Packing and doubling in metric spaces with curvature bounded above
aa r X i v : . [ m a t h . M G ] S e p Packing and doubling in metric spaceswith curvature bounded above
Nicola Cavallucci, Andrea Sambusetti
Abstract.
We study locally compact, locally geodesically complete, locally CAT ( κ ) spaces(GCBA κ -spaces). We prove a Croke-type local volume estimate only depending on thedimension of these spaces. We show that a local doubling condition, with respect tothe natural measure, implies pure-dimensionality. Then, we consider GCBA κ -spaces sat-isfying a uniform packing condition at some fixed scale r or a doubling condition atarbitrarily small scale, and prove several compactness results with respect to pointedGromov-Hausdorff convergence. Finally, as a particular case, we study convergence andstability of M κ -complexes with bounded geometry. Contents ( κ ) and GCBA-spaces . . . . . . . . . . . . . . . . . . . . 92.2 Contraction maps and almost-convexity radius . . . . . . . . 102.3 Tangent cone and the logarithmic map . . . . . . . . . . . . . 112.4 Dimension and natural measure . . . . . . . . . . . . . . . . . 142.5 Gromov-Hausdorff convergence . . . . . . . . . . . . . . . . . 15 M κ -complexes 37 M κ -complexes . . . . . . . . . . . . . . . . . . . 377.2 Compactness of M κ -complexes . . . . . . . . . . . . . . . . . 45 A Ultralimits 50 Introduction
Metric spaces with curvature bounded from above are currently one of themain topics in metric geometry. They have been studied from various pointsof view during the last decades. In general, these metric spaces can be verywild and the local geometry can be difficult to understand. Under basicadditional assumptions (local compactness and local geodesic completeness)it is possible to control much better the local and asymptotic properties ofthese spaces, as proved by Kleiner and Lytchak-Nagano. In particular, underthese assumptions, the topological dimension coincides with the Hausdorffdimension, the local dimension can be detected from the tangent cones, andthere exists a decomposition of X in k-dimensional subspaces X k (contain-ing dense open subsets locally bilipschitz equivalent to R k and admitting aregular Riemannian metric), and a canonical measure µ X , coinciding withthe restriction of the k-dimensional Hausdorff measure on each X k , whichis positive and finite on any open relatively compact subset (cp. the foun-dational works [Kle99] , [LN19], [LN18]). Following [LN19], we will call forshort GCBA-spaces the locally geodesically complete, locally compact andseparable metric spaces satisfying some curvature upper bound, i.e. whichare locally CAT ( κ ) for some κ . When we want to emphasize in a statementthe role of κ , we will write GCBA κ .GCBA-spaces arise in a natural way as generalizations and limits of Rie-mannian manifolds with sectional curvature bounded from above. However,many geometric results in the Riemannian setting, such as convergence andfiniteness theorems, Margulis’ lemma etc. also require lower bounds on thecurvature. For instance, by Bishop-Gromov’s Theorem, a lower bound onthe Ricci curvature implies a bound of the complexity of the manifold as ametric space: namely, a lower bound Ric X ≥ − a implies a uniform estimateof the packing function at any fixed scale r .We recall that a metric space ( X, d ) satisfies the P -packing conditionat scale r > if all balls of radius r contain at most P points that are r -separated from each other (this can be equivalently expressed in termsof coverings with balls, see Sec.4). Also, we will say that a metric-measuredspace ( X, d, µ ) satisfies a D -doubling condition up to scale r > if for any < r ≤ r and for any x ∈ X it holds µ ( B ( x, r )) µ ( B ( x, r )) ≤ D . From a metric-geometry perspective, the original interest in studying metricspaces satisfying a packing condition at arbitrarily small scales is Gromov’sfamous Precompactness Theorem [Gro81]. Another major outcome involvingpacking is Gromov’s celebrated result on groups with polynomial growth, asextended by Breuillard-Green-Tao [BGT11] (cp. also the previous results[Kle10] and [ST10]), which shows that a uniform bound of the packing or2oubling constant for X at arbitrarily large scale (or even at fixed, sufficientlylarge scale with respect to the diameter) yields an even stronger limitationon the complexity of the fundamental group of X , that is almost-nilpotency.We will see soon (cp. Theorem C below) that, for GCBA-spaces, a doublingcondition for the canonical measure µ X at arbitrarily small scales also hasinteresting consequences on the local structure of X .The first key-result of the paper is a Croke-type local volume estimatefor GCBA-spaces of dimension bounded above, for balls of radius smallerthan the almost-convexity radius : Theorem A (Theorem 3.1) . For any complete
GCBA space X of dimension ≤ n and any ball of radius r < min { ρ ac ( X ) , } it holds : µ X ( B ( x, r )) ≥ c n · r n (1) where c n is a constant depending only on the dimension n . The almost-convexity radius ρ ac ( x ) of a geodesic space X at a point x isdefined as the supremum of the radii r such that for any y, z ∈ B ( x, r ) andany t ∈ [0 , it holds: d ( y t , z t ) ≤ t · d ( y, z ) where y t , z t denote points along geodesics [ x, y ] and [ x, z ] at distance td ( x, y ) and td ( x, z ) respectively from x . The almost-convexity radius of X is cor-respondingly defined as ρ ac ( X ) = inf x ∈ X ρ ac ( x ) . It is not difficult to showthat every GCBA-space X always has positive almost-convexity radius atevery point: namely, if X is locally CAT ( κ ) and x ∈ X , then ρ ac ( x ) is al-ways greater than or equal to the CAT ( κ ) - radius ρ cat ( x ) (see Section 2 forall details and the relation with the contraction and the logarithmic maps).However, the almost-convexity radius is a more flexible geometric invariantthan the CAT ( κ ) -radius, much alike the injectivity radius for Riemannianmaniolds, since a space X might have a large curvature κ concentrated in avery small region around x , so that it may happen that ρ ac ( x ) is much largerthan the CAT ( κ ) -radius at x .We stress the fact that no explicit upper bound on the curvature is as-sumed for the estimate (1); the condition GCBA is only needed to ensuresufficient regularity of the space (and the existence of a natural measure tocompute volumes).For all subsequent results, we will take as standing assumptions a com-plete, GCBA-space with a uniform upper bound on the packing constantat some fixed scale r smaller than the almost-convexity radius, or a dou-bling condition up to an arbitrary small scale. These classes of metric spacesare large enough to contain many interesting examples besides Riemannianmanifolds, and small enough to be, as we will see, compact in the Gromov-Hausdorff sense. 3otice that, for Riemannian manifolds, a local doubling or a packing condi-tion at some scale r > are much weaker assumption than a lower boundof the Ricci curvature (see [BCGS17], Sec.3.3, for different examples anda comparison of Ricci, packing and doubling conditions). However, thereare a lot of non-manifolds examples in these classes of metric spaces. Thesimplest ones are simplicial complexes with locally constant curvature (alsocalled M κ -complexes , cp. [BH13]) and "bounded geometry" in an appropri-ate sense: they will be studied in detail in Section 7. Other interesting classesof spaces satisfying a uniform packing condition at fixed scale are the classof Gromov-hyperbolic spaces with bounded entropy , admitting a cocompactgroup of isometries (as shown in [BCGS17], [BCGS]), or the class of (uni-versal coverings of) compact, non-positively curved manifolds with boundedentropy, admitting acylindrical splittings (see [CSb]). See also [CSa] for ap-plications in the non-cocompact case.In Sections 3 and 4 we will see how the packing or covering conditionsand the upper bound on the curvature interact. While in geodesic metricspaces it is always possible to extend the packing condition at some scaleto bigger scales (cp. Lemma 4.7), it is not possible in general to extendthe packing condition uniformly to smaller scales, see