ X .The next result states that in a GCB-space the packing function Pack ( R, r ) is well controlled by the packing condition at any fixed scale r : Proposition 2.5 (Theorem 4.2 and Lemma 4.5 of [CS20]) . Let ( X, σ ) be a GCB -space that is P -packed at scale r . Then:(i) X is proper and for all r ≤ r it is P -packed at scale r ;(ii) for every < r ≤ R it holds: Pack ( R, r ) ≤ P (1 + P ) Rr − , if r ≤ r ; Pack ( R, r ) ≤ P (1 + P ) Rr − , if r > r . (Notice that, in [CS20], the above estimates are proved with an additionalmultiplicative factor . This is due to the fact that, in [CS20], the contractionmaps in the spaces under consideration are rR -Lipschitz, while here thesemaps are precisely rR -Lipschitz.)Some basic examples of GCB-spaces that are P -packed at scale r are:i) complete, simply connected Riemannian manifolds X with pinched,nonpositive sectional curvature κ ′ ≤ K ( X ) ≤ κ ≤ ;ii) complete, geodesically complete, CAT ( κ ) -spaces X with κ ≤ of di-mension at most n and volume of balls of radius at most V ;iii) simply connected M κ -complexes of bounded geometry without free faces,for κ ≤ .By bounded geometry we mean that the M κ -complex X has positive injec-tivity radius, valency ≤ V and size ≤ S (the size of X is by definition thelargest R such that every simplex of X is contained in a b all of radius R and contains a ball of radius R ). 14he second and the third class above are discussed in detail in [CS20].Very special cases of (ii) and (iii) are geodesically complete metric treesand CAT (0) -cube complexes with bounded valency (i.e. such that the num-ber of edges and cubes, respectively, having a common vertex is uniformlybounded by some constant V ), see Proposition 2.6 below; for instance, allthose admitting a cocompact group of isometries have bounded valency.The examples above are all clearly Gromov-hyperbolic when κ < . But theycan manifest a Gromov-hyperbolicity behaviour even without such a sharpbound; for instance it is classical that any proper cocompact CAT (0) -spacewithout -flats is Gromov-hyperbolic (see [Gro87], [BH13]).For cube complexes with bounded valency we have the following hyper-bolicity criterion. Recall that one says that a cube complex X has L -thinrectangles if all Euclidean rectangles [0 , a ] × [0 , b ] isometrically embedded in X satisfy min { a, b } ≤ L . Then: Proposition 2.6.
Let X be a simply connected cube complex with no freefaces, dimension ≤ n and valency ≤ V , endowed with its canonical ℓ -metric(the length metric which makes any d -cube of X isometric to [ − , d ):(i) if X has positive injectivity radius, then it is a complete, geodesicallycomplete, CAT (0) -space which is P -packed at scale r = , for a pack-ing constant P = P ( V ) ;(ii) if moreover X has L -thin rectangles, then it is Gromov-hyperbolic withhyperbolicty constant δ = 4 · Ram ⌈ L + 1 ⌉ (where Ram ( m ) denotes theRamsey number of m ).Proof. The barycenter subdivision of the cube-complex gives a M -complexstructure to X with valency bounded uniformly in terms of V and n and,clearly, with uniformly bounded size. Moreover since X has positive injec-tivity radius and it has no free fraces then the same is true for the metricinduced by the complex structure (which is isometric), therefore we can ap-ply Proposition 7.13 of [CS20] to conclude (i) (the fact that X is globallyCAT (0) follows from the simply connectedness assumption).The proof of (ii) is presented in Theorem 3.3 of [Gen16].Theorem 1.1 and its consequences will apply to all these cases. In partic-ular notice that the quantitative Tits Alternative with specification is neweven for hyperbolic, CAT (0) cube complexes (cp. [SW05], [GJN20]). Consider a proper metric space X and a discrete group of isometries Γ of X .In general the nilradius of the action can well be zero. However under thepacking assumption it is always bounded away from zero: Theorem 2.7 (Corollary 11.17 of [BGT11]) . Let X be a proper metric space such that Cov ( B ( x, , ≤ C for all x ∈ X .Then there exists a constant ε M = ε M ( C ) > , only depending on C , suchthat for every discrete group of isometries Γ of X one has nilrad (Γ , X ) ≥ ε M .
15n analogy with the case of Riemannian manifolds, the constant ε M is calledthe (generalized) Margulis constant of X . Combining with Proposition 2.5,we immediately get a Margulis contant for the class of complete GCB-spaceswhich are P -packed at some scale r : Corollary 2.8.
Given P , r > there exists ε = ε ( P , r ) > such thatfor any GCB -space X which is P -packed at scale r and for any discretegroup of isometries Γ of X one has nilrad (Γ , X ) ≥ ε .Proof. By Proposition 2.5 X is proper. We rescale the metric by a factor r .As the packing property is invariant under rescaling, we have Pack ( , ) ≤ P by assumption. Hence Cov (4 , ≤ Pack (4 , ) ≤ P (1 + P ) , as follows from(2) and from Proposition 2.5. Applying Theorem 2.7 we find a Margulisconstant ε M for the space r X , only depending on P ; then, the constant ε ( P , r ) = 2 r · ε M satisfies the thesis. Let X be a geodesic space. Given three points x, y, z ∈ X , the Gromovproduct of y and z with respect to x is defined as ( y, z ) x = 12 (cid:0) d ( x, y ) + d ( x, z ) − d ( y, z ) (cid:1) . The space X is said δ -hyperbolic if for every four points x, y, z, w ∈ X thefollowing hold: ( x, z ) w ≥ min { ( x, y ) w , ( y, z ) w } − δ (3)or, equivalently, d ( x, y ) + d ( z, w ) ≤ max { d ( x, z ) + d ( y, w ) , d ( x, w ) + d ( y, z ) } + 2 δ. (4)The space X is Gromov hyperbolic if it is δ -hyperbolic for some δ ≥ .The above formulations of δ -hyperbolicity are convenient when interested intaking limits (since they are preserved under ultralimits). However, we willalso make use of other classical characterizations of δ -hyperbolicity, depend-ing on which one is more useful in the context.Recall that a geodesic triangle in X is the union of three geodesic segments [ x, y ] , [ y, z ] , [ z, x ] and is denoted by ∆( x, y, z ) . For every geodesic trianglethere exists a unique tripod ∆ with vertices ¯ x, ¯ y, ¯ z such that the lengths of [¯ x, ¯ y ] , [¯ y, ¯ z ] , [¯ z, ¯ x ] equal the lengths of [ x, y ] , [ y, z ] , [ z, x ] respectively. Thereexists a unique map f ¯∆ from ∆( x, y, z ) to the tripod ∆ that identifiesisometrically the corresponding edges, and there are exactly three points c x ∈ [ y, z ] , c y ∈ [ x, z ] , c z ∈ [ x, y ] such that f ¯∆ ( c x ) = f ¯∆ ( c y ) = f ¯∆ ( c z ) = c ,where c is the center of the tripod ∆ . By definition of f ¯∆ it holds: d ( x, c z ) = d ( x, c y ) , d ( y, c x ) = d ( y, c z ) , d ( z, c x ) = d ( z, c y ) . The triangle ∆( x, y, z ) is called δ -thin if for every u, v ∈ ∆( x, y, z ) suchthat f ¯∆ ( u ) = f ¯∆ ( v ) it holds d ( u, v ) ≤ δ ; in particular the mutual distances16etween c x , c y and c z are at most δ . It is well-known that every geodesictriangle in a geodesic δ -hyperbolic metric space (as defined above) is δ -thin,and moreover satisfies the Rips’ condition : [ y, z ] ⊂ B ([ x, y ] ∪ [ x, z ] , δ ) . (5)Moreover these last conditions are equivalent to the above definition of hy-perbolicity, up to slightly increasing the hyperbolicity constant δ in (3).As a consequence of the δ -thinness of triangles we have the following: let x, y, z ∈ X , f ¯∆ : ∆( x, y, z ) → ¯∆ the tripod approximation and c x , c y , c z asbefore. Then ( y, z ) x = d ( x, c z ) = d ( x, c y ) and d ( x, c x ) ≤ d ( x, c y ) + 4 δ (6) All Gromov-hyperbolic spaces, in this paper, we will be supposed proper; wewill however stress this assumption in the statements where it is needed.
We fix a δ -hyperbolic metric space X and a base point x of X .The Gromov boundary of X is defined as the quotient ∂ G X = { ( y n ) n ∈ N ⊆ X | lim n,m → + ∞ ( y n , y m ) x = + ∞} / ∼ , where ( y n ) n ∈ N is any sequence of points in X and ∼ is the equivalence relationdefined by ( y n ) n ∈ N ∼ ( z n ) n ∈ N if and only if lim n,m → + ∞ ( y n , z m ) x = + ∞ .We will write y = [( y n )] ∈ ∂ G X for short, and we say that ( y n ) converges to y .Clearly, this definition does not depend on the basepoint x .The Gromov product can be extended to points y, z ∈ ∂ G X by ( y, z ) x = sup ( y n ) , ( z n ) lim inf n,m → + ∞ ( y n , z m ) x where the supremum is over all sequences such that ( y n ) ∼ y and ( z n ) ∼ z .For any x, y, z ∈ ∂ G X it continues to hold ( x, y ) x ≥ min { ( x, z ) x , ( y, z ) x } − δ. (7)Moreover, for all sequences ( y n ) , ( z n ) converging to y, z respectively it holds ( y, z ) x − δ ≤ lim inf n,m → + ∞ ( y n , z m ) x ≤ ( y, z ) x . (8)In a similar way is defined the Gromov product between a point y ∈ X anda point z ∈ ∂ G X . This product satisfies a condition analogue of (8).Any ray γ defines a point γ + = [( γ ( n )) n ∈ N ] of the Gromov boundary ∂ G X : we say that γ joins γ (0) = y to γ + = z , and we denote it by [ y, z ] .Notice that any point y ∈ X can be joined to any point z = [( z n )] ∈ ∂ G X :in fact, the sequence ( z n ) must be unbounded (as ( z n , z n ) x is unbounded), so17he geodesic segments [ y, z n ] converge uniformly on compact sets, by proper-ness of X , to a geodesic ray γ = [ y, z ] . Analogously, given different points z = [( z n )] , z ′ = [( z ′ n )] ∈ ∂ G X there always exists a geodesic line γ joining z to z ′ , i.e. such that γ | [0 , + ∞ ) and γ | ( −∞ , join γ (0) to z, z ′ respectively (justconsider the limit γ of the segments [ z n , z ′ n ] ; notice that all these segmentsintersect a ball of fixed radius centered at x , since ( z n , z ′ m ) x is uniformlybounded above). We call z and z ′ the positive and negative endpoints of γ ,respectively, denoted γ ± . We will also write, for short, ∂γ := { γ + , γ − } .As X is not uniquely geodesic, it may happen that there are severalgeodesic rays joining a point of X to some point z ∈ ∂ G X , or several geodesiclines joining two points of the boundary. However, the following standarduniform estimates hold: Lemma 3.1 (Prop. 8.10 of [BCGS17]) . Let X be a δ -hyperbolic space.(i) let γ, ξ be two geodesic rays with γ + = ξ + : then there exist t , t ≥ with t + t = d ( γ (0) , ξ (0)) such that d ( γ ( t + t ) , ξ ( t + t )) ≤ δ , ∀ t ≥ ;(ii) let γ, ξ be two geodesic lines with γ + = ξ + and γ − = ξ − : then for all t ∈ R there exists s ∈ R such that d ( γ ( t ) , ξ ( s )) ≤ δ . Recall that a subset C ⊆ X ∪ ∂ G X is said convex if for every x, y ∈ C there exists at least one geodesic (segment, ray, line) joining x to y that isincluded in C . Given any closed, convex subset C of X and a point x ∈ X ,a projection of x to C is a point c ∈ C such that d ( x, C ) = d ( x, c ) . Since C is closed and X is proper, it is clear that there exists at least a projection.A fundamental tool in the study of projections in δ -hyperbolic spaces is thefollowing: Lemma 3.2 (Projection Lemma, cp. Lemma 3.2.7 of [CDP90]) . Let X be a δ -hyperbolic space, and let x, y, z ∈ X . For any geodesic segment [ x, y ] we have: ( y, z ) x ≥ d ( x, [ y, z ]) − δ. Therefore if C is a convex subset and x is a projection of x on C then ( x , c ) x ≥ d ( x, x ) − δ for all c ∈ C . This easily implies that the projection x satisfies, for all c ∈ C : ( x, c ) x ≤ δ (9)One can then extend the definition of projection to boundary points, usingthis relation, as follows: we say that x is a projection of x ∈ ∂ G X on C if ( x, c ) x ≤ δ for all c ∈ C. In the next lemma we summarize the properties of projections we need.Recall that, since C is convex and closed, then it is naturally a geodesic, δ -hyperbolic, proper metric space; furthermore the Gromov boundary ∂ G C of C canonically embeds into ∂ G X . 18 emma 3.3. Let X be a proper, δ -hyperbolic metric space and C be a closed,convex subset of X . Let x, x ′ ∈ X ∪ ∂ G X \ ∂ G C . The following facts hold:(a) there exists at least one projection of x on C ;(b) if x , x are two projections of x on C then d ( x , x ) ≤ δ ;(c) if x and x ′ are respectively projections of x and x ′ on C , then d ( x , x ′ ) ≤ d ( x, x ′ ) + 12 δ .Proof. We first show the existence of a projection for points x ∈ ∂ G X \ ∂ G C .Let ( x n ) be a sequence converging to x and let c n be a projection of x n on C .First of all we claim that the sequence ( c n ) is bounded. As the sequence ( c n ) is in C and x / ∈ ∂ G C , then ( c n ) is not equivalent to ( x n ) . In particular ( x n , c n ) c ≤ D for some ≤ D < + ∞ and some c ∈ C . This means d ( c , x n ) + d ( c , c n ) − d ( c n , x n ) ≤ D. As x ∈ C and c n is a projection of x n on C , we have d ( c , x n ) ≥ d ( c n , x n ) and therefore d ( c , c n ) ≤ D for all n . Therefore the sequence c n converges,up to a subsequence, to a point c ∈ C . Notice that for any n and any c ′ ∈ C we have ( x n , c ′ ) c n ≤ δ . Applying (8) we get for all c ′ ∈ C ( x, c ′ ) c ≤ lim sup n → + ∞ ( x n , c ′ ) c + δ ≤ lim sup n → + ∞ ( x n , c ′ ) c n + d ( c n , c ) + δ ≤ δ. This proves (a). Assertion (b) is an easy consequence of the definition, as ( x, x ) x ≤ δ, ( x, x ) x ≤ δ, so d ( x , x ) = ( x, x ) x + ( x, x ) x ≤ δ .Finally the proof of (c) can be found in [CDP90], Corollary 10.2.2. Remark 3.4.
We record here a consequence of the proof above: if ( x n ) is a sequence of points converging to a point x ∈ ∂ G X \ ∂ G C and c n is aprojection of x n on C for all n , then, up to a subsequence, the limit point ofthe sequence c n is a projection of x on C .We now recall the Morse property of geodesic segments in a Gromov-hyperbolic space. A map α : [0 , l ] → X is a (1 , ν ) -quasigeodesic segment if for any t, t ′ ∈ [0 , l ] it holds: | t − t ′ | − ν ≤ d ( α ( t ) , α ( t ′ )) ≤ | t − t ′ | + ν. The points α (0) and α ( l ) are called the endpoints of α . Proposition 3.5 (Morse Property) . Let X be a δ -hyperbolic space and let α be a (1 , ν ) -quasigeodesic segment. The following facts hold:(a) for any geodesic segment β joining the endpoints of α we have d H ( α, β ) ≤ ν + 12 δ , where d H is the Hausdorff distance;(b) for any (1 , ν ) -quasigeodesic segment β with the same endpoints of α andfor any time t where both α and β are defined it holds d ( α ( t ) , β ( t )) ≤ ν + 48 δ . The proof of the first part can be found in [Bow06], while the second part isclassical and follows from a straightforward computation.19e can now state the contracting property of projections we will use later:
Proposition 3.6 (Contracting Projections) . Let X be a proper, δ -hyperbolicspace, C ⊆ X any closed convex subset and Y ⊆ X another convex subset.The following facts hold:(a) if the projections c, c ′ on C of y and y ′ ∈ Y satisfy d ( c, c ′ ) > δ , then [ y, c ] ∪ [ c, c ′ ] ∪ [ c ′ , y ′ ] is a (1 , δ ) -quasigeodesic segment;(b) if d ( Y, C ) > δ , then any two projections on C of points of Y are atdistance at most δ ;(c) if α, β are geodesic with ∂α ∩ ∂β = ∅ , then any projection b + of β + on α satisfies d ( b + , β ) ≤ max { δ, d ( α, β ) + 19 δ } .Proof. By assumption we have ( y, c ′ ) c ≤ δ and ( y ′ , c ) c ′ ≤ δ , i.e. d ( y, c ′ ) ≥ d ( y, c ) + d ( c, c ′ ) − δ, d ( y ′ , c ) ≥ d ( y ′ , c ′ ) + d ( c, c ′ ) − δ. (10)We apply the four-points condition (4) to ( y, c ′ , y ′ , c ) obtaining d ( y, c ′ ) + d ( y ′ , c ) ≤ max { d ( y, y ′ ) + d ( c, c ′ ) , d ( y, c ) + d ( y ′ , c ′ ) } + 2 δ. Assuming d ( c, c ′ ) > δ we get by (10) d ( y, c ′ )+ d ( y ′ , c ) ≥ d ( y, c )+ d ( c, c ′ )+ d ( y ′ , c ′ )+ d ( c, c ′ ) − δ>d ( y, c )+ d ( y ′ , c ′ )+2 δ and the 4-points condition becomes: d ( y, c ′ )+ d ( y ′ , c ) ≤ d ( y, y ′ )+ d ( c, c ′ ) + 2 δ. Using again (10) we get d ( y, c ) + d ( c, c ′ ) + d ( y ′ , c ′ ) + d ( c, c ′ ) − δ ≤ d ( y, y ′ ) + d ( c, c ′ ) + 2 δ which proves (a). We suppose now d ( Y, C ) > δ and that there are twopoints y, y ′ ∈ Y with projections c, c ′ on C such that d ( c, c ′ ) > δ . Then,the path [ y, c ] ∪ [ c, c ′ ] ∪ [ c ′ , y ′ ] is a (1 , δ ) -quasigeodesic segment by (a), andit is at Hausdorff distance at most δ from any geodesic segment [ y, y ′ ] , byLemma 3.5. As Y is convex, one of these geodesic segments is included in Y , so c is at distance at most δ from Y . This contradiction proves (b).In order to prove (c) we observe that α, β are two closed, convex subsets of X .We divide the proof in two cases. Case 1: d ( α, β ) > δ . Then let x ∈ α and y ∈ β be points minimizingthe distance between α and β ; in particular, x is a projection of y on α .By Remark 3.4 and by (b) there exists a projection b + of β + on α that fallsat distance at most δ from x . Therefore we have d ( b + , β ) ≤ d ( b + , x ) + d ( x , β ) ≤ δ + d ( α, β ) . The thesis for all possible projections of β + on α follows from Lemma 3.3.(b). Case 2: d ( α, β ) ≤ δ . In this case we parameterize β in such a way that β (0) is at distance at most δ from α . Then let t = max { t ∈ [0 , + ∞ ) s.t. d ( β ( t ) , α ) ≤ δ } , let y = β ( t ) and let x be any projection of y on α . The convex subset [ β ( t ) , β + ] of β is at distance > δ from α , so arguing as before we havethat any projection b + of β + on α is at distance at most δ from x . Then,again d ( b + , β ) ≤ d ( b + , x ) + d ( x , y ) ≤ δ .20 .3 Helly’s Theorem A subset C ⊆ X ∪ ∂ G X is said λ -quasiconvex , where λ ≥ , if for every x, y ∈ C there exists at least one geodesic (segment, ray or line) joining x to y that is included in B ( C, λ ) . The subset C is called starlike with respect to apoint x ∈ C if for all x ∈ C there exists at least one geodesic (segment, rayor line) [ x , x ] entirely included in C . For instance a convex set is starlikewith respect to all of its points. The proof of the following lemma can befound in [DKL18] and [CDP90]: Lemma 3.7 (Lemma 3.3 of [DKL18] and Proposition 10.1.2 of [CDP90]) . Let X be δ -hyperbolic and let C ⊆ X ∪ ∂ G X be starlike with respect to x .Then C is δ -quasiconvex and B ( C, λ ) is δ -quasiconvex for all λ ≥ . We state now the version of Helly’s Theorem which we will need. Theproof we give here follows the one given by [BF18], with the minor modifi-cations needed to deal with quasiconvex subsets instead of convex ones.
Proposition 3.8 (Helly’s Theorem) . Let X be a δ -hyperbolic space and let ( C i ) i ∈ I be a family of λ -quasiconvex subsets of X such that C i ∩ C j = ∅ ∀ i, j .Then: \ i ∈ I B ( C i , δ + 15 λ ) = ∅ . The proof is a direct consequence of the following lemma.
Lemma 3.9.
Let X be a δ -hyperbolic space, let C , C ⊆ X be two λ -quasiconvex subsets with non-empty intersection and let x ∈ X be fixed.Assume that we have points x ∈ C and x ∈ C which satisfy: d ( x , x i ) ≤ d ( x , C i ) + δ for i = 1 , d ( x , x ) ≥ d ( x , x ) − δ Then, d ( x , C ) ≤ δ + 15 λ .Proof. Let u ∈ C ∩ C . By the Projection Lemma 3.2 we have ( u, x ) x ≥ d ( x , [ u, x ]) − δ. Moreover, d ( x , [ u, x ]) ≥ d ( x , B ( C , λ )) ≥ d ( x , C ) − λ ≥ d ( x , x ) − λ − δ. So ( u, x ) x ≥ d ( x , x ) − λ − δ . Computing the Gromov product we get d ( u, z ) + 10 δ + 2 λ ≥ d ( x , z ) + d ( x , u ) . The same conclusion holds for x . Hence the two paths α = [ u, x ] ∪ [ x , x ] and β = [ u, x ] ∪ [ x , x ] are (1 , δ + 2 λ ) -quasigeodesic segments with sameendpoints. Applying Proposition 3.5 we conclude that for any t where thetwo paths are defined it holds: d ( α ( t ) , β ( t )) ≤ δ + 12 λ. We estimate now the distance between x and the geodesic segment [ u, x ] .21et t , t be such that α ( t ) = x and β ( t ) = x . If t ≤ t then β ( t ) belongs to [ u, x ] and t is a common time for both geodesic segments.Therefore we can conclude that d ( x , [ u, x ]) ≤ δ + 12 λ . We considernow the case t ≥ t . Since d ( x , x ) ≥ d ( x , x ) − δ we know that t = d ( u, x ) ≤ d ( x , u ) − d ( x , x ) + 10 δ + 2 λ ≤ d ( x , u ) − d ( x , x ) + 11 δ + 2 λ ≤ d ( x , x ) + d ( x , u ) − d ( x , x ) + 11 δ + 2 λ = t + 11 δ + 2 λ. Therefore we get d ( x , x ) = d ( α ( t ) , β ( t )) ≤ d ( α ( t ) , α ( t )) + d ( α ( t ) , β ( t )) ≤ δ + 2 λ + 108 δ + 12 λ = 119 δ + 14 λ. In any case we have d ( x , [ u, x ]) ≤ δ + 14 λ . In conclusion d ( x , C ) ≤ d ( x , B ( C , λ )) + λ ≤ d ( x , [ u, x ]) + λ ≤ δ + 15 λ. Proof of Proposition 3.8.
We choose a point x ∈ X . Let x i ∈ C i be as in theprevious lemma, say with d ( x , x ) ≥ d ( x , x i ) − δ for all i ∈ I . Applyingthe lemma to any couple C , C i we find that the point x belongs to theintersection of all the desired neighbouroods of C i . δ -hyperbolic spaces We record here some basic properties about discrete groups of isometries ofa Gromov-hyperbolic metric space.First recall that the isometries of X are classified into three types accordingto the behaviour of their orbits (cp. for instance [CDP90]):• an isometry g is elliptic if it has bounded orbits; when g acts discretely,this is the same as asking that it is a torsion element, cp. [BCGS17];• an isometry g is parabolic if there exists a point g ∞ ∈ ∂ G X such thatfor all x ∈ X the sequences ( g k x ) k ≥ and ( g k x ) k ≤ converge to g ∞ ;• an isometry g is hyperbolic if the map k g k x is a quasi-isometry ∀ x ∈ X , i.e. there exist L, C > such that for any k, k ′ ∈ Z it holds L | k − k ′ | − C ≤ d ( g k x, g k ′ x ) ≤ L | k − k ′ | + C. In this case there exist two points g − = g + in ∂ G X such that forany x ∈ X the sequence ( g k x ) k ≥ converges to g + and the sequence ( g k x ) k ≤ converges to g − . 22lso, recall that the asymptotic displacement of an isometry g is defined asthe limit (which exists and does not depend on the choice of x ∈ X ): k g k = lim n → + ∞ d ( x, g n x ) n . It is well known that for any isometry g of X and for any k ∈ Z ∗ it holds k g k k = k k g k and that g is hyperbolic if and only if k g k > (see [CDP90]).The following lemma is well known, and will be used to bound the displace-ment of an isometry by its asymptotic displacement: Lemma 3.10 ([BCGS17], [CDP90]) . For any isometry g and every x ∈ X we have: d ( x, g n x ) ≤ d ( x, gx ) + ( n − · | g | + 4 δ · log n . for all n > . Any isometry of X acts naturally on ∂ G X . Namely there exists a nat-ural topology on X ∪ ∂ G X extending the metric topology of X . If X is proper the sets ∂ G X and X ∪ ∂ G X are compact with respect to thistopology and any isometry of X acts as a homeomorphism on X ∪ ∂ G X .If g is parabolic then g ∞ is the unique fixed point of the action of g on X ∪ ∂ G X , while if g is hyperbolic then g − , g + are the only fixed points ofthe action of g on X ∪ ∂ G X . The set of fixed points of an isometry g on theGromov boundary is denoted by Fix ∂ ( g ) . For any k ∈ Z ∗ = Z \ { } we havethat an isometry g is elliptic (resp.parabolic, hyperbolic) if and only if g k iselliptic (resp.parabolic, hyperbolic); moreover, if g is parabolic or hyperbolicit holds Fix ∂ ( g k ) = Fix ∂ ( g ) .The limit set Λ(Γ) of a discrete group of isometries Γ of a proper, δ -hyperbolic space X is the set of accumulation points of the orbit Γ x on ∂ G X , where x is any point of X ; it is the smallest Γ -invariant closed setof the Gromov boundary, cp [Coo93]. The group Γ is called elementary if ≤ . For an elementary discrete group Γ there are three possibilities(cp. [Gro87], [CDP90], [DSU17], [BCGS17]):• Γ is elliptic , i.e. ; then d ( x, Γ x ) < ∞ for all x ∈ X , so theorbit of Γ is finite by discreteness;• Γ is parabolic , i.e. ; then in this case Γ contains onlyparabolic or elliptic elements and all the parabolic elements have thesame fixed point at infinity;• Γ is lineal , i.e. ; in this case Γ contains only hyperbolic orelliptic elements and all the hyperbolic elements have the same fixedpoints at infinity.So if two non-elliptic isometries a, b generate a discrete elementary group thenthey are either both parabolic or both hyperbolic and they have the sameset of fixed points in ∂ G X ; conversely if a, b are two non-elliptic isometries of X generating a discrete group h a, b i such that Fix ∂ ( a ) = Fix ∂ ( b ) , then h a, b i is elementary. We also recall the following property of elementary subgroupsof general Gromov-hyperbolic spaces:23 emma 3.11. Let X be a Gromov-hyperbolic space and let Γ be a discretegroup of isometries of X . Then for any non-elliptic g ∈ Γ there exists aunique maximal, elementary subgroup of Γ containing g .Proof. The maximal, elementary subgroup of Γ containing g is: { g ′ ∈ Γ | g ′ · Fix ∂ ( g ) = Fix ∂ ( g ) } . It is well known that any virtually nilpotent group of isometries Γ of X is elementary (since any non-elementary group Γ contains a free subgroup).Conversely if Γ is elliptic it is virtually nilpotent since it is finite. Alsoany lineal group is virtually cyclic (cp. Proposition 3.29 of [Cou16]), hencevirtually nilpotent. On the other hand there are examples of non-virtuallynilpotent, even free non abelian, parabolic groups acting on simply connectedRiemannian manifolds with curvature ≤ − (see [Bow93], Sec. 6). However,under a mild packing assumption it is possible to conclude that any parabolicgroup is virtually nilpotent. Some version of this fact is probably known tothe experts and we present here the proof for completeness; we thank S.Gallot for explaining it to us. Proposition 3.12.
Let X be a proper, geodesic, Gromov hyperbolic spacethat is P -packed at some scale r . Then any finitely generated, discrete,parabolic group of isometries Γ of X is virtually nilpotent. Hence, we deduce:
Corollary 3.13 (Elementary groups are virtually nilpotent) . Let X be a proper, geodesic, Gromov hyperbolic space, P -packed at scale r .Then a discrete, finitely generated group of isometries of X is elementary ifand only if it is virtually nilpotent. We remark that the scale of the packing is not important: it plays the roleof an asymptotic bound on the complexity of the space. The proof is basedon the following fundamental result proved in [BGT11]:
Theorem 3.14 (Corollary 11.2 of [BGT11]) . For every p ∈ N there exists N ( p ) ∈ N such that the following holds for every group Γ and every finite,symmetric generating set S of Γ : if there exists some A ⊂ Γ such that S N ( p ) ⊂ A and card ( A · A ) ≤ p · card ( A ) then Γ is virtually nilpotent.Proof of Proposition 3.12. Let S be a finite, symmetric generating set of Γ .Moreover let Λ(Γ) = { z } and let γ be any geodesic ray such that γ + = z .Finally set Σ R ( x ) := { g ∈ Γ s.t. d ( gx, x ) ≤ R } . Step 1. Setting R = max { r , δ } and p = P (1 + P ) R r − , we have forany x ∈ X : card (cid:0) Σ R ( x ) · Σ R ( x ) (cid:1) ≤ p · card (Σ R ( x )) Actually by Lemma 4.7 of [CS20] we havePack (cid:18) R , R (cid:19) ≤ Pack (9 R , r ) ≤ P (1 + P ) R r − . (cid:0) Σ R ( x ) · Σ R ( x ) (cid:1) card (Σ R ( x )) ≤ card (Σ R ( x )) card (Σ R ( x )) ≤ P (1 + P ) R r − which is our claim. Remark that p does not depend on the point x . Step 2. There exists T = T ( S, γ, p ) such that d ( γ ( T ) , gγ ( T )) ≤ δ for all g ∈ S N ( p ) (where N ( p ) is the value associated to p given by Theorem 3.14). Let ρ = max s ∈ S d ( γ (0) , sγ (0)) . So we have d ( γ (0) , gγ (0)) ≤ N ( p ) ρ for all g ∈ S N ( p ) . Let g ∈ S N ( p ) . By definition we have gz = z , so ( gγ ) + = z .Then by Lemma 3.1 there exist t , t ≥ such that t + t = d ( γ (0) , gγ (0)) and d ( γ ( t + t ) , gγ ( t + t )) ≤ δ for all t ≥ . Therefore, d ( γ ( s + t − t ) , gγ ( s )) ≤ δ (11)for all s ≥ T := max { t , t } ≤ N ( p ) ρ . In the following we may assume t ≥ t and call ∆ = t − t . If ∆ ≤ δ , we apply the previous esti-mate to s = T and we get d ( γ ( T ) , gγ ( T )) ≤ δ , so the claim is true.Otherwise, we have ∆ > δ , and we consider the triangle with vertices A = γ ( T + ∆) , B = γ ( T + 2∆) and C = gA . Let ( ¯ A, ¯ B, ¯ C ) be the corre-sponding tripod with center ¯ o with edge lengths ρ = ℓ ([ ¯ A, ¯ o ]) , σ = ℓ ([ ¯ C, ¯ o ]) and τ = ℓ ([ ¯ B, ¯ o ]) . We therefore have: ρ + τ = ∆ , σ + τ ≤ δ and σ + ρ = d ( A, gA ) . This implies that ρ − σ = ( ρ + τ ) − ( σ + τ ) ≥ ∆ − δ ≥ . In particular, if m is the midpoint of [ A, gA ] and ¯ m the corresponding point on the tripod,we have d ( ¯ m, ¯ A ) ≤ d (¯ o, ¯ A ) , so there exists a point m ′ ∈ [ A, B ] such that d ( m, m ′ ) ≤ δ . Applying Lemma 8.21 of [BCGS17] we deduce d ( m ′ , gm ′ ) ≤ d ( m, gm ) + 8 δ ≤ δ (as g is elliptic or parabolic). Moreover, as m ′ = γ ( s + ∆) for some s ≥ T (since ∆ ≥ ) we have by (11) that d ( m ′ , gγ ( s )) ≤ δ . By the triangleinequality we deduce ∆ = d ( gγ ( s ) , gm ′ ) ≤ δ. Therefore also in this case we get d ( γ ( T ) , gγ ( T )) ≤ δ . Conclusion.
We have S N ( p ) ⊂ Σ R ( γ ( T )) , where T is the constant of step 2.So we apply Theorem 3.14 and conclude that Γ is virtually nilpotent. δ -hyperbolic GCB-spaces Throughout this section ( X, σ ) will be a proper, δ -hyperbolic GCB -space.We will consider only σ -invariant isometries (called σ -isometries) of X :they are isometries g such that for all x, y ∈ X it holds σ g ( x ) g ( y ) = g ( σ xy ) . .1 Special properties in the convex case The boundary at infinity of X , denoted by ∂X , is defined as the set ofgeodesic rays modulo the equivalence relation ≈ , where γ ≈ ξ if and only if d ∞ ( γ, ξ ) = sup t ∈ [0 , + ∞ ) d ( γ ( t ) , ξ ( t )) < + ∞ endowed with the topology of uniform convergence on compact subsets.The σ -boundary at infinity of X , denoted ∂ σ X , is the set of σ -geodesicrays modulo ≈ . In general the natural inclusion ∂ σ X ⊆ ∂X can be strict,but these boundaries coincide when X is proper and δ -hyperbolic: Lemma 4.1.