Example 4.3. Anotherkey-result of the paper is that this extension is possible when the metricspace has curvature bounded from above and is locally geodesically complete.In particular, the local geometry at scales smaller than r is controlled bythe packing condition: Theorem B (Extract from Theorem 4.9) . Let X be a complete, geodesic, GCBA -space with almost-convexity radius ρ ac ( X ) ≥ ρ > . Then, the following conditions are equivalent:(a) there exist P and r ≤ ρ / such that X satisfies the P -packing con-dition at scale r ;(b) there exist n and V , R > such that X has dimension ≤ n and µ X ( B ( x, R )) ≤ V for all x ∈ X ;(c) there exist two functions c ( r ) , C ( r ) such that for any x ∈ X and forany < r < ρ : < c ( r ) ≤ µ X ( B ( x, r )) ≤ C ( r ) < + ∞ . For Riemannian manifolds of dimension n , the measure µ X coincides withthe n -dimensional Hausdorff measure, so (b) corresponds simply to a uniformupper bound on the Riemannian volume of balls of some fixed radius R ,a condition that it is sometimes easier to verify than the bounded packing.The proof of Theorem 4.9 is essentially based on universal estimates frombelow and from above of the volume of small balls of X in terms of dimensionand of the packing constants. We will prove these estimates in Section 3.We want to point out that, while the estimate (1) and Theorem 4.9 are4ew, many of the ideas behind these results are already implicitely presentin [LN19].In Section 5, we investigate the relation between the local doubling con-dition with respect to the natural measure µ X and the local structure ofGCBA-spaces. It is easy to show that a local doubling condition implies thepacking. However, it turns out that the doubling property is much strongerand characterizes GCBA-spaces which are purely dimensional spaces , i.e.those whose points have all the same dimension. Indeed, we prove: Theorem C (Extract from Corollary 5.5 & Theorem 5.2) . Let X be a complete, geodesic, GCBA -space with almost-convexity radius ρ ac ( X ) ≥ ρ > . The following conditions are equivalent:(a) there exists D > such that the natural measure µ X is D -doublingup to some scale r > ;(b) X is purely n -dimensional for some n , and there exist constants P and r ≤ ρ / such that X satisfies the P -packing condition at scale r . The families of spaces with uniformly bounded diameter, satisfying apacking condition for some universal function P = P ( r ) and all < r ≤ r , are classically called uniformly compact ; actually, one can always ex-tract from them convergent subsequences for the Gromov-Hausdorff distance(see [Gro81]). Moreover, it is classical that an upper bound on the curva-ture is stable under Gromov-Hausdorff convergence, provided that the cor-responding CAT ( κ ) -radius is uniformly bounded below. Starting from theresults proved above, it is possible to decline Gromov’s Precompactness The-orem for GCBA-spaces as follows. Consider the classes GCBA κ pack ( P , r ; ρ ) , GCBA κ vol ( V , R ; ρ , n ) of complete, geodesic, GCBA-spaces with curvature ≤ κ , almost-convexityradius ρ ac ( X ) ≥ ρ > and satisfying, respectively, condition (a) or (b) ofTheorem B.Let also denote by GCBA κ vol ( V ; ρ , n =0 ) the class of complete, geodesic, GCBA-spaces with curvature ≤ κ , totalmeasure µ X ( X ) ≤ V , almost-convexity radius ρ ac ( X ) ≥ ρ > and dimension precisely equal to n . Then: Beware that the doubling constant which is used in [LN19] is a different notion, whichis purely metric and does not depend on the measure. Mnemonically, we write before the semicolon the parameters which are relative to thepacking condition or to the condition on the natural measure µ X heorem D (Theorem 6.1, Corollary 6.9 & 6.7) . (a) The classes GCBA κ pack ( P , r ; ρ ) and GCBA κ vol ( V , r ; ρ , n ) are com-pact with respect to the pointed Gromov-Hausdorff convergence;(b) the class GCBA κ vol ( V ; ρ , n =0 ) is compact with respect to the Gromov-Hausdorff convergence and contains only finitely many homotopy types. We will also see that a uniform packing at some scale r is also a necessary condition for compactness (see Theorem 6.4 for the precise statement).As our spaces are locally CAT ( κ ) with CAT ( κ ) -radius uniformly boundedbelow (see inequality (2) in Sec. 2), it is not surprising that the limit spaceis again locally CAT ( κ ) . Less trivially, as a part of the proof of the compact-ness, we need to show that the conditions on the measure, on the almost-convexity radius and on the dimension are stable under Gromov-Hausdorfflimits.So, let us highlight the following results, which are consequence of the es-timates in Theorems A and B, and are part of the compactness theorem.They will be proved in Section 6: Theorem E (Proposition 6.2 & Proposition 6.5) . Let ( X n , x n ) be GCBA κ -spaces converging to ( X, x ) with respect to the pointedGromov-Hausdorff topology. Then:(a) ρ ac ( X ) ≥ lim sup n →∞ ρ ac ( X n ) ;(b) if ρ ac ( X n ) ≥ ρ > for all n , then dim ( X ) ≤ lim n → + ∞ dim ( X n ) and the equality holds if and only if the distance from x n to the max-imal dimensional subspace X max n of X n stays uniformly bounded when n → ∞ .(The second assertion refines Lemma 2.1 of [Nag18], holding for CAT ( κ ) -spaces). Therefore, GCBA spaces with curvature uniformly bounded from above andalmost convexity radius uniformly bounded below can collapse only if themaximal dimensional subspaces go to infinity. We will see such an examplein Section 6.On the other hand, the lower-semicontinuity of the natural measure ofballs and of the total volume will follow from [LN19], where it is proved thatif ( X n ) n ≥ is a sequence of GCBA-spaces converging to X , then the naturalmeasures µ X n converge weakly to the natural measure µ X (see Lemma 2.7and the proof of Corollary 5.7 for details). We will see in Section 5 that,under the stronger assumptions that the natural measure is doubling up tosome arbitrarily small scale, then the volume of balls is actually continuous (cp. Corollary 5.7).Once proved that the bound on the total volume is stable under Gromov-Hausdorff convergence and that this implies the uniform boundedness of thespaces in our class, the homotopy finiteness stated in (ii) is a particular6ase of Petersen’s finiteness theorem [Pet90]; actually, as the CAT ( κ ) -radiusis uniformly bounded below, these spaces have a common local geometriccontractibility function LGC ( r ) = r for r ≤ ρ .It is not difficult (see Section 6) to check that also the doubling propertyis stable under pointed Gromov-Hausdorff convergence and so is the propertyof being pure dimensional. Namely, let us also consider the classes (with thesame conventions as before)GCBA κ doub ( D , r ; ρ ) GCBA κ vol ( V ; ρ , n pure ) of complete, geodesic, GCBA-spaces X with curvature ≤ κ , almost-convexityradius ρ ac ( X ) ≥ ρ > and which are, respectively, either D -doublingup to scale r , or purely n -dimensional with total measure µ X ( X ) ≤ V .We then deduce the following additional compactness results: Theorem F (Extract from Corollaries 6.9 & 6.7) . The classes
GCBA κ doub ( D , r ; ρ ) and GCBA κ vol ( V ; ρ , n pure ) are compactwith respect to, respectively, pointed and unpointed Gromov-Hausdorff con-vergence. Moreover GCBA κ vol ( V ; ρ , n pure ) contains only finitely many ho-motopy types. The proof of these and other compactness and stability results is presentedin Section 6.Finally, in Section 7 we specialize our results to study the convergence andstability of M κ -complexes with bounded geometry. We will first establishsome basic relations relating the injectivity radius to the size and valency ofthe complexes. Recall that the valency of a M κ -complex X is the maximumnumber of simplices having a same vertex in common, and the size of thesimplices of a X is defined as the smallest radius R > such that any simplexcontains a ball of radius R and is contained in a ball of radius R ; we referto Sec. 7.1 for further definitions and details. Then, we prove: Theorem G (Proposition 7.12 , Sec.7) . Let X be a M κ -complex whose simplices have size bounded by R , with valencyat most N and no free faces. Then the following conditions are equivalent:(a) X is a complete GCBA -space with curvature ≤ κ ;(b) X satisfies the link condition at all vertices;(c) X is locally uniquely geodesic;(d) X has positive injectivity radius;(e) X has injectivity radius ≥ ι , for some ι depending only on R and N . The equivalence of the first four conditions is well-known for M κ -complexeswith finite shape (that is, whose geometric simplices, up to isometry, varyin a finite set), see [BH13]. The last condition is new and we will use it toexhibit other examples of compact families of GCBA-spaces. Namely, let M κ ( R , N ) , M κ ( R ; V , n )