Let σ be a convex, consistent, reversible geodesic bicombing ofa proper, δ -hyperbolic metric space X . Then:(i) geodesic bigons are uniformly thin, i.e. for every geodesic segments γ, ξ between x, y ∈ X it holds d ( γ ( s ) , ξ ( s )) ≤ δ , for all ≤ s ≤ d ( x, y ) ;(ii) for every geodesic ray γ and every x ∈ X there exists a unique σ -geodesic ray starting at x which is equivalent to γ , so ∂ σ X = ∂X ;(iii) there exists a natural homeomorphism between ∂ σ X and ∂ G X ;(iv) for every geodesic line γ there exists a σ -geodesic line γ ′ staying atbounded distance from γ .Proof. Claim (i) is well known in δ -hyperbolic spaces.The uniqueness in (ii) follows by the convexity of σ . Actually, if ξ, ξ ′ are σ -geodesic rays from x such that d ∞ ( ξ, ξ ′ ) ≤ D , then for any s > we have d ( ξ ( st ) , ξ ′ ( st )) ≤ sd ( ξ ( t ) , ξ ′ ( t ) ≤ sD for all t ; taking t ≫ it follows that d ( ξ ( s ) , ξ ′ ( s )) ≤ sDt is arbitrarily small, hence zero. To show the existence,we can suppose γ is a σ -geodesic ray. Indeed for a sequence t n → + ∞ weconsider the σ -geodesic segments [ γ (0) , γ ( t n )] . By the properness of X theyconverge to a geodesic ray γ ′ with origin γ (0) staying at bounded distancefrom γ by (i), which is in fact a σ -geodesic ray since the bicombing σ iscontinuous (Lemma 2.1). Now for a generic x ∈ X we consider a sequence of σ -geodesic segments ξ n = [ x, γ ( t n )] for t n → + ∞ . As before they converge toa σ -geodesic ray ξ with origin x . By convexity we get d ( ξ ( T ) , γ ) ≤ d ( x, γ (0)) for all T ≥ , so ξ is asymptotic to γ . From (ii) and the classical homeomor-phism between ∂X and ∂ G X (see Lemma III.3.13 of [BH13]) we conclude(iii).Finally (iv) follows as explained in Section 3.1, by taking the limit of σ -geodesic segments ξ n = [ γ ( − n ) , γ ( n )] . By the continuity of σ , this gives a σ -geodesic line, which is equivalent to γ by (i).Recall that the displacement function of an isometry g is defined as d g ( x ) = d ( x, gx ) , and the minimal displacement of g is ℓ ( g ) = inf x ∈ X d ( x, gx ) . In general the asymptotic displacement always satisfies k g k ≤ ℓ ( g ) ; howeveron a GCB-space ( X, σ ) we always have k g k = ℓ ( g ) for σ -invariant isometries,26ee [Des15], Proposition 7.1. In particular, if ( X, σ ) is a δ -hyperbolic GCB-space all parabolic σ -isometries have zero minimal displacement (notice thatthis is false for arbitrary convex spaces, cp. [Wu18]) and ℓ ( g ) > if and onlyif g is of hyperbolic type. Moreover by Lemma 2.4 every elliptic σ -isometry g of X has a fixed point (the center of any bounded orbit g n x of g , whichis clearly invariant by g ); reciprocally, every isometry with a fixed point isclearly elliptic. We can therefore restate the classification of isometries of aproper, Gromov-hyperbolic GCB-space ( X, σ ) as follows:• a σ -isometry g is elliptic if and only if ℓ ( g ) = 0 and the value of minimaldisplacement is attained for some x ∈ X ;• a σ -isometry g is parabolic if and only if ℓ ( g ) = 0 and the minimaldisplacement is not attained;• a σ -isometry g is hyperbolic if and only if ℓ ( g ) > and the minimaldisplacement is attained.It is well known that on a Busemann space any hyperbolic isometry has anaxis, i.e. a geodesic joining g − to g + on which g acts as a translation by ℓ ( g ) .This is also true for GCB-spaces, as proved in [Des15], Proposition 7.1.Less trivially (compare with Example 7.7 of [Des15]), a hyperbolic isometry g of a Gromov-hyperbolic GCB-space also has a σ -axis , that is an axis whichis a σ -geodesic: Lemma 4.2.
Let ( X, σ ) be a proper, δ -hyperbolic GCB -space and let g be a σ -invariant isometry of hyperbolic type. Then there exists a σ -axis of g .Proof. We consider the minimal set Min ( g ) , that is the subset of points where d g attains its minimum. This is a σ -convex, closed and g -invariant subset.Define ϕ g : X → X as the midpoint of the σ -geodesic segment [ g − x, gx ] from g − x to gx . By Lemma 7.5 of [Des15] we know that there exists a σ -axis of g if and only if there is some x ∈ Min ( g ) such that d ( x, ϕ g ( x )) = 0 .Moreover, by Proposition 7.6 of [Des15], there always exists a sequence ofpoints x k ∈ Min ( g ) such that lim k → + ∞ d ( x k , ϕ g ( x k )) = 0 .Fix now a arbitrary axis γ of g . By Proposition 7.1 in [Des15], for every k there exists an axis of g passing through x k , so by Lemma 3.1 we deducethat there exists t k ∈ R such that d ( x k , γ ( t k )) ≤ δ . Since γ is a geodesicwe have | t k − d ( γ (0) , x k ) | ≤ δ . Taking n k = ⌈ t k ℓ ( g ) ⌉ we obtain d ( g − n k x k , γ (0)) ≤ d ( g − n k x , g − n k γ ( t k )) + d ( g − n k γ ( t k ) , γ (0)) ≤ δ + ℓ ( g ) . Then, the sequence ( g − n k x k ) belongs to the closed ball B ( γ (0) , δ + ℓ ( g )) ,so there exists a subsequence converging to some point x ∞ . Finally, since g is σ -invariant, we deduce that ϕ g ( g − n k x k ) = g − n k ϕ g ( x k ) and so d ( g − n k x k , ϕ g ( g − n k x k )) = d ( x k , ϕ g ( x k )) −→ k → + ∞ , which implies that d ( x ∞ , ϕ g ( x ∞ )) = 0 , hence the existence of a σ -axis.27 .2 The Margulis domain We are interested in the sublevel sets of d g . Given ε > the subset M ε ( g ) = { x ∈ X s.t. d ( x, gx ) ≤ ε } is called the Margulis domain of g with displacement ε . As d g is σ -convex,the Margulis domain is a closed and σ -convex subset of X . Finally, letMin ( g ) = M ℓ ( g ) ( g ) be the subset of points of X where d g attains its minimum (which is emptyfor a parabolic isometry g ). Lemma 4.3.
The Margulis domain M ε ( g ) of any σ -invariant isometry g of X , if non-empty, is starlike with respect to any point z ∈ Fix ∂ ( g ) ∪ Min ( g ) .Proof. Fix a point x ∈ M ε ( g ) and z ∈ Fix ∂ ( g ) . Assume that the σ -geodesicray [ x, z ] is not contained in M ε ( g ) . Then there exists a point y ∈ [ x, z ] suchthat d ( y, gy ) ≥ ε + η for some η > . Let L = d ( x, y ) and consider the points y n along [ x, z ] at distance nL from x . By σ -convexity of the displacementfunction, we have d ( y n , gy n ) ≥ ε + nη. We observe that the points gy n belongto the σ -geodesic ray [ gx, gz ] , defining the point gz of the boundary. The tworays [ x, z ] and [ gx, gz ] are not parallel, hence gz = z which is a contradictionsince z ∈ Fix ∂ ( g ) . The case where z ∈ Min ( g ) follows directly from the σ -convexity of the displacement function and the minimality of z .The generalized Margulis domain of g at level ε is the set M ε ( g ) = [ i ∈ Z ∗ M ε ( g i ) . It clearly is a g -invariant subset of X . We remark that for all ε > theunion is finite when g is elliptic or hyperbolic, while it is infinite when g isof parabolic type. Lemma 4.4.
The generalized Margulis domain M ε ( g ) is δ -quasiconvexand connected.Proof. As a consequence of Lemma 4.3, the domain M ε ( g ) is starlike withrespect to any x ∈ Min ( g ) ∪ Fix ∂ ( g ) . So, by Lemma 3.7, it is δ -quasiconvex.The last assertion is trivial if g is elliptic or hyperbolic: in that case M ε ( g ) is a finite union of connected sets with a common point. If g is parabolic wefix a point x ∈ M ε ( g ) and any y ∈ M ε ( g ) , so y ∈ M ε ( g i ) for some i = 0 .Since ℓ ( g ) = 0 we can take a point x ′ ∈ M ε/ | i | ( g ) . By σ -convexity we havethat the σ -geodesic segment [ x, x ′ ] is entirely contained in M ε ( g ) . Moreover d ( x ′ , g i x ′ ) ≤ ε , so the σ -geodesic segment [ y, x ′ ] is contained in M ε ( g i ) . Asa consequence the curve [ x, x ′ ] ∪ [ x ′ , y ] is contained in M ε ( g ) . We concludethat M ε ( g ) is connected since every of its points can be connected to thefixed point x . 28ne of the key ingredients in the proof of Theorem 1.1 and in the appli-cations are the following lower and upper uniform estimates of the distancebetween the boundaries of two different generalized Margulis domains. Hy-perbolicity in used only in the upper estimate, while the packing conditionis essential to both. Proposition 4.5.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r , and let < ε ≤ ε . Let g be any σ -invariant non-ellipticisometry, x ∈ X \ M ε ( g ) and assume M ε ( g ) = ∅ . Then:(i) d ( x, M ε ( g )) ≥
12 ( ε − ε ) ;(ii) if ε ≤ r then d ( x, M ε ( g )) > L ε ( ε ) = log (cid:16) ε − (cid:17) P ) · ε − . Notice that the estimate (ii) is significative only for ε small enough,and has a different geometrical meaning from (i): it says that the M ε ( g ) contains a large ball around any point x ∈ M ε ( g ) , of radius which is largerand larger as ε tends to zero. Proof.
The first estimate is simple and does not need any additional condi-tion on the metric space X . Let ¯ x be a projection of x on the closure M ε ( g ) of the generalized Margulis domain. By definition for all η > there existssome nontrivial power g η of g such that d (¯ x, g η ¯ x ) ≤ ε + η . So: ε ≤ d ( x, g η x ) ≤ d ( x, ¯ x ) + d (¯ x, g η ¯ x ) + d ( g η ¯ x, g η x ) ≤ d ( x, M ε ( g )) + ε + η. The estimate follows from the arbitrariness of η .Let us now prove (ii). Let again ¯ x ∈ M ε ( g ) with d ( x, ¯ x )= d ( x, M ε ( g ))= R .For any η > let g η be some nontrivial power of g satisfying d (¯ x, g η ¯ x ) ≤ ε + η . Then again d ( x, g kη x ) ≤ R + d (¯ x, g kη ¯ x ) ≤ R + | k | ( ε + η ) so d ( x, g kη x ) ≤ R + 1 for all k such that | k | ≤ / ( ε + η ) . Therefore we haveat least n ( ε , η ) = 1 + 2 ⌊ / ( ε + η ) ⌋ points in the orbit Γ x inside the ball B ( x, R + 1) . We deduce that if n ( ε , η ) > Pack (2 R + 1 , ε ) two of thesepoints stay at distance less than ε one from the other, which implies that x ∈ X ε , a contradiction. Therefore, n ( ε , η ) = 1 + 2 ⌊ / ( ε + η ) ⌋ ≤ Pack (2 R + 1 , ε ) ≤ P (1 + P ) R +1 ε − by Proposition 2.5 (since ε ≤ r ), which implies that R = d ( x, y ) is greaterthan the function L ε ( ε ) in (ii) by the arbitrariness of η .The upper bound is more tricky: 29 roposition 4.6. Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r and let < ε ≤ ε . Then there exists K , only depending on P , r , δ, ε and ε , such that for every non-elliptic, σ -invariant isometry g of X such that M ε ( g ) = ∅ it holds: sup x ∈M ε ( g ) d ( x, M ε ( g )) ≤ K . Proof.
Let x ∈ M ε ( g ) , so by definition there exists i with d ( x, g i x ) ≤ ε .If x ∈ M ε ( g ) there is nothing to prove. Otherwise we can find a point ¯ x of ∂ M ε ( g ) such that d ( x, ¯ x ) = d ( x, M ε ( g )) . We set τ = max { ε , δ } and N = Pack (cid:0) τ, ε (cid:1) , which is a number depending only on P , r , δ and ε ,by Proposition 2.5. Step 1: we prove that there exists an integer k ≤ N +1 such that d (¯ x, g k · i ¯ x ) > τ and d ( x, g k · i x ) ≤ K := ε + 84 τ + 2( N + 1) δ (12)If this was not true, then for all k ≤ N +1 such that d ( x, g k · i x ) ≤ K we would have d (¯ x, g k · i ¯ x ) ≤ τ . Let p be the largest integer such that d (¯ x, g p · i ¯ x ) ≤ τ for all ≤ p ≤ p . We affirm that p ≥ N + 1 .Actually, p ≥ because d ( x, g i x ) ≤ ε ≤ K , hence d (¯ x, g i ¯ x ) ≤ τ byassumption. Also, by Lemma 3.10, we get d ( x, g i · i x ) ≤ d ( x, g i − · i x )+ ℓ ( g i − · i )+2 δ = d ( x, g i − · i x )+2 i − i ℓ ( g )+2 δ. and, iterating, d ( x, g p · i x ) ≤ d ( x, g i x ) + (2 p − i ℓ ( g ) + 2 pδ ≤ (2 p − i ℓ ( g ) + 2 pδ + ε for every ≤ p ≤ N + 1 . So, if p ≤ N we would have: d ( x, g ( p · i x ) ≤ d ( x, g p · i x ) + 2 p · i ℓ ( g ) + 2 δ ≤ d ( x, g p · i x ) + d (¯ x, g p · i ¯ x ) + 2 δ ≤ (2 p − i ℓ ( g ) + 2 p δ + ε + 42 τ + 2 δ ≤ τ + 2( p + 1) δ + ε < K since p i ℓ ( g ) ≤ d (¯ x, g p · i ¯ x ) ≤ τ by definition. Hence by assumption d (¯ x, g ( p · i ¯ x ) ≤ τ and p would not be maximal.Moreover, since ¯ x is in the boundary of ∂ M ε ( g ) , then inf i ∈ Z ∗ d (¯ x, g i ¯ x ) ≥ ε . Indeed if d (¯ x, g i ¯ x ) = ε − η for some i ∈ Z ∗ and some η > , then it is easyto show that for any y ∈ B (¯ x, η/ we would have d ( y, g i y ) < ε ; hence, B (¯ x, η/ ⊂ M ε ( g ) and ¯ x would not belong to ∂ M ε ( g ) .Then the points g p · i ¯ x , for p = 1 , . . . , N + 1 , are ε -separated. But, as theybelong all to the ball B (¯ x, τ ) , they should be at most N and this is acontradiction. This proves the first step. Step 2: for any k ≤ N +1 satisfying the conditions (12), we have: d ( x, g k · i x ) ≥ d ( x, ¯ x ) + d ( g k · i x, g k · i ¯ x ) (13)Indeed let us write y = g k · i x and ¯ y = g k · i ¯ x . By definition the point ¯ x d ( x, ¯ x ) = d ( x, M ε ( g )) ; so, from the δ -quasiconvexity of M ε ( g ) (Lemma 4.4), we deduce that d ( x, [¯ x, ¯ y ]) ≥ d ( x, B ( M ε ( g ) , δ )) = d ( x, ¯ x ) − δ. Moreover from the Projection Lemma 3.2 we have d ( x, [¯ x, ¯ y ]) ≤ (¯ x, ¯ y ) x + 4 δ. Combining these estimates and expanding the Gromov product we obtain d ( x, ¯ y ) ≥ d ( x, ¯ x ) + d (¯ x, ¯ y ) − δ. (14)Similarly, using that d ( y, ¯ y ) = d ( y, M ε ( g )) (as M ε ( g ) is g -invariant), weobtain d ( y, ¯ x ) ≥ d ( y, ¯ y ) + d (¯ y, ¯ x ) − δ. (15)Adding these last two inequalities and using that d (¯ x, ¯ y ) > τ ≥ δ wededuce d ( x, ¯ y ) + d ( y, ¯ x ) > d ( x, ¯ x ) + d ( y, ¯ y ) + 2 δ. Therefore applying the four-points condition (4) to x, ¯ y, ¯ x, y we find d ( x, ¯ y ) + d (¯ x, y ) ≤ max { d ( x, ¯ x ) + d ( y, ¯ y ); d ( x, y ) + d (¯ x, ¯ y ) } + 2 δ = d ( x, y ) + d (¯ x, ¯ y ) + 2 δ It follows: d ( x, y ) ≥ d ( x, ¯ y ) + d (¯ x, y ) − d (¯ x, ¯ y ) − δ ≥ d ( x, ¯ x ) + d (¯ x, ¯ y ) − δ + d ( y, ¯ y ) + d (¯ y, ¯ x ) − δ − δ ≥ d ( x, ¯ x ) + d ( y, ¯ y ) , where we have used again (14), (15) and that d (¯ x, ¯ y ) > τ ≥ δ (the firstcondition in (12)). Moreover, the second condition in (12) now yields d ( x, ¯ x ) + d ( y, ¯ y ) ≤ K The conclusion follows observing that d ( y, ¯ y ) = d ( x, ¯ x ) , so that d ( x, M ε ( g )) ≤ d ( x, ¯ x ) ≤ K/ which is the announced bound, depending only on P , r , δ, ε and ε .The distance between two (non generalized) Margulis domains of a non-elliptic isometry can also be bounded uniformly in δ -hyperbolic GCB-spaces,but the bound is not explicit. We will use this estimate to study the limitof sequences of isometries in Section 6.4. Proposition 4.7.
For any given ε, δ> there exists c ( ε, δ ) > satisfying thefollowing property. Let ( X, σ ) be a proper, δ -hyperbolic GCB -space X and g be a non-elliptic, σ -invariant isometry of X with M ε ( g ) = ∅ : then, for all x ∈ X it holds d ( x, gx ) ≥ c ( ε, δ ) · d ( x, M ε ( g )) . In particular for all < ε ≤ ε we get sup x ∈ M ε ( g ) d ( x, M ε ( g )) ≤ ε c ( ε , δ ) =: K ( ε , ε , δ ) . roof. Suppose by contradiction that the thesis is not true. Then for every n ∈ N there exist a proper, δ -hyperbolic GCB-space ( X n , σ n ) , a non-elliptic, σ n -invariant isometry g n of X n such that M ε ( g n ) = ∅ and a point x n ∈ X n such that < d ( x n , g n x n ) ≤ n d ( x n , M ε ( g n )) , (where the first inequality follows from the assumption that g n is not elliptic).Observe that this implies that x n / ∈ M ε ( g n ) for every n .For every n , let y n be the projection of x n on the σ n -convex set M ε ( g n ) .We fix a non-principal ultrafilter ω and we consider the ultralimit X ω ofthe sequence ( X n , y n ) . The space X ω is δ -hyperbolic (the stability of thehyperbolicity constant follows from (3)) and admits a structure σ ω of GCB-space, by Lemma 2.2. The sequence of isometries ( g n ) is admissible, as forevery n it holds d ( g n y n , y n ) = ε , so it defines a limit isometry g ω = ω - lim g n of X ω (cp. Proposition A.5 of [CS20]); moreover, the isometry g ω is σ ω -invariant, as follows from the definition of the limit structure σ ω and of g ω .We consider the sequence of σ n -geodesic segments γ n = [ y n , x n ] . We claimthat this sequence converges to a σ ω -geodesic ray of X ω : by Proposition A.5and Lemma A.6 of [CS20] it is enough to show that ω - lim d ( y n , x n ) = + ∞ .Since y n ∈ M ε ( g n ) and x n is not in this set we deduce d ( y n , x n ) ≥ n · d ( x n , g n x n ) ≥ n · ε for ω -a.e. ( n ) , so γ ω = ω - lim γ n is a σ ω -geodesic ray.We claim now that d ( g ω γ ω ( T ) , γ ω ( T )) ≤ ε for all T ≥ . For this, fix T ≥ .We showed before that the segment γ n is defined at time T , for ω -a.e. n .Clearly, the sequence of points ( γ n ( T )) defines the point γ ω ( T ) of X ω .Now, for every arbitrary η > , we will find an upper bound for all theintegers n such that d ( γ n ( T ) , g n γ n ( T )) > ε + η . By the arbitrariness of η , this will imply the claim. Let m ∈ N be the smallest integer such that mT ≥ d ( x n , y n ) and, for every integer k ∈ { , . . . , m − } consider thepoint γ n ( kT ) . From the σ n -convexity of the displacement function it holds d ( γ n ( kT ) , g n γ n ( kT )) > ε + kη. In particular, again by σ n -convexity of thedisplacement function, we get mT ≥ n · d ( x n , g n x n ) ≥ n · d ( γ n (( m − T ) , g n γ n (( m − T )) ≥ n ( ε + ( m − η ) . so n ≤ mm − Tη ≤ Tη =: n η , and the claim is proved.This imples that γ + ω is a fixed point at infinity of g ω . Therefore if g ω isnot elliptic one of the two sequences { g kω y ω } , { g − kω y ω } , where k ∈ N , mustconverge to γ + ω . Let now k ∈ Z be fixed. Clearly g kω y ω = ω - lim g kn y n .Moreover the point g kn y n belongs to M ε ( g n ) for ω -a.e. ( n ) , and since y n isthe projection of x n on this σ n -convex set we get ( x n , g kn y n ) y n ≤ δ by (9).Since this is true for ω -a.e. ( n ) we deduce that ( γ + ω , g kω y ω ) y ω ≤ δ . Sincethis holds for every k ∈ Z , the sequences { g kω y ω } and { g kω y ω } defined for k ∈ N do not converge to γ + ω , which implies g ω is elliptic. This means thereis a point z ω = ω - lim z n such that d ( z ω , g ω z ω ) < ε and so d ( z n , g n z n ) < ε ω -a.e. ( n ) . Moreover there exists L ≥ such that d ( y n , z n ) ≤ L for ω -a.e. ( n ) . Let us consider the σ -geodesic segment [ z n , x n ] and let us denoteby w n the point along this geodesic with d ( w n , g n w n ) = ε . Clearly we have d ( x n , w n ) ≥ d ( x n , y n ) and d ( w n , z n ) ≤ L . Moreover by convexity along [ z n , x n ] we get ε + d ( x n , w n ) L ε ≤ d ( x n , g n x n ) ≤ n d ( x n , y n ) ≤ n d ( x n , w n ) implying ε L ≤ n which is clearly impossible if n is big enough, a contradic-tion. In this section we will prove Theorem 1.1.First remark that it is possible to assume ℓ ( a ) = ℓ ( b ) =: ℓ .Indeed we can take b ′ = bab − and h a, b ′ i is still a discrete and non-elementarygroup, which moreover is torsion-free if h a, b i was torsion-free. Futhermore ℓ ( b ′ ) = ℓ ( a ) and the length of b ′ as a word of a, b is .The proof will then be divided into two cases: ℓ ≤ ε and ℓ > ε , where ε = ε ( P , r ) is the Margulis constant given by Corollary 2.8. The proofin the first case does not use the torsionless assumption and produces a truefree subgroup; it heavily draws, in this case, from techniques introduced in[DKL18] and [BCGS17]. On the other hand, in the case where ℓ > ε theproofs of the statements (i) and (ii) diverge. In this last case producing a freesub-semigroup is quite standard, while producing a free subgroup is muchmore complicated and for this we need to properly modify the argument of[DKL18] to use it in our context. ℓ ≤ ε . We assume here a, b non-elliptic, σ -isometries with ℓ ( a ) = ℓ ( b ) = ℓ ≤ ε of a δ -hyperbolic GCB -space ( X, σ ) which is P -packed at scale r , and thatthe group h a, b i is non-elementary and discrete. In particular a and b areboth parabolic or both hyperbolic. In order to find a free subgroup in this case we will use a criterion which isthe generalization of Proposition 4.21 of [BCGS17] to non-elliptic isometries.Recall that, given two isometries a, b ∈ Isom ( X ) , the Margulis constant ofthe couple ( a, b ) is the number L ( a, b ) = inf x ∈ X inf ( p,q ) ∈ Z ∗ × Z ∗ n max { d ( x, a p x ) , d ( x, b q x ) } o . Proposition 5.1.
Let a, b be two non-elliptic σ -isometries of X of the sametype such that h a, b i is discrete and non-elementary. If a, b satisfy L ( a, b ) > max { ℓ ( a ) , ℓ ( b ) } + 56 δ then h a, b i is a free group. L ( a, b ) bythe distance of the corresponding generalized Margulis domains: Lemma 5.2.
Let a, b two σ -isometries of X and let L > :(a) if d ( M L ( a ) , M L ( b )) > then L ( a, b ) ≥ L ;(b) conversely, if L ( a, b ) > L then M L ( a ) ∩ M L ( b ) = ∅ . Proof. If d ( M L ( a ) , M L ( b )) > then M L ( a ) ∩ M L ( b ) = ∅ . In particular forevery x ∈ X and for all p, q ∈ Z ∗ we have d ( x, a p x ) > L or d ( x, b q x ) > L .Taking the infimum over x ∈ X we get L ( a, b ) ≥ L , proving (a).Suppose now L ( a, b ) > L and M L ( a ) ∩ M L ( b ) = ∅ . Take x in the intersec-tion. In particular ∀ η > there exist x η ∈ M L ( a ) , y η ∈ M L ( b ) such thatfor some ( p η , q η ) ∈ Z ∗ × Z ∗ d ( x, x η ) < η, d ( x, y η ) < η, d ( x η , a p η x η ) ≤ L, d ( y η , b q η y η ) ≤ L. By the triangle inequality we get d ( x, a p η x ) , d ( x, b q η x ) ≤ L + 2 η . As this istrue for every η > we get L ( a, b ) ≤ L . This contradiction proves (b). Proof of Proposition 5.1.
We will assume that a, b are parabolic isometries,the hyperbolic case being covered in [BCGS17]. The aim is to show thatthere exists x ∈ X such that d ( a p x, b q x ) > max { d ( x, a p x ) , d ( x, b q x ) } + 2 δ ∀ ( p, q ) ∈ Z ∗ × Z ∗ ; (16)this will imply that a and b are in Schottky position by Proposition 4.6 of[BCGS17], so the group h a, b i is free.With this in mind, choose L and < ε < δ with L ( a, b ) > L > δ + 2 ε ,and set ℓ = δ + ε . Since L ( a, b ) > L then M L ( a ) ∩ M L ( b ) = ∅ by Lemma5.2. Moreover the two Margulis domains are non-empty since L > .Now fix points x ∈ M l ( a ) and y ∈ M l ( b ) which ε -almost realize thedistance between the two generalized Margulis domains, that is: d ( x , y ) ≥ d ( x , y ) − ε ∀ y ∈ M l ( b ) d ( y , x ) ≥ d ( y , x ) − ε ∀ x ∈ M l ( a ) . Then we can find a point x ∈ [ x , y ] such that: d ( x, a p x ) > L and d ( x, b q x ) > L ∀ ( p, q ) ∈ Z ∗ × Z ∗ . (17)Indeed the sets M L ( a ) and M L ( b ) are non-empty, closed and disjoint;moreover the former contains x and the latter contains y . Then theirunion cannot cover the whole geodesic segment [ x , y ] . Any x on this seg-ment which does not belong to M L ( a ) ∪ M L ( b ) satisfies our requests.34s x and a p x belong to M l ( a ) and x ∈ X \ M L ( a ) we deduce by Lemma4.5 that d ( x, x ) ≥ d ( x, M l ( a )) − ε ≥ L − ℓ − ε > δ ∀ p ∈ Z ∗ (18)(notice that x ∈ [ x , y ] and x is a ε -almost projection of y on M l ( a ) )and the same is true for d ( x, a p x ) .Now choose points u ∈ [ x, a p x ] , u ′ ∈ [ x, a p x ] and u ′′ ∈ [ x, x ] at distance δ from x (notice that this is possible as d ( x, a p x ) > L > δ and by (18)).Consider the approximating tripod f ¯∆ : ∆( x, x , a p x ) → ¯∆ and the preim-age c ∈ f − (¯ c ) ∩ [ x , a p x ] of its center ¯ c . By Lemma 4.4 we deduce that d ( c, M l ( a )) ≤ δ and then, by (6) and Lemma 4.5, that ( a p x , x ) x ≥ d ( x, c ) − δ ≥ d ( x, M l ( a )) − δ > δ = d ( x, u ′ ) = d ( x, u ′′ ) . So f ¯∆ ( u ′ )= f ¯∆ ( u ′′ ) and, by the thinness of ∆( x, x , a p x ) , we get d ( u ′ , u ′′ ) ≤ δ .Since d ( x, a p x ) > L and d ( x, a p x ) > d ( a p x, a p x ) we immediately deduce ( a p x , a p x ) x > L > d ( u, x ) = d ( u ′ , x ) . So, again by thinness of the triangle ∆( x, a p x , a p x ) , we have d ( u, u ′ ) ≤ δ .Therefore d ( u, u ′′ ) ≤ δ . One analogously proves that, choosing v ∈ [ x, b q x ] , v ′ ∈ [ x, b q y ] and v ′′ ∈ [ x, y ] at distance δ from x , we have d ( v, v ′′ ) ≤ δ .Therefore (as x belongs to the geodesic segment [ x , y ] ) we deduce that d ( u, v ) ≥ d ( u ′′ , x ) + d ( x, v ′′ ) − d ( u, u ′′ ) − d ( v, v ′′ ) ≥ δ. Comparing with the tripod ¯∆ ′ which approximates the triangle ∆( a p x, x, b q x ) ,we deduce by the δ -thinness that f ¯∆ ′ ( u ) = f ¯∆ ′ ( v ) . It follows that ( a p x, b q x ) x < d ( x, u ) = d ( x, v ) = 11 δ. One then computes: d ( a p x, b q x ) = d ( a p x, x ) + d ( x, b q x ) − a p x, b q x ) x ≥ max { d ( a p x, x ) , d ( x, b q x ) } + min { d ( a p x, x ) , d ( x, b q x ) } − δ ≥ max { d ( a p x, x ) , d ( x, b q x ) } + L − δ which implies (16), by definition of L .We continue the proof of Theorem . in the case ℓ ( a ) = ℓ ( b ) = ℓ ≤ ε .Set b i = b i ab − i . Then ℓ ( b i ) = ℓ for all i , and b i is of the same type of a ,in particular it is non-elliptic. Moreover for any i = j the group h b i , b j i isdiscrete (as a subgroup of a discrete group) and non-elementary.Indeed otherwise there would exist a subset F ⊂ ∂X fixed by both b i , b j ,so b i Fix ∂ ( a ) = Fix ∂ ( b i ab − i ) = F = Fix ∂ ( b j ab − j ) = b j Fix ∂ ( a ) . This impliesthat b i − j ( F ) = F , hence F ⊆ Fix ∂ ( b ) and, as these sets have the same cardi-nality, they coincide. Therefore we deduce that Fix ∂ ( a ) = b − i ( F ) = Fix ∂ ( b ) ,which means that the group h a, b i is elementary, a contradiction.35ow, since for any i = j the group h b i , b j i is discrete and non-elementary,then by definition of the Margulis constant ε we have M ε ( b i ) ∩ M ε ( b j ) = ∅ . (otherwise there would exist a point x ∈ X and powers k, h such that d ( x, b ki x ) ≤ ε and d ( x, b hj x ) ≤ ε , and h b ki , b hj i would be virtually nilpo-tent, hence elementary; but we have just seen that this implies that h a, b i is elementary, a contradiction). Moreover each Margulis domain M ε ( b i ) isnon-empty, since we assumed ℓ = ℓ ( b i ) ≤ ε .We now need the following Lemma 5.3.
Let B the set of all the conjugates b i = b i ab − i , for i ∈ Z .For any fixed L > , the cardinality of every subset S of B such that d ( M ε ( b i h ) , M ε ( b i k )) ≤ L ∀ b i h , b i k ∈ S is bounded from above by a constant M only depending on P , r , δ and L .Proof. Let S = { b i , · · · , b i M } ⊆ B satisfying d ( M ε ( b i h ) , M ε ( b i k )) ≤ L forall b i h , b i k ∈ S . Fix η > and consider the closed ( L + η ) -neighbourhoodsof the generalized Margulis domains B k = B ( M ε ( b i k ) , L + η ) , for b i k ∈ S .Since the domains are starlike, then B k is δ -quasiconvex for all k , byLemma 3.7. Moreover B h ∩ B k = ∅ for every h, k . Indeed, chosen points x i h ∈ M ε ( b i h ) and x i k ∈ M ε ( b i k ) which η -almost realize the distancebetween these domains, then the midpoint of any geodesic segment [ x i h , x i k ] is in B h ∩ B k . Therefore we can apply Helly’s Theorem (Proposition 3.8) tofind a point x at distance at most δ from each B k . So d ( x , M ε ( b i k )) ≤ R = 419 δ + L η, for k = 1 , . . . , M. Notice that x belongs at most to one of the domains M ε ( b i k ) , since theyare pairwise disjoint. So for each of the remaining M − domains we can findpoints x k ∈ M ε ( b i k ) ∩ B ( x , R + η ) and p k ∈ Z ∗ such that d ( x k , b p k i k x k ) = ε .For this consider any x ′ k ∈ M ε ( b i k ) ∩ B ( x , R + η ) : by definition there exists p k ∈ Z ∗ such that d ( x ′ k , b p k i k x ′ k ) ≤ ε . On the other hand d ( x , b p ik i k x ) > ε since x / ∈ M ε ( b i k ) . Then, by continuity of the displacement function ofthe isometry b p k i k , we can find a point x k along a geodesic segment [ x , x ′ k ] such that d ( x k , b p k i k x k ) = ε precisely. Remark that x k ∈ B ( x , R + η ) as itbelongs to the geodesic [ x , x ′ k ] .Now, since ℓ ( b i k ) ≤ ε for all k , we can apply Proposition 4.6 and get d (cid:16) x k , M ε ( b i k ) (cid:17) ≤ K for some K depending only on P , r , δ and ε , so (by Corollary 2.8)ultimately only on P , r and δ .So for each k we have some point y k ∈ X such that d ( y k , b q k i k y k ) ≤ ε and d ( x k , y k ) ≤ K + η q k ∈ Z ∗ . Set R = R + η + K + η , so that y k ∈ B ( x , R ) . Weremark now that the ball B ( y k , ε ) is contained in M ε ( b i k ) : indeed for every z ∈ B ( y k , ε ) it holds d ( z, b q k i k z ) ≤ d ( z, y k ) + d ( y k , b q k i k y k ) + d ( b q k i k y k , b q k i k z ) ≤ ε . Finally we set R = R + ε : then we have B ( y k , ε ) ⊂ B ( x , R ) for all k .All the balls B ( y k , ε ) are pairwise disjoint, so the points y k are ε -separated.Hence the cardinality M of the set S satisfies M ≤ Pack (cid:0) R , ε (cid:1) =: M ,which is a number depending only on P , r , δ , L and η by Proposition 2.5.Taking for instance the constant M obtained for η = 1 , we get the an-nounced bound.To conclude the proof of the theorem in this case we will apply theprevious lemma for an appropriate value of L . By Proposition 4.6 we get sup x ∈M ε δ ( b i ) d ( x, M ε ( b i )) ≤ K ( P , r , δ, ε ) = K ′ ( P , r , δ ) , where again Corollary 2.8 bounds the value of ε in terms of P and r .We set L = 2 K ′ and apply Lemma 5.3: so there exist i, j ≤ M ( P , r , δ ) such that d ( M ε ( b i ) , M ε ( b j )) > K ′ . In particular d ( M ε +56 δ ( b i ) , M ε +56 δ ( b j )) > . (otherwise we would find x i ∈ M ε +56 δ ( b i ) and x j ∈ M ε +56 δ ( b j ) at ar-bitrarily small distance; but there exists also points y i ∈ M ε ( b i ) and y j ∈ M ε ( b j ) with d ( x i , y i ) ≤ K ′ and d ( x j , y j ) ≤ K ′ , which would yield d ( M ε ( b i ) , M ε ( b j )) ≤ K ′ , a contradiction).Applying b − i , we deduce that d ( M ε +56 δ ( a ) , M ε +56 δ ( b j − i ab i − j )) > . This implies, by Lemma 5.2, that L ( a, b j − i ab i − j ) ≥ ε + 56 δ > max { ℓ ( a ) , ℓ ( b j − i ab i − j ) } + 56 δ. By Proposition 5.1 we then deduce that the subgroup generated by a and w = b j − i ab i − j is free. Remark that the length of w is bounded above by M that is a function depending only on P , r and δ . ℓ > ε . We assume here that a, b are hyperbolic σ -isometries of a δ -hyperbolic GCB -space ( X, σ ) which is P -packed at scale r , satisfying ℓ ( a ) = ℓ ( b ) = ℓ > ε ,and such that the group h a, b i is non-elementary and discrete. The proof of assertion (i) in Theorem 1.1 in this case stems directly fromProposition 4.9 of [BCGS17] (free sub-semigroup theorem for isometries withminimal displacement bounded below), since Corollary 2.8 bounds ε in37erms of P and r . So, we will focus here on the proof of assertion (ii),therefore assuming moreover h a, b i torsionless .We fix σ -axes α and β of a and b respectively. The proof of assertion (ii)of Theorem 1.1 will break down into three subcases, according to the value ofthe distance d = d ( α, β ) between the minimal sets: the case where d ≤ ε ,the case ε < d ≤ δ and the case d > δ . In all cases we will use aping-pong argument which we will explain in the next subsection. The aim of this subsection is to prove the following:
Proposition 5.4.
Let ( X, σ ) be a proper, δ -hyperbolic GCB -space and let a, b be hyperbolic σ -isometries of X with minimal displacement ℓ ( a ) = ℓ ( b ) = ℓ .Let α, β be two σ -axes for a and b respectively, satisfying ∂α ∩ ∂β = ∅ .Finally let x − , x + be respectively projections of β − , β + on α and supposethat x + follows x − along the (oriented) geodesic α . Assume d ( x − , x + ) ≤ M :then the group h a N , b N i is free for any N ≥ ( M + 77 δ ) /ℓ . For any x ∈ α and any T ≥ we will denote for short by x ± T the pointsalong α at distance T from x , according to the orientation of α ; we will usethe analogous notation y ± T for the points on β such that d ( y, y ± T ) = T .By assumption, we have x + = x − + T for some T ≥ .Finally, for T > we define T -neighbourhoods of α + and α − as: A + ( T ) = { z ∈ X s.t. d ( z, x + + T ) ≤ d ( z, x + ) } A − ( T ) = { z ∈ X s.t. d ( z, x − − T ) ≤ d ( z, x − ) } (19)and their analogues B ± ( T ) = { z ∈ X s.t. d ( z, y ± ± T ) ≤ d ( z, y ± ) } for β .The proof will stem from a series of technical lemmas. Lemma 5.5.
For any T ≥ t ≥ one has: A + ( T ) ⊂ { z ∈ X | ( α + , z ) x + ≥ t } A − ( T ) ⊂ { z ∈ X | ( α − , z ) x − ≥ t } . Analogously, B ± ( T ) ⊂ { z ∈ X | ( β ± , z ) y ± ≥ t } for T ≥ t ≥ .Proof. Let z ∈ A + ( T ) , i.e. d ( z, x + + T ) ≤ d ( z, x + ) . For any S ≥ T we have d ( z, x + + S ) ≤ d ( z, x + + T ) + ( S − T ) ≤ d ( z, x + ) + ( S − T ) . Hence ( α + , z ) x + ≥ lim inf S → + ∞ (cid:2) d ( z, x + ) + S − ( d ( z, x + ) + S − T ) (cid:3) = T ≥ t. The proof for B ± ( T ) is analogous. Lemma 5.6.
For any T ≥ t + 8 δ one has: B ± ( T ) ⊂ { z ∈ X | ( β ± , z ) x ± ≥ t } . roof. Let z ∈ B + ( T ) , i.e. d ( z, y + ) ≥ d ( z, y + + T ) . We have, again, ∀ S ≥ T , d ( z, y + + S ) ≤ d ( z, y + + T ) + ( S − T ) ≤ d ( z, y + ) + ( S − T ) . Since y + is also a projection of x + on the geodesic segment [ y + , y + + S ] , byLemma 3.2 it holds ( y + , y + + S ) x + ≥ d ( x + , y + ) − δ . Expanding the Gromovproduct we get d ( x + , y + + S ) ≥ d ( x + , y + ) + S − δ. Therefore z, β + ) x + ≥ lim inf S → + ∞ (cid:2) d ( x + , z ) + d ( x + , y + + S ) − d ( z, y + + S ) (cid:3) ≥ lim inf S → + ∞ (cid:2) d ( x + , z ) + d ( x + , y + ) + S − δ − d ( z, y + ) − ( S − T ) (cid:3) ≥ T − δ ≥ t. The proof for B − ( T ) is the same. Lemma 5.7.
For any z ∈ X it holds: ( α + , z ) x − ≥ ( α ± , z ) x + − δ, ( α + , β − ) x − ≥ ( α + , β − ) x + − δ, ( α − , z ) x + ≥ ( α ± , z ) x − − δ, ( α − , β + ) x + ≥ ( α − , β + ) x − − δ. Proof.
We have ( α + , z ) x − ≥ lim inf S → + ∞ ( x + + S, z ) x − . On the other handfor any S ≥ we get (as x + follows x − along α ): x + + S, z ) x − = d ( x + + S, x − ) + d ( x − , z ) − d ( x + + S, z )= d ( x + , x + + S ) + d ( x + , x − ) + d ( x − , z ) − d ( x + + S, z ) ≥ d ( x + , x + + S ) + d ( z, x + ) − d ( x + + S, z )= 2( x + + S, z ) x + . So, by (8), we get ( α + , z ) x − ≥ lim inf S → + ∞ ( x + + S, z ) x + ≥ ( α + , z ) x + − δ .Taking any sequence z n converging to β − then proves the second formula.The proof for ( α − , z ) x + and ( α − , β + ) x + is analogous. Lemma 5.8.
For any u ∈ X it holds: ( β + , u ) x − ≥ ( β + , u ) x + − δ, ( β − , u ) x + ≥ ( β − , u ) x − − δ. Proof.
Take a sequence ( y i ) defining β + and let x i be a projection of y i on α .By Remark 3.4 we know that, up to a subsequence, the sequence x i convergesto x ∞ which is a projection on α of β + . So d ( x ∞ , x + ) ≤ δ by Lemma 3.3.For any ε > and for every i large enough, by Lemma 3.2, we have d ( y i , x − ) ≥ d ( y i , x i ) + d ( x i , x − ) − δ ≥ d ( y i , x + ) + d ( x + , x − ) − δ − ε. Therefore we get β + , u ) x − ≥ lim inf i → + ∞ (cid:2) d ( x − , u ) + d ( x − , y i ) − d ( u, y i ) (cid:3) ≥ lim inf i → + ∞ (cid:2) d ( x − , u ) + d ( x + , y i ) + d ( x − , x + ) − δ − d ( u, y i ) (cid:3) ≥ lim inf i → + ∞ (cid:2) d ( x + , y i ) + d ( x + , u ) − d ( u, y i ) − δ (cid:3) ≥ β + , u ) x + − δ. emma 5.9. We have: ( α + , β + ) x + ≤ δ, ( α − , β + ) x + ≤ δ, ( α − , β − ) x − ≤ δ, ( α + , β − ) x − ≤ δ Proof.
Take a sequence ( y i ) defining β + and call x i a projection of y i on α .As before, x i converges, up to a subsequence, to a projection x ∞ of β + on α ,and d ( x ∞ , x + ) ≤ δ . For any ε > , for any S ≥ and for every i largeenough, by Lemma 3.2, we have d ( y i , x + ± S ) ≥ d ( y i , x i ) + d ( x i , x + ± S ) − δ ≥ d ( y i , x + ) + d ( x + , x + ± S ) − δ − ε. Therefore we get α + , β + ) x + ≤ lim inf i,S → + ∞ (cid:2) d ( x + , x + + S ) + d ( x + , y i ) − d ( x + + S, y i ) (cid:3) + 2 δ ≤ δ. The same computation with − S instead of S proves the second inequality.The inequalities involving β − are proved in the same way. Lemma 5.10.
The subsets A + ( T ) , A − ( T ) , B + ( T ) and B − ( T ) are pairwisedisjoint for T > δ .Proof. Fix some t > δ . We claim that the subsets A + ( T ) , A − ( T ) , B + ( T ) and B − ( T ) are pairwise disjoint provided that T ≥ t + 8 δ > δ .We first prove that A + ( T ) ∩ A − ( T ) = ∅ . If z ∈ A + ( T ) ∩ A − ( T ) then ( α + , z ) x + ≥ t > δ and ( α − , z ) x − ≥ t > δ by Lemma 5.5. Then byLemma 5.7 we have ( α − , z ) x + > δ . Thus we obtain a contradiction since δ ≥ ( α + , α − ) x + ≥ min { ( α + , z ) x + , ( α − , z ) x + } − δ > δ. Let us prove now that A + ( T ) ∩ B + ( T ) = ∅ . If z ∈ A + ( T ) ∩ B + ( T ) then wehave ( α + , z ) x + > δ and ( β + , z ) x + > δ by Lemma 5.5 and Lemma 5.6.We then obtain again a contradiction by Lemma 5.9, as δ ≥ ( α + , β + ) x + ≥ min { ( α + , z ) x + , ( β + , z ) x + } − δ > δ. We now prove that A + ( T ) ∩ B − ( T ) = ∅ . Actually if z ∈ A + ( T ) ∩ B − ( T ) then ( α + , z ) x + > δ and ( β − , z ) x − > δ by Lemma 5.5 and Lemma 5.6.Moreover by Lemma 5.8 we have ( β − , z ) x + > δ and combining Lemma5.9 with Lemma 5.7 we deduce that ( β − , α + ) x + ≤ ( β − , α + ) x − + δ ≤ δ. So we again get a contradiction since δ ≥ ( α + , β − ) x + ≥ min { ( α + , z ) x + , ( β − , z ) x + } − δ > δ. The proof of B + ( T ) ∩ B − ( T ) = ∅ can be done as for A + ( T ) ∩ A − ( T ) = ∅ ,using Lemma 5.7. The remaining cases can be proved similarly.We are now in position to prove Proposition 5.4.40 roof of Proposition 5.4. We define the sets: A + = { z ∈ X s.t. d ( z, a N x − ) ≤ d ( z, x + ) } ,A − = { z ∈ X s.t. d ( z, a − N x + ) ≤ d ( z, x − ) } and their analogues B ± = { z ∈ X s.t. d ( z, b N y ∓ ) ≤ d ( z, y ± ) } for b .By assumption we have d ( x − , a N x − ) = N ℓ ( a ) ≥ d ( x − , x + ) + 65 δ .In particular A + ⊂ A + (65 δ ) as defined in (19), as follows directly from the σ -convexity of the distance function from the σ -geodesic line α using the factthat x + + 65 δ is between x + and a N x − (according to the chosen orientationof α ), and A − ⊂ A − (65 δ ) .Moreover we have d ( y − , y + ) ≤ d ( x − , x + ) + 12 δ by Lemma 3.3, and we canprove in the same way that B + ⊂ B + (65 δ ) and B − ⊂ B − (65 δ ) .Then by Proposition 5.10 the sets A + , A − , B + , B − are pairwise disjoint.We will prove now the following relations: a N ( X \ A − ) ⊂ A + , a − N ( X \ A + ) ⊂ A − ,b N ( X \ B − ) ⊂ B + , b − N ( X \ B + ) ⊂ B − Indeed if z ∈ X \ A − then d ( z, a − N x + ) > d ( z, x − ) ; applying a N to both sideswe get d ( a N z, x + ) > d ( a N z, a N x − ) , i.e. a N u ∈ A + . The other relations areproved in the same way. As a consequence we have, for all k ∈ N ∗ , a kN ( X \ A − ) ⊂ A + , a − kN ( X \ A + ) ⊂ A − ,b kN ( X \ B − ) ⊂ B + , b − kN ( X \ B + ) ⊂ B − . It is then standard to deduce by a ping-pong argument that the group gener-ated by a N and b N is free. Actually no nontrivial reduced word w in { a N , b N } can represent the identity, since it sends any point of X \ ( A + ∪ A − ∪ B + ∪ B − ) into the complement A + ∪ A − ∪ B + ∪ B − (notice that the former set is non-empty, as X is connected). d ( α, β ) ≤ ε . Recall that we are assuming a, b σ -isometries with ℓ ( a ) = ℓ ( b ) = ℓ > ε . Let x ∈ α, y ∈ β be points with d ( x , y ) = d ( α, β ) . Denote by π α and π β the projection maps on α and β respectively, and call x ± the projections of β ± on α . As in subsection 5.2.1, up to replacing b with b − we can assumethat x + follows x − along α .Let now [ z − , z + ] be the set of points of α at distance d = ε < ℓ from β .It is a nonempty, finite geodesic segment since the distance function from β is convex when restricted to α , and ∂α ∩ ∂β = ∅ (the group h a, b i beingnon-elementary). Clearly, it holds: d ( z − , β ) = d ( z + , β ) = d. Call L the length of [ z − , z + ] . The following estimate of this length is due toDey-Kapovich-Liu: since the proof is scattered in different papers (it appearsin the discussion after Lemma 4.5 in [DKL18], using an argument of [Kap01]for trees), we consider worth to recall it for completeness:41 roposition 5.11. With the notations above it holds: L < ℓ . For ≤ T ≤ L let α T denote the central segment of [ z − , z + ] of length T (so, α L = [ z − , z + ] ). Then: Lemma 5.12.
For any x ∈ α L − ℓ we have that either d ( π β ( ax ) , π β ( bx )) ≤ d and d ( π β ( a − x ) , π β ( b − x )) ≤ d (20) or d ( π β ( ax ) , π β ( b − x )) ≤ d and d ( π β ( a − x ) , π β ( bx )) ≤ d. (21) Moreover the conditions (20) and (21) cannot hold together, and one of thetwo hold on the whole interval α L − ℓ .Proof. By assumption, as ℓ ( a ) = ℓ , the points a ± x belong to α L . Then,by definition of α L , we have d ( π β ( x ) , x ) ≤ d and d ( π β ( a ± x ) , a ± x ) ≤ d .Therefore, we have | d ( π β ( x ) , π β ( a ± x )) − ℓ ( a ) | ≤ d. As π β ( x ) , π β ( a ± x ) belong to β and b translates π β ( x ) along β precisely by ℓ = ℓ ( b ) = ℓ ( a ) , itfollows that there exists τ, τ ′ ∈ { , − } such that d ( π β ( ax ) , π β ( b τ x )) = d ( π β ( ax ) , b τ π β ( x )) ≤ d,d ( π β ( a − x ) , π β ( b τ ′ x )) ≤ d. Moreover, since d ( π β ( ax ) , π β ( a − x )) ≥ ℓ − d , the above relations cannothold together with τ = τ ′ when ℓ − d > d . Therefore, for our choice of d < ℓ we have τ ′ = − τ and (20) is proved.Since d ( π β ( bx ) , π β ( b − x )) = 2 ℓ > d , the relations (20) and (21) cannot holdat the same time. Finally the last assertion follows from the connectednessof the interval α L − ℓ . Lemma 5.13.
Let η ≥ and x ∈ B ( α L − ℓ , η ) . Then either d ( bx, π α ( ax )) ≤ η + 6 d and d ( b − x, π α ( a − x )) ≤ η + 6 d (22) or d ( bx, π α ( ax )) ≤ η + 6 d and d ( b − x, π α ( a − x )) ≤ η + 6 d (23) Moreover the first (resp. second) condition occurs if and only if the first(resp. second) condition in Lemma 5.12 holds.Proof.
We fix x ∈ X such that d ( x, α L − ℓ ) ≤ η . Since every point of α L − ℓ is at distance at most d from β then d ( x, π β ( x )) ≤ η + d . We assume that(20) holds and we prove the first one in (22), the other cases being similar.We have: d ( bx, π α ( ax )) ≤ d ( bx, π β ( bx ))+ d ( π β ( bx ) , π β ( π α ( ax )))+ d ( π β ( π α ( ax )) , π α ( ax )) . The first term equals d ( x, π β ( x )) ≤ η + d . We observe that from the choiceof L − ℓ we have π α ( ax ) ∈ α L , so the third term is smaller than or equal to d . For the second term we have d ( π β ( bx ) , π β ( π α ( ax ))) ≤ d ( π β ( bx ) , π β ( bπ α ( x ))) + d ( π β ( bπ α ( x )) , π β ( aπ α ( x ))) ≤ d ( π β ( x ) , x ) + d ( x, π β ( π α ( x ))) + 2 d ≤ η + 4 d. where we used (20) to estimate the term d ( π β ( bπ α ( x )) , π β ( aπ α ( x ))) .In conclusion, d ( bx, π α ( ax )) ≤ η + 6 d .42 emma 5.14. If L ≥ ℓ then for any commutator g of { a ± , b ± } and any x ∈ α L/ we have d ( x, gx ) ≤ d .Proof. We give the proof for [ a, b ] = aba − b − , the other cases are similar.Assume that (22) holds. We have d ( b − x, a − x ) ≤ d by Lemma 5.13.Calling x ′ = a − b − x , we have d ([ a, b ] x, x )= d ( bx ′ , a − x ) ≤ d ( bx ′ , π α ( b − x ))+ d ( π α ( b − x ) , a − x ) . (24)By the second inequality in (22) we have d ( x ′ , α ) = d ( b − x, α ) ≤ d ; henceapplying the first one in (22) to x ′ yields d ( bx ′ , π α ( b − x )) ≤ d . The secondterm in (24) is less than or equal to d ( π α ( b − x ) , b − x ) + d ( b − x, a − x ) ≤ d ( b − x, a − x ) ≤ d. So d ([ a, b ] x, x ) ≤ d . The proof in case (23) holds is analogous. Proof of Proposition 5.11. If L ≥ ℓ , then by Lemma 5.14 there exists apoint x ∈ α L/ which is displaced by all the commutators of a ± , b ± by lessthan d < ε . In particular [ a ± , b ± ] and [ a ± , b ∓ ] belong to the sameelementary group. This in turns implies that h a, b i is elementary. Indeed, ifone commutator is the identity then h a, b i is abelian, hence elementary. Ifthe commutators are all different from the identity, then they do not havefinite order (since h a, b i is assumed to be torsion-free); so, as they belong tothe same elementary group, there exists a subset F ⊆ ∂X made of one or twopoints that is fixed by all the commutators. But since a − [ a, b ] a = [ b, a − ] and b − [ a, b ] b = [ b − , a ] we deduce that a − ( F ) = F and b − ( F ) = F .Therefore F is invariant for both a and b , hence h a, b i should be elementary.This contradiction concludes the proof.Let now x ∈ [ z − , z + ] be any point whose distance from β is at most ε .It exists, by assumption on the distance between the geodesics α and β .Now the distance function d β ( · ) = d ( · , β ) restricted to α is convex with d β ( x ) ≤ ε and d β ( z ± ) = d = ε . Furthermore by Proposition 3.6 the valueof d β at x + and at x − does not exceed M = max { δ, ε + 19 δ } . Thereforeby convexity we deduce d ( x, x + ) ≤ d β ( x + ) − d β ( x ) d β ( z + ) − d β ( x ) · d ( x, z + ) ≤ Mε · d ( x, z + ) and the same estimate holds for d ( x, x − ) . Thus: d ( x − , x + ) ≤
74 max (cid:26) δε , δε + 1 (cid:27) · L =: M As L ≤ ℓ , we conclude the proof in this case using Proposition 5.4.Recall that ℓ ( a ) = ℓ > ε , so it is enough to choose N = (cid:24) δε + 1 (cid:25) which is bounded above by a function depending only on P , r and δ .43 .2.3 Proof of Theorem 1.1(ii), when ε < d ( α, β ) ≤ δ. Recall that we are assuming a, b σ -isometries with ℓ ( a ) = ℓ ( b ) = ℓ > ε . We use the same notations as in 5.2.2: α (0) = x , β (0) = y and d ( α, β ) = d ( x , y ) . Finally the points x + , x − are respectively projections of β + , β − on α , and x + follows x − along α .The strategy here is to use the packing assumption to reduce the proof tothe previous subcase. For this, we set P := Pack (cid:18) δ, ε (cid:19) + 1 . By assumption d ( x , y ) = d ( x , β ) ≤ δ . We define [ z − , z + ] as the (non-empty) subsegment of α of points whose distance from β is at most δ .Moreover let w − and w + be some projections of z − and z + on β , respectively.There are two possibilities: d ( w − , w + ) ≥ P ℓ or the opposite.Assume that we are in the first case. Let w be the midpoint of thesegment [ w − , w + ] . For every i = 1 , . . . , P we consider the isometry b i . Then: d ( b i z − , β ) = d ( z − , β ) = 60 δ, d ( b i z + , β ) = d ( z + , β ) = 60 δ. Notice that the points w i − = b i w − and w i + = b i w + are respectively pro-jections of b i z − and b i z + on β . So d ( w − , w i − ) = i · ℓ ≤ P · ℓ ≤ d ( w − , w ) .In particular w i − belongs to the segment [ w − , w ] for all ≤ i ≤ P .Hence w belongs to the segment [ w i − , w i + ] . Since the distance from b i α of w i − and w i + is ≤ δ then, by convexity, we get d ( w, b i α ) ≤ δ . Hence thereexists a point z i ∈ b i α such that d ( z i , w ) ≤ δ . If the distance between anytwo of these points z i was greater than ε then the subset S = { z , · · · , z P } would be a ε -separated subset of B ( w, δ ) , but this is in contrast with thedefinition of P . So there must be two different indices ≤ i, j ≤ P such that d ( z i , z j ) ≤ ε . Therefore: d ( α, b j − i α ) = d ( b i α, b j α ) ≤ ε . Clearly b j − i α is a σ -axis of b j − i ab i − j and the group h a, b j − i ab i − j i is againdiscrete, non-elementary and torsion-free. From the proof of Theorem 1.1 inthe case where there are σ -axes at distance ≤ ε given in Subsection 5.2.2,we then deduce that there exists an integer N ( δ, P , r ) such that the group h a N , ( b j − i ab i − j ) N i is free.Remark that the length of ( b j − i ab i − j ) N , as a word in { a, b } , is at most P N ,and this number is bounded above in terms of δ, P and r .Assume now that we are in the case where d ( w − , w + ) < P ℓ . Then: d ( z − , z + ) ≤ P ℓ + 120 δ. Starting from this inequality we want to bound the distance between theprojections x − , x + of β − and β + on α , in order to apply Proposition 5.4.We look again at the distance d β from β : we know that d β ( x ) ≤ δ and d β ( z − ) = d β ( z + ) = 60 δ . Moreover d ( x , z + ) ≤ d ( z − , z + ) ≤ P ℓ + 120 δ .By Proposition 3.6.c, we know that d β ( x + ) ≤ max { δ, δ + d ( α, β ) } = 49 δ .44hen we can conclude that d ( x , x + ) ≤ d ( x , z + ) ≤ P ℓ + 120 δ , by convexityof the function d β restricted to α . The same estimate holds for x − , so d ( x − , x + ) ≤ P ℓ + 240 δ =: M . We can therefore conclude that group h a N , b N i is free by Proposition 5.4, forany N ≥ P + 317 δ/ℓ . Again we remark that N can be bounded from aboveby a function depending only on P , r and δ . d ( α, β ) > δ . Recall that we are assuming a, b σ -isometries with ℓ ( a ) = ℓ ( b ) = ℓ > ε . We use the same notations as in 5.2.2: α (0) = x , β (0) = y and d ( α, β ) = d ( x , y ) . Finally the points x + , x − are respectively projections of β + , β − on α , and x + follows x − along α . By Remark 3.4 the points x − , x + can bechosen in this way: for any time t ≥ we denote by x t a projection on α of β ( t ) . The limit point of a convergent subsequence of ( x t ) , for t → + ∞ ,defines a projection x + of β + on α . The point x − can be similarly chosenas the limit point of a convergent subsequence of ( x t ) for t → −∞ . ByProposition 3.6.b we have d ( x t , x t ′ ) ≤ δ for any t, t ′ ∈ R and this impliesthat d ( x − , x + ) ≤ δ . We can then apply again Proposition 5.4 to concludethat the group h a N , b N i is free for N ≥ δℓ , and the least N with thisproperty can be bounded as before by a function depending only on P , r and δ . In this last section we will always assume that ( X, σ ) is a δ -hyperbolic, GCB-space that is P -packed at scale r . We prove here the universal lower bounds for the entropy of X and for thealgebraic entropy of any finitely generated, non-elementary discrete group Γ of σ -isometries of X .Recall that we defined in Section 1.1 the nilpotence radius of Γ at x asnilrad (Γ , x ) = sup { r ≥ s.t. Γ r ( x ) is virtually nilpotent } and the nilradius of the action as nilrad (Γ , X ) = inf x ∈ X nilrad (Γ , x ) . Proof of Corollary 1.2.
We fix any symmetric, finite generating set S of Γ .Clearly there exist a, b ∈ S such that h a, b i is non-elementary, or Γ would beelementary. By Theorem 1.1(i), there exists a free semigroup h v, w i + where v, w ∈ S N , with N depending on δ, P , r . In particular card ( S kN ) ≥ k forall k , so Ent (Γ , S ) ≥ log 2 N = C . Since this holds for any S , this proves (i).The first inequality in (ii) is classic (see for instance [ ? ], Lemma 7.2).In order to show the second inequality notice that, if ν = nilrad (Γ , X ) ,there exist a point x ∈ X and g , g ∈ Γ generating a non-virtually nilpotent45ubgroup such that max { ℓ ( g ) , ℓ ( g ) } ≤ ν . Since X is packed, by Corollary3.13 the subgroup h g , g i is non-elementary, so we apply Theorem 1.1(i)and deduce the existence of a free semigroup h v, w i + where v, w ∈ S N ,for N = N ( P , r , δ ) . As card (cid:0) h g , g i x ∩ B ( x, kN ν ) (cid:1) ≥ k , we haveEnt (Γ , X ) ≥ log 2 Nν = C · ν − . Recall that we defined in Section 1.1 the minimal free displacement sys ⋄ (Γ , x ) at x as the infimum of d ( x, gx ) when g runs over the subset Γ \ Γ ⋄ of thetorsionless elements of Γ , and the free systole of the action assys ⋄ (Γ , X ) = inf x ∈ X sys ⋄ (Γ , x ) . Corollary 6.1.
Let ( X, σ ) be a δ -hyperbolic GCB -space, P -packed at scale r > . Then for any non-elementary discrete group of σ -isometries Γ of X it holds: sys ⋄ (Γ , x ) ≥ min (cid:26) ε , H e − H · nilrad (Γ ,x ) (cid:27) (25) where H = H ( P , r , δ ) is a constant depending only on P , r and δ . The proof is a modification of the proof of Theorem 6.20 of [BCGS17].We will also need the following:
Lemma 6.2 (see [Cav21b], Lemma 3.3 & Lemma 7.2) . Let ( X, σ ) be a δ -hyperbolic GCB -space, P -packed at scale r > , and let Γ be a discrete group of σ -isometries of X . Then Ent (Γ , X ) ≤ log(1 + P ) r =: E . Proof of Corollary 6.1.
Suppose that sys ⋄ (Γ , x ) < ε : then we can choose anon-elliptic element a ∈ Γ such that d ( x, ax ) = sys ⋄ (Γ , x ) < nilrad (Γ , x ) .Indeed the infimum in the definition of sys ⋄ (Γ , x ) is attained since the actionis discrete and nilrad (Γ , x ) ≥ ε by definition of ε . If nilrad (Γ , x ) = + ∞ there is nothing to prove. Otherwise we fix any arbitrary ε > and set R = (1 + ε ) · nilrad (Γ , x ) . By definition Γ R ( x ) is not virtually nilpotent andcontains a . This implies that there exists b ∈ Γ R ( x ) such that d ( x, bx ) ≤ R and h a, b i is not elementary. Indeed Γ R ( x ) is finitely generated by some b , . . . , b k with d ( x, b i x ) ≤ R for all i = 1 , . . . , k (since Γ is discrete); if h a, b i i was elementary for all i then each b i would belong to the maxi-mal, elementary subgroup containing g (Lemma 3.11), hence Γ R ( x ) wouldbe elementary and virtually nilpotent (by Corollary 3.13), a contradiction.We can therefore apply Theorem 1.1 and infer that the semigroup h a τN , w i + is free, where w is a word on a and b of length at most N = N ( P , r , δ ) ,and τ ∈ {± } . We now use Lemma 6.22.(ii) of [BCGS17] to deduce thatEnt (Γ , X ) · min (cid:8) d ( x, a τN x ) , d ( x, wx ) (cid:9) ≥ e − Ent (Γ ,X ) · max { d ( x,a τN x ) ,d ( x,wx ) } . d ( x, wx ) ≤ N R and d ( x, a ± N x ) ≤ N · sys ⋄ (Γ , x ) < N · nilrad (Γ , x ) , thisimplies, by Lemma 6.2 and by the arbitrariness of ε , that E N · sys ⋄ (Γ , x ) ≥ e − E N · nilrad (Γ ,x ) . The conclusion follows by setting H = E · N .We define the upper nilradius of Γ acting on X as the supremum ofnilrad (Γ , x ) over the ε -thin subsetnilrad + (Γ , X ) = sup x ∈ X ε nilrad (Γ , x ) where ε = ε ( P , r ) always denotes the generalized Margulis constant.By convention we set nilrad + (Γ , X ) = −∞ if X ε = ∅ . Then, by taking in(25) the supremum over all x ∈ X ε we deduce the formula:sys ⋄ (Γ , X ) ≥ min (cid:26) ε , H e − H · nilrad + (Γ ,X ) (cid:27) (26)which proves in particular Corollary 1.3 when Γ is torsionless.Notice that the upper nilradius can well be infinite, even when X ε = ∅ .For instance, for every elementary group Γ , we have nilrad + (Γ , X ) = + ∞ byCorollary 3.13, since ( X, σ ) is packed. Here are some non-trivial examples: Example 6.3.
Let ( X, σ ) be a δ -hyperbolic GCB-space that is P -packedat scale r > . Let Γ be a discrete group of σ -isometries of X containing aparabolic element: then sup x ∈ X r nilrad (Γ , x ) = + ∞ for all r > .Actually let g be a parabolic element of Γ , in particular ℓ ( g ) = 0 . We take asequence of points x n ∈ X such that d ( x n , gx n ) ≤ n . Thus sys ⋄ (Γ , x n ) ≤ n .So for any fixed r > we have points x n such that sys ⋄ (Γ , x n ) ≤ r and thenilpotence radius at x is larger and larger by the corollary.Clearly if Γ is a cocompact group one has nilrad + (Γ , X ) ≤ diam (Γ \ X ) .However there are many non-compact examples where the upper nilradius isfinite and the estimate (26) is non-trivial: Example 6.4 (Quasiconvex-cocompact groups) . A discrete group Γ of isometries of a δ -hyperbolic space X is called quasiconvex-cocompact if it acts cocompactly on the quasiconvex-hull of its limit set Λ(Γ) .The latter is defined as the union of all the geodesic lines joining two pointsof
Λ(Γ) and is clearly a Γ -invariant subset of X denoted by QC-Hull (Λ(Γ)) .We define the codiameter of Γ acting on X as the infimum D > such thatthe orbit Γ x is D -dense in QC-Hull (Λ(Γ)) for all x ∈ QC-Hull (Λ(Γ)) .Consider now a quasiconvex-cocompact group of σ -isometries Γ of a δ -hyper-bolic GCB-space ( X, σ ) that is P -packed at scale r . It is classical that,for all x ∈ QC-Hull (Λ(Γ)) , the subset Σ D ( x ) of elements g of Γ such that d ( x, gx ) ≤ D generates Γ .We affirm that there exists s = s ( P , r , δ ; D ) such that sys ⋄ (Γ , X ) ≥ s .47ndeed since the action is quasiconvex-cocompact then any element of Γ iseither elliptic or hyperbolic. Therefore the infimum defining the free sys-tole equals the infimum over points belonging to the axes of all hyperbolicelements of Γ . Any such axis is in QC-Hull (Λ(Γ)) by definition, sosys ⋄ (Γ , X ) = inf x ∈ QC-Hull (Λ(Γ)) sys ⋄ (Γ , x ) . By (25) we conclude that sys ⋄ (Γ , X ) ≥ min n ε , H e − H D o =: s . Now, for any point x ∈ X ε by definition there exists g ∈ Γ such that d ( x, gx ) ≤ ε and we denote one of its axis by Ax ( g ) . By applying Proposi-tion 4.6 to all ε ∈ [ s , ε ] we deduce d ( x, Ax ( g )) ≤ K ( P , r , δ, ℓ ( g ) , ε ) = K ( P , r , δ ; D ) =: K. Moreover the axis of g belongs to QC-Hull (Λ(Γ)) , therefore it is easy toconclude that nilrad (Γ , x ) ≤ K + 2 D . This shows that nilrad + (Γ , X ) isfinite, bounded by K + D ) . Example 6.5 (Abelian covers) . Let Γ be a torsionless group of σ -isometries of a δ -hyperbolic GCB-space ( X, σ ) that is P -packed at scale r > . Let Γ = [Γ , Γ] be the commutatorsubgroup. We affirm thatnilrad + (Γ , X ) ≤ · nilrad + (Γ , X ) (in particular nilrad + (Γ , X ) is finite for any quasiconvex-cocompact Γ ).Actually, assume that nilrad + (Γ , X ) < D and let a, b ∈ Γ elements whichgenerate a non-virtually nilpotent (hence non-elementary) subgroup and sat-isfy d ( x, ax ) < D , d ( x, bx ) < D . Then also the elements a ′ = a − [ a, b ] a and b ′ = b − [ a, b ] b generate a non-elementary (hence non-virtually nilpotent)subgroup of Γ , by the same argument used in the last lines of the proof ofProposition 5.11. However a ′ and b ′ belong to Γ and both move x less than D , which proves the claim.Notice that Γ \ X is not compact provided that the abelianization Γ / Γ of Γ is infinite.Corollary 6.1 says that given g ∈ Γ \ Γ ⋄ and x ∈ X an upper boundof the displacement of this point by any other element of Γ which does notgenerates with g an elementary subgroup yields a corresponding lower boundof the displacement d ( x, gx ) . Reversing the inequality (25) we obtain: Lemma 6.6.
Let ε be smaller than the generalized Margulis constant ε .For any given x ∈ X ⋄ ε the group Γ R ε ( x ) is elementary for R ε = 1 H · log (cid:18) ε · H (cid:19) where H = H ( P , r , δ ) is the constant given in Corollary 6.1. .3 Lower bound for the diastole The estimate of the diastole given in Corollary 1.4 stems from the applica-tion of the classical Tits Alternative combined with Breuillard-Green-Tao’sgeneralized Margulis Lemma. We state here the version allowing torsion,from which Corollary 1.4 easily follows.Recall that the free diastole of Γ acting on X is dias ⋄ (Γ , X ) = sup x ∈ X sys ⋄ (Γ , x ) ,and that the free r -thin subset of X is defined as X ⋄ r = { x ∈ X | ∃ g ∈ Γ \ Γ ⋄ s.t. d ( x, gx ) < r } . We then have:
Corollary 6.7.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r > . Then for any non-elementary, discrete group of σ -isometries Γ of X we have: dias ⋄ (Γ , X ) ≥ ε where ε = ε ( P , r ) always denotes the generalized Margulis constant.Proof of Corollary 6.7. Consider the free r -thin subset X ⋄ r . We first showthat if r ≤ ε ( P , r ) then X ⋄ r is not connected. By definition ∀ x ∈ X ⋄ r the group Γ r ( x ) is virtually nilpotent and contains a torsionless element a .For any such x we denote by N r ( x ) the unique maximal elementary subgroupof Γ containing Γ r ( x ) given by Lemma 3.11. Moreover, since there are onlyfinitely many g ∈ Γ such that d ( x, gx ) < r , there exists η > such thatfor all g ∈ Γ r ( x ) it holds d ( x, gx ) ≤ r − η . In particular if d ( x, x ′ ) < η then d ( x ′ , ax ′ ) < r , so x ′ ∈ X ⋄ r too, and Γ r ( x ) ⊂ Γ r ( x ′ ) . This implies thatthe map x N r ( x ) is locally constant. So if X ⋄ r was connected we wouldhave that N r ( x ) is a fixed elementary subgroup N r which does not dependon x ∈ X ⋄ r . We show now that N r is a normal subgroup of Γ : indeed ∀ g ∈ Γ and ∀ x ∈ X ⋄ r we have that gx ∈ X ⋄ r and Γ r ( gx ) = g Γ r ( x ) g − , therefore gN r g − = N r . As Γ is non-elementary there exists a non-elliptic isometry b ∈ Γ such that the group h N r , b i is non-elementary, by the maximality of N r .Now, the group N r is normal in Γ and amenable (since by Corollary 3.13it is virtually nilpotent), therefore its cyclic extension h N r , b i is amenable.However we know that any non-elementary group contains a free group (cp.[Gro87]) and this is impossible for an amenable group. This shows that X ⋄ r is not connected and in particular X ⋄ r = X . Therefore there exists a point x ∈ X such that d ( x, gx ) ≥ r for every g ∈ Γ ⋄ . In this last section we investigate the convergence under ultralimits of groupactions on δ -hyperbolic GCB-spaces that are P -packed at scale r .We denote by GCB ( P , r ) the class of pointed GCB-spaces ( X, x, σ ) thatare P -packed at scale r , and by GCB ( P , r , δ ) the subclass made of δ -hyperbolic spaces. We refer to [CS20] and [Cav21a] for details about ultra-limits and the relation with Gromov-Hausdorff convergence.49hese classes are closed under ultralimits and compact under pointed Gromov-Hausdorff convergence. For GCB ( P , r ) , this property follows exactly as inthe proof of Theorem 6.1 in [CS20]. The closure of GCB ( P , r , δ ) then fol-lows from the fact that the δ -hyperbolicity condition, as expressed in (3), isclearly stable under ultralimits.We start with a general construction. Let ( X n , x n ) be any sequence ofmetric spaces and suppose to have for every n a group of isometries Γ n of X n .We fix a non-principal ultrafilter ω . We define a limit group of isometries Γ ω of X ω . We say that a sequence of isometries ( g n ) , with each g n ∈ Γ n , is admissible if there exists M < + ∞ such that d ( x n , g n x n ) ≤ M for ω -a.e. ( n ) .Every admissible sequence ( g n ) defines a limit isometry g ω = ω - lim g n of X ω by the formula g ω ( y ω ) = ω - lim( g n ( y n )) where y ω = ω - lim y n is a generic point of X ω , see [CS20], Proposition A.5.We then define: Γ ω := { ω - lim g n | ( g n ) admissible sequence, g n ∈ Γ n ∀ n } . The following lemma is straightforward:
Lemma 6.8.
The composition of admissible sequences of isometries is anadmissible sequence of isometries and the limit of the composition is thecomposition of the limits. (Indeed, if g ω = ω - lim g n , h ω = ω - lim h n belong to Γ ω then their compositionbelong to Γ ω , as d ( g n h n · x n , x n ) ≤ d ( g n h n · x n , g n · x n )+ d ( g n · x n , x n ) < + ∞ ).Analogously one proves that ( id n ) belongs to Γ ω and defines the identitymap of X ω , and that if g ω = ω - lim g n belongs to Γ ω then also the sequence ( g − n ) defines an element of Γ ω , which is the inverse of g ω .So we have a well defined composition law on Γ ω , that is for g ω = ω - lim g n and h ω = ω - lim h n we set g ω ◦ h ω = ω - lim( g n ◦ h n ) With this operation Γ ω is a group of isometries of X ω and it is called theultralimit group of the sequence of groups Γ n . Remark that the limit groupdepends on the choice of base points x n in X n (in particular, sequences ofisometries ( g n ) whose displacement at x tend to infinity do not appear inthe limit group).Clearly if we have a sequence ( X n , x n , σ n ) , where σ n is a GCB on X n forevery n , and a sequence of groups of σ n -isometries Γ n of X n , then Γ ω is aset of σ ω -isometries of X ω , where σ ω is the GCB defined in Lemma 2.2.In the following proposition we describe the possible ultralimits of anadmissible sequence of isometries in our setting: Proposition 6.9.
Let ( X n , x n , σ n ) be a sequence of spaces in GCB ( P , r , δ ) .Let ω be a non-principal ultrafilter, and let ( g n ) be any admissible sequenceof σ n -isometries.(i) If g n is of hyperbolic type with σ n -axis γ n for ω -a.e. ( n ) then i.1) if ω - lim d ( x n , γ n ) < + ∞ then g ω is elliptic when ω - lim ℓ ( g n ) = 0 , andhyperbolic with σ ω -axis γ ω = ω - lim γ n and ℓ ( g ω ) = ω - lim ℓ ( g n ) otherwise;(i.2) if ω - lim d ( x n , γ n ) = + ∞ then g ω is either elliptic or parabolic.(ii) If g n is parabolic for ω -a.e. ( n ) then g ω is either elliptic or parabolic. Notice that any two σ n -axes of a hyperbolic isometry are at uniformlybounded distance from each other, by the δ -hyperbolicity assumption, cp.Lemma 3.5, so (i) does not depend on the particular choice of γ n . Moreoverthe ultralimit of a sequence of σ n -geodesics γ n at uniformily bounded dis-tance from the base points x n is again a σ ω -geodesic of X ω (cp. PropositionA.5 of [CS20] and the definition of σ ω in Lemma 2.2). Example 6.10.
Case (i.2) can actually occur for the limit g ω of a sequence ofhyperbolic isometries g n . Let for instance Γ n be a Schottky group of X = H generated by two hyperbolic isometries a n , b n with non-intersecting axes.The convex core of the quotient space ¯ X n = G n \ H is a hyperbolic pairof pants, with boundary given by three periodic geodesics α n , β n and γ n .These geodesics correspond, respectively, to the projections of the axes of theelements a n , b n and c n = a n · b n (up to replacing b with its inverse). Lettingthe length ℓ ( γ n ) tend to zero (which means pulling two of the isometry circlesof a and b closer and closer), the sequence of hyperbolic isometries c n tendsto a parabolic isometry, and ¯ X n tends to a surface with one cusp. Proof of Proposition 6.9.
We start from g n of hyperbolic type with axis γ n for ω -a.e. ( n ) .Assume first that ω - lim d ( x n , γ n ) = C < + ∞ . If ω - lim ℓ ( g n ) = 0 thenany point of the limit σ ω -geodesic γ ω is a fixed point of g ω , so g ω is elliptic.Otherwise ω - lim ℓ ( g n ) = ℓ > and it is immediate that g ω translates γ ω by ℓ ,hence it is of hyperbolic type with σ ω -axis γ ω .Suppose now that g n is of hyperbolic type with σ n -axis γ n for ω -a.e. ( n ) andthat ω - lim d ( x n , γ n ) = + ∞ . Let M ≥ be an upper bound for d ( x n , g n x n ) for every n . A direct application of Proposition 4.7 gives ω - lim ℓ ( g n ) = 0 ,otherwise the distance between x n and the axis γ n of g n would be uniformlybounded. Suppose that g ω is hyperbolic: in this case ℓ ( g ω ) = ℓ would bestrictly positive. Applying again Proposition 4.7 we would find, for ω -a.e. ( n ) ,a point p n ∈ X n satisfying d ( p n , x n ) ≤ K ( ℓ , M , δ ) , d ( p n , g n p n ) ≤ ℓ . The first condition implies that the sequence ( p n ) defines a point p ω of X ω ,while the second condition implies that d ( p ω , g ω p ω ) ≤ ℓ which is impossible,so g ω is no of hyperbolic type.Finally, suppose that g n is of parabolic type for ω -a.e. ( n ) . If g ω was hyper-bolic of translation length ℓ > then arguing as before there would exist apoint p n ∈ X n satisfying d ( p n , x n ) ≤ K ( ℓ , M , δ ) , d ( p n , g n p n ) ≤ ℓ . Theorem 6.11.
Let ( X n , x n , σ n ) be a sequence of spaces in GCB ( P , r , δ ) ,and Γ n a sequence of torsion-free, discrete groups of σ n -isometries of X n .Let ω be a non-principal ultrafilter and Γ ω the limit group of σ ω -isometriesof X ω . Then one of the following mutually esclusive possibilities holds:(a) ∀ L ≥ ∃ r > such that ω - lim d ( x n , ( X n ) r ) > L . In this case thegroup Γ ω acts discretely on X ω and it has no torsion;(b) ∃ L ≥ such that ∀ r > it holds ω - lim d ( x n , ( X n ) r ) ≤ L . In this casethe group Γ ω is elementary (possibly non-discrete).Proof. We start from case (a). By Proposition 2.5 we know that X ω is proper.Let g ω = ω - lim g n be an element of Γ ω and y ω = ω - lim y n be a point of X ω .By definition of y ω there exists L ≥ such that d ( x n , y n ) ≤ L for all n .By assumption, there exists r such that d ( y n , g n y n ) ≥ r for ω -a.e. ( n ) , if g n = id for ω -a.e. ( n ) . This implies d ( y ω , g ω y ω ) ≥ r , so sys (Γ ω , y ω ) ≥ r for all y ω ∈ B ( x ω , L ) . Since X ω is proper we conclude that Γ ω is discrete.Moreover it is torsion-free: indeed any elliptic element g ω = ( g n ) of Γ ω musthave a fixed point y ω , hence, as just proved, g n is the identity for ω -a.e. ( n ) ;so g ω = id necessarily, since sys (Γ ω , y ω ) is strictly positive.We now study case (b). In this case for all r > there exists a point y n ∈ X n with d ( x n , y n ) ≤ L and sys (Γ n , y n ) ≤ r for all ω -a.e. ( n ) . Observe that forall r ≤ ε the group Γ R r ( y n ) is elementary, with R r → + ∞ when r → byLemma 6.6. Now, by definition, for every g ω = ω - lim g n of Γ ω there exists M such that d ( x n , g n x n ) ≤ M for ω -a.e. ( n ) , so d ( y n , g n y n ) ≤ L + M ≤ R r provided that r is small enough. This implies that g n belongs, for ω -a.e. ( n ) ,to a fixed elementary subgroup Γ ′ n < Γ n that does not depend on the ele-ment g ω under consideration. Then, there are two possibilities: for ω -a.e. ( n ) either Γ ′ n is of hyperbolic type or it is of parabolic type.Assume that the isometries in Γ ′ n are all hyperbolic for ω -a.e. ( n ) , so thereexists a common σ n -axis γ n for all of them. If ω - lim d ( x n , γ n ) < + ∞ and ω - lim ℓ ( g n ) > , then we are in case (i.1) of Proposition 6.9: the limit g ω is hyperbolic with σ ω -axis γ ω = ω - lim γ n , for all g ω ∈ Γ ω , hence this groupis elementary (and discrete). In all the other cases, Proposition 6.9 impliesthat Γ ω does not contain any hyperbolic isometry, so the group is elemen-tary by Gromov’ classification of groups acting on hyperbolic spaces (cp.[Gro87],[DSU17]).We examine now the case when the limit group is discrete: Corollary 6.12.
Same assumptions as in Theorem 6.11.Let ¯ X n = Γ n \ X n be the quotient metric spaces, let p n : X n → ¯ X n theprojection maps and let ¯ x n = p n ( x n ) . Then: i) if the groups Γ n are non-elementary then one can always suitably choosethe base points x n ∈ X n so that case (a) of Theorem 6.11 occurs, hencethe ultralimit group Γ ω is discrete and torsion-free;(ii) if case (a) of Theorem 6.11 occurs then the ultralimit space ¯ X ω of thesequence ( ¯ X n , ¯ x n ) is isometric to Γ ω \ X ω .Proof. By Corollary 6.7, if the groups Γ n are non-elementary it is alwayspossible to choose x n ∈ X n in order that the pointwise systole of the Γ n at x n are uniformly bounded away from zero, i.e. sys (Γ n , x n ) ≥ ε > for every n . The fact that case (a) occurs for this choice of the base points is then adirect consequence of Proposition 4.5.(ii).To show (ii), notice that the projections p n : X n → ¯ X n form an admissiblesequence of -Lipschitz maps and then, by Proposition A.5 of [CS20], theyyield a limit map p ω : X ω → ¯ X ω defined as p ω ( y ω ) = ω - lim p n ( y n ) , for ω - lim y n = y ω . The map p ω is clearly surjective. We want to show that it is Γ ω -equivariant. We fix g ω = ω - lim g n ∈ Γ ω and y ω = ω - lim y n ∈ X ω . Then: p ω ( γ ω y ω ) = ω - lim p n ( γ n y n ) = ω - lim p n ( y n ) = p ω ( y ω ) . Therefore we have a well defined, surjective quotient map ¯ p ω : Γ ω \ X ω → ¯ X ω .We will now show that it is a local isometry. We fix an arbitrary point y ω = ω - lim y n ∈ X ω , consider its class [ y ω ] ∈ Γ ω \ X ω and set L = d ( x ω , y ω ) .By assumption there exists r , depending only on L , such that sys (Γ n , y n ) ≥ r for ω -a.e. ( n ) . In particular the systole of Γ ω at y ω is at least r , so the quo-tient map X ω → Γ ω \ X ω is an isometry between B ( y ω , r ) and B ([ y ω ] , r ) .Moreover for ω -a.e. ( n ) we have that B ( p n ( y n ) , r ) is isometric to B ( y n , r ) .By Lemma A.8 of [CS20] we know that ω - lim B ( p n ( y n ) , r ) is isometric to B ( p ω ( y ω ) , r ) = B (¯ p ω ( y ω ) , r ) and that ω - lim B ( y n , r ) is isometric to B ( y ω , r ) .Therefore B (¯ p ω ( y ω ) , r ) is isometric to B ([ y ω ] , r ) . By Proposition 3.28, Sec.I,of [BH13] we conclude that the map ¯ p ω is a locally isometric covering map.To conclude, it is enough to show that ¯ p ω is injective. Let [ z ω ] , [ y ω ] ∈ Γ ω \ X ω .Then we have ¯ p ω ([ z ω ]) = ¯ p ω ([ y ω ]) if and only if p ω ( z ω ) = p ω ( y ω ) . This isequivalent to ω - lim d ( p n ( z n ) , p n ( y n )) = 0 and, as the systole of y n is uni-formly bounded away from zero, this means ω - lim d ( z n , g n y n ) = 0 for some g n ∈ Γ n and for ω -a.e. ( n ) . We observe that the sequence ( g n ) is admissible,therefore it defines an element g ω = ω - lim g n ∈ Γ ω satisfying d ( z ω , g ω y ω ) = 0 .This implies that [ z ω ] = [ y ω ] and therefore ¯ p ω is an isometry.Non-elementary ultralimit groups are characterized in the next result: Theorem 6.13.
Same assumptions as in Theorem 6.11.(i) If there exist two sequences of admissible isometries ( g n ) , ( h n ) in Γ n of the same type such that h g n , h n i is non-elementary for ω -a.e. ( n ) then the group h g ω , h ω i is non-elementary;(ii) the group Γ ω is non-elementary if and only if there exist two sequencesof admissible isometries ( g n ) , ( h n ) such that h g n , h n i is non-elementaryfor ω -a.e. ( n ) . Γ ω non-elementary implies that it is also discrete and torsion-free, by Theorem 6.11. Proof.
Let ( g n ) , ( h n ) be as in (i) and let M ≥ such that for all n it holds d ( x n , g n x n ) , d ( x n , h n x n ) ≤ M . We first show that there are no elliptics in h g ω , h ω i . Actually, assume that f ω ∈ h g ω , h ω i is elliptic, with f ω = ω - lim f n ,for some admissible sequence of isometries f n ∈ h g ω , h ω i . Then, there wouldexist a point y ω = ω - lim y n with f ω y ω = y ω . So, for all r > and for ω -a.e. ( n ) the following conditions would hold, for some L ≥ : d ( x n , y n ) ≤ L, d ( y n , f n y n ) ≤ r. The first condition implies that d ( y n , g n y n ) ≤ d ( y n , x n ) + d ( x n , g n x n ) + d ( g n x n , g n y n ) ≤ L + M and similarly for h n . If r is small enough we then deduce that h g n , h n i iselementary by Lemma 6.6, a contradiction.Assume now that the elements g ω , h ω are both parabolic. If h g ω , h ω i waselementary then they would have the same fixed point at infinity z . We thenchoose ε > small enough so that R ε ≥ δ + ε , where R ε is the quantitydefined in Lemma 6.6. As ℓ ( g ω ) = ℓ ( h ω ) = 0 there exist points y ω = ω - lim y n , w ω = ω - lim w n of X ω such that d ( y ω , g ω y ω ) < ε and d ( w ω , h ω w ω ) < ε . ByLemma 3.1 we can also find points y ′ ω ∈ [ y ω , z ] and w ′ ω ∈ [ w ω , z ] such that d ( y ω , w ω ) ≤ δ . By convexity and by the triangular inequality we deduce: d ( y ′ ω , g ω y ′ ω ) < ε, d ( y ′ ω , h ω y ′ ω ) < δ + ε ≤ R ε . Similar estimates hold for g n and h n for ω -a.e. ( n ) , implying that h g n , h n i iselementary for ω -a.e. ( n ) , a contradiction.A similar argument works when one element is parabolic, say g ω , and theother, h ω , is hyperbolic. In this case, if h g ω , h ω i was elementary, the fixedpoint of g ω would coincide with one point at infinity z of an axis η of h ω . We choose ε > so that R ε > ℓ + 16 δ , where R ε is again thenumber given by Lemma 6.6 and ℓ is the minimal displacement of h ω .We then take a point z ω = ω - lim z n of X ω such that d ( z ω , g ω z ω ) < ε , apoint y ω = ω - lim y n on η , and a point z ′ ω = ω - lim z ′ n ∈ [ z ω , z ] such that d ( y ω , z ′ ω ) ≤ δ . By convexity of σ ω (since g ω is a σ ω -isometry) and thetriangular inequality we get d ( z ′ ω , g ω z ′ ω ) < ε, d ( z ′ ω , h ω z ′ ω ) ≤ δ + ℓ < R ε . and again similar estimates hold for g n , h n for ω -a.e. ( n ) , showing that h g n , h n i is elementary for ω -a.e. ( n ) , a contradiction.It remains to consider the case where both g ω and h ω are of hyperbolic type.Suppose they have the same σ ω -axis. By Lemma 6.9 we know that this σ ω -axis is the ultralimit of some σ n -axis γ n of g n , and of some σ n -axis η n of h n as54ell; therefore ω - lim γ n = ω - lim η n . This means that for all C > and for ω -a.e. ( n ) the set of points of γ n that are at distance at most ε from η n is a sub-segment of length at least C , where ε is the generalized Margulis constant.By Proposition 5.11 we conclude that ℓ ( g n ) ≥ C . Therefore the sequence ( g n ) is not admissible, a contradiction. This implies that g ω and h ω do nothave the same axis, therefore h g ω , h ω i is not elementary. This proves (i).In order to prove (ii), assume first that ( g n ) , ( h n ) are two admissible se-quences such that h g n , h n i is not elementary for ω -a.e. ( n ) . Up to replacing h n with h n g n h − n we may suppose that g n , h n are of the same type, andstill admissible. So Γ ω is not elementary by (i). Conversely, assume that Γ ω is not elementary. Then, it contains at least a hyperbolic element g ω and, by Theorem 6.11 we know it is discrete. Therefore, by discreteness andnon-elementarity, there exists another element h ω ∈ Γ ω such that h g ω , h ω i is not elementary. Up to replacing h ω with h ω g ω h − ω we may again supposethat h ω is of hyperbolic type (and the group genarated by g ω and this el-ement remains non-elementary). Hence the σ ω -axis γ ω , η ω respectively of g ω = ω - lim g n and h ω = ω - lim h n are not the same. By Lemma 6.9 theelements g n , h n are hyperbolic for ω -a.e. ( n ) , and have σ n -axes γ n , η n suchthat γ ω = ω - lim γ n , and η ω = ω - lim η n . Now, if h g n , h n i was elementary for ω -a.e. ( n ) then we could choose γ n = η n for ω -a.e. ( n ) and therefore γ ω = η ω ,a contradiction. This shows that h g n , h n i is not elementary for ω -a.e. ( n ) .Recall from the introduction that, for any fixed choice of parameters P , r , δ, ∆ > , we called GCB ( P , r , δ ; ∆) the class of spaces that are quotients of a space ( X, x, σ ) ∈ GCB ( P , r , δ ) bya discrete, torsion-free group of σ -isometries Γ of X with nilrad + (Γ , X ) ≤ ∆ (notice, then, that the group Γ is non-elementary by assumption).We will now prove that this class is closed under ultralimits, hence compactunder pointed, equivariant Gromov-Hausdorff convergence. Proof of Theorem 1.7.
Let ( ¯ X n , ¯ x n , ¯ σ n ) = Γ n \ ( X n , x n , σ n ) be any sequencein this class, with ( X n , x n , σ n ) ∈ GCB ( P , r , δ ) . As recalled at the begin-ning of Section 6.4 the ultralimit ( X ω , x ω , σ ω ) of the sequence ( X n , x n , σ n ) is again an element of GCB ( P , r , δ ) . Then, to prove that our class isclosed under ultralimits, we need only to show that the bound of the uppernilradius is satisfied also by the limit group Γ ω acting on X ω . Indeed itwill be enough to apply Corollary 6.12 to ensure that the quotient spaces ( ¯ X n , ¯ x n , ¯ σ n ) converge to Γ ω \ ( X ω , x ω , σ ω ) , that belongs to GCB ( P , r , δ ; ∆) .By the estimate (26) we know that sys ( X n , Γ n ) ≥ s ( P , r , δ, ∆) for all n .Therefore we are in case (a) of Theorem 6.11 and Γ ω is a discrete and torsion-free group of isometries of X ω .Now, assume first that sys (Γ n , X n ) is greater than or equal to the the gen-eralized Margulis constant ε for ω -a.e. ( n ) . Then sys (Γ ω , X ω ) ≥ ε , sonilrad + (Γ ω , X ω ) = −∞ ≤ ∆ , and the conclusion holds.Otherwise, sys (Γ n , X n ) < ε for ω -a.e. ( n ) . In this case we take any55 ω = ω - lim y n such that s = sys (Γ ω , y ω ) < ε . By the discreteness of Γ ω thereexists g ω = ω - lim g n ∈ Γ ω such that d ( y ω , g ω y ω ) = s . We fix ε < ε − s and wededuce that d ( y n , g n y n ) < s + ε < ε for ω -a.e. ( n ) , so nilrad + (Γ n , y n ) ≤ ∆ .This means that for all ε > there is h n ∈ Γ n such that d ( y n , h n y n ) ≤ ∆ + ε and h h n , g n i is not elementary. To conclude, we need to show that h g ω , h ω i is not elementary. Assume the contrary: then, h ω has the same type and thesame fixed points at infinity as g ω . If they were hyperbolic, then by Lemma6.9 also g n , h n would be hyperbolic for ω -a.e. ( n ) , and by Theorem 6.13we would obtain that h g n , h n i is elementary for ω -a.e. ( n ) , a contradiction.On the other hand, if both are parabolic then we have two possibilities: either g n , h n are of the same type for ω -a.e. ( n ) , and arguing as before would giveagain a contradiction; or g n is hyperbolic and h n is parabolic for ω -a.e. ( n ) .In this last case we consider the elementary group h g ω , h ω g ω h − ω i and applyTheorem 6.13 as before to deduce that the group h g n , h n g n h − n i is elemen-tary for ω -a.e. ( n ) . Therefore h n Fix ∂ ( g n ) = Fix ∂ ( h n gh − n ) = Fix ∂ ( g n ) , so thefixed point of h n coincides with one of the fixed points of g n , which contra-dicts the fact that Γ n is discrete. This shows that h g ω , h ω i is not elementary.By the arbitrariness of ε we then obtain nilrad + (Γ ω , y ω ) ≤ ∆ .We will finally prove Theorem 1.6. For this, we need to explain howcommutators, normal subgroups and quotients behave under ultralimits.In the next lemmas, we assume that a non-principal ultrafilter ω is given: Lemma 6.14.
Let Γ ′ n , Γ n be isometry groups of pointed metric spaces ( X n , x n ) ,with Γ ′ n < Γ n , and let Γ ′ ω , Γ ω be the limit groups of isometries of X ω . Then:(i) [Γ ω , Γ ω ] ⊆ [Γ n , Γ n ] ω ;(ii) if Γ n is abelian (resp. m -step nilpotent, m -step solvable) for ω -a.e. ( n ) ,then Γ ω is abelian (resp. m -step nilpotent, m -step solvable);(iii) if Γ ′ n ⊳ Γ n for ω -a.e. ( n ) , then Γ ′ ω is a normal subgroup of Γ ω .Proof. Let g ω = [ a ω, , b ω, ] · · · [ a ω,l , b ω,l ] ∈ [Γ ω , Γ ω ] . We can write any a ω,i as ω - lim a n,i and any b ω,i as ω - lim b n,i , for admissible sequences a n,i , b n,i ∈ Γ n .Moreover by Lemma 6.8, since l is finite, we get [ a ω, , b ω, ] · · · [ a ω,l , b ω,l ] = ω - lim([ a n, , b n, ] · · · [ a n,l , b n,l ]) , so g ω ∈ [Γ n , Γ n ] ω , showing (i).Let us prove (ii): if Γ n is abelian for ω -a.e. ( n ) , then by Lemma 6.8 we getdirectly that also Γ ω is abelian. The nilpotent and the solvable case areproved by induction on m . If m = 0 , we are in the abelian case. Assumenow that the claim holds for ( m − -step nilpotent (resp. solvable) groups,and let Γ n be m -step nilpotent (resp. solvable) for ω -a.e. ( n ) . The groups [Γ n , Γ n ] are ( m − -step nilpotent (resp. solvable) for ω -a.e. ( n ) ; so, by theinduction hypothesis, [Γ n , Γ n ] ω is ( m − -step nilpotent (resp. solvable).But then, by (i), also [Γ ω , Γ ω ] is ( m − -step nilpotent (resp. solvable) andtherefore Γ ω is m -step nilpotent (resp. solvable).56inally, let us show (iii). The limit group Γ ′ ω is contained in Γ ω , by definition.We take g ω = ω - lim g n ∈ Γ ω and h ω = ω - lim h n ∈ Γ ′ ω . We know that for ω -a.e. ( n ) there exists h ′ n ∈ Γ ′ n such that g n h n = h ′ n g n , as Γ ′ n is normal in Γ n .Since ( h ′ n ) = ( g n h n g − n ) is an admissible sequence, it defines a limit isometry h ′ ω ∈ Γ ′ ω . Moreover by Lemma 6.8 we get h ′ ω = g ω h ω g − ω , so Γ ′ ω is normal. Lemma 6.15.
Let ( X n , x n , σ n ) be pointed spaces belonging to GCB ( P , r , δ ) ,and let Γ ′ n , Γ n be groups of σ n -isometries of X n with Γ ′ n ⊳ Γ n for ω -a.e. ( n ) .Suppose that the groups Γ n and X n satisfy condition (a) of Theorem 6.11.Then there is a natural isomorphism between the ultralimit Q ω of the groups Q n = Γ n / Γ ′ n acting on the quotient, pointed spaces ( X ′ n , x ′ n ) = Γ ′ n \ ( X n , x n ) and the group Γ ω / Γ ′ ω (observe that these are isometry groups of the spaces X ′ ω and Γ ′ ω \ X ω respectively, which are isometric by Corollary 6.12).Proof. The groups Q n = Γ n / Γ ′ n act naturally on the pointed, quotient spaces ( X ′ n , x ′ n ) , where x ′ n is the image of x n under the projection map X n → X ′ n ,and their ultralimit Q ω acts on the ultralimit ( X ′ ω , x ′ ω ) of the ( X ′ n , x ′ n ) .We define Φ : Q ω → Γ ω / Γ ′ ω as follows. An element q ω ∈ Q ω is a sequenceof admissible isometries ( q n = g n Γ ′ n ) of Q n acting on X ′ n . As Γ n and X n satisfy condition (a) of Theorem 6.11, we have d ( q n x ′ n , x ′ n ) ≤ L for ω -a.e. ( n ) .We then choose an admissible sequence ( g n ) of isometries of X n belong-ing to the class q n , and define Φ( q ω ) as the class of ω - lim g n modulo Γ ′ ω .It clearly does not depend on the choice of the admissible representative ( g n ) , and yields a surjective homomorphism. The injectivity of Φ followsdirectly from the assumptions on Γ ′ n . Indeed, if q ω = ω - lim q n and ( g n ) isan admissible sequence of isometries of X n belonging to the classes ( q n ) ,then Φ( q ω ) ∈ Γ ′ ω implies that there exists an admissible sequence ( g ′ n ) ofisometries in Γ ′ n such that ω - lim g ′ n = ω - lim g n . In particular ∀ ε > we have d ( g n x n , g ′ n x n ) ≤ ε for ω -a.e. ( n ) . By the assumption (a) of 6.11 we deducethat g n = g ′ n , hence g n ∈ Γ ′ n , for ω -a.e. ( n ) . Thus ω - lim q n is the identity. Proof of Theorem 1.6.
We have a sequence of pointed spaces ( ¯ X n , ¯ x n ) withpointed, universal coverings ( X n , x n , σ n ) belonging to GCB ( P , r , δ ) , and asequence of normal coverings π n : ˆ X n → ¯ X n . This means that ¯ X n = Γ n \ X n and ˆ X n = Γ ′ n \ X n , where Γ n , Γ ′ n are discrete, torsionless groups of isometriesof X n , and Γ ′ n are normal subgroups of Γ n . We denote by ¯ p n : X n → ¯ X n and ˆ p n : X n → ˆ X n the natural universal covering maps, so π n ◦ ˆ p n = ¯ p n for all n .We take a non-principal ultrafilter ω , fix points x n , ˆ x n , ¯ x n of X n , ˆ X n , ¯ X n re-spectively, with ˆ p n ( x n ) = ˆ x n and ¯ p n ( x n ) = ¯ x n , consider the pointed spaces ( X n , x n ) , ( ˆ X n , ˆ x n ) , ( ¯ X n , ¯ X n ) and the corresponding ultralimits X ω , ˆ X ω , ¯ X ω .We consider the limit groups Γ ω , Γ ′ ω of σ ω -isometries of X ω . Since thegroups Γ n are cocompact with codiameter ≤ D we know from Example 6.4that sys (Γ n , X n ) ≥ s ( P , r , δ, D ) > . Moreover, since Γ ′ n < Γ n , the sameestimate holds for Γ ′ n . Therefore we are in case (a) of Theorem 6.11, bothfor the sequence (Γ n ) and for (Γ ′ n ) . We conclude that Γ ω and Γ ′ ω are discreteand torsion-free, and moreover Γ ′ ω ⊳ Γ ω by Lemma 6.14.(iii).Now, calling π ω and ˆ p ω , ¯ p ω the ultralimit maps of the sequences ( π n ) , (ˆ p n ) , (¯ p n )
57e have by definition π ω ◦ ˆ p ω = ¯ p ω . Moreover by Corollary 6.12.(ii) we knowthat ˆ p ω and ¯ p ω are covering maps with automorphism groups Γ ′ ω and Γ ω respectively. It is then straightforward to conclude that π ω is a covering.Since Γ ′ ω is normal in Γ ω , by the classical covering theory we know thatthe automorphism group of π ω coincides with the group Γ ω / Γ ′ ω , which isisomorphic to the ultralimit group Q ω of the Q n = Γ n / Γ ′ n , by Lemma 6.15.Furthermore, as every Q n is abelian (resp. m -step nilpoten or solvable), thenthe same holds for the ultralimit Q ω and for Γ ω / Γ ′ ω , by Lemma 6.14.(ii). Fi-nally, we observe that the group Γ ′ ω is not elementary. Indeed, if Γ ′ ω waselementary then it would be cyclic (being a torsionless subgroup of Γ ω with-out parabolics), and by the exact sequence → Γ ′ ω → Γ ω → Γ ω / Γ ′ ω → , with both Γ ′ ω and Γ ω / Γ ′ ω solvable, we would deduce that Γ ω is solvable. Thiscontradicts the fact that Γ ω contains a free group. The closure then followsonce again by Corollary 6.12. References [AL06] G.N. Arzhantseva and I.G. Lysenok. A lower bound on the growthof word hyperbolic groups.
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