7e the class of M κ -complexes X without free faces, with positive injec-tivity radius (but nor a-priori uniformly bounded below), simplices of sizebounded by R and, respectively, valency bounded by N or total volumebounded by V and dim ( X ) ≤ n . It is immediate to check that, for suitable N = N ( R , V , n ) , the class M κ ( R ; V , n ) is a subclass of M κ ( R , N ) ,made of compact M κ -complexes, namely with a uniformly bounded numberof simplices (cp. proof of Theorem 7.16); hence, it contains only finitelymany M κ -complexes, up to simplicial homeomorphism . On the other hand,we prove: Theorem H (Extract from Theorem 7.14 & Corollary 7.16, Sec.7) . The classes M κ ( R , N ) and M κ ( R ; V , n ) are compact, respectively, underpointed and unpointed Gromov-Hausdorff convergence. Moreover, there areonly finitely many M κ -complexes of diameter ≤ ∆ in M κ ( R , N ) , up tosimplicial homeomorphisms. All the assumptions in this result are necessary. Indeed, we will see how,dropping the bounds on the valency or on the size of the simplices, we donot have neither finiteness nor compactness (see Example 7.17).We think that Theorems D, F and H mark quite well the advantage of thesynthetic condition of curvature ≤ κ over sectional curvature bounds, byidentifying classes which are closed under Gromov-Hausdorff convergence,in contrast with the the classical convergence theorems of Riemannian ge-ometry.The Appendix is devoted to recall, for the reader’s convenience, somebasics of ultrafilters and ultraconvergence of metric spaces, which is a toolheavily used all along the paper. Acknowledgments.
We would like to thank S. Gallot, G. Besson and A. Lytchakfor the useful and stimulating discussions during the preparation of this work.
First of all we fix the notation. The open and the closed ball of radius R centered at x in a metric space X will be denoted by B X ( x, R ) and B X ( x, R ) respectively; if the metric space is clear from the context, we will simplywrite B ( x, R ) and B ( x, R ) . The closed annulus with center at x and radii r < r will be denoted by A ( x, r , r ) . If ( X, d ) is a metric space and λ is a positive real number we denote by λX the metric space ( X, λd ) ,where ( λd )( x, y ) = λd ( x, y ) for any x, y ∈ X , i.e. the rescaled metric space.We denote with B λX ( x, r ) the ball of center x and radius r with respect tothe metric λd . The identity map from ( X, d ) to ( X, λd ) is denoted by dil λ .A geodesic is a curve γ : I → X , where I is an interval in R , such that forany t ≤ s ∈ I it holds d ( γ ( t ) , γ ( s )) = | t − s | . If I = [ a, b ] we say that γ is ageodesic joining x = γ ( a ) to y = γ ( b ) . A generic geodesic joining two points x, y ∈ X will be denoted by [ x, y ] , even if there are more geodesics joining x y . A curve is a local geodesic if it is a geodesic around any point in itsinterval of definition.Finally, we stress the fact that we consider pointed Gromov-Hausdorff con-vergence only for complete metric spaces : so, every time we write ( X n , x n ) → ( X, x ) in the pointed Gromov-Hausdorff sense we mean that X n and X arecomplete. This condition is not restrictive; indeed if ( X n , x n ) converges to ( X, x ) , then it converges also to the completion ( ˆ X, ˆ x ) . As a consequence if ( X n , x n ) is a sequence of proper metric spaces converging to ( X, x ) , then X is proper (see Corollary 3.10 of [Her16]). ( κ ) and GCBA-spaces We recall the definition of locally CAT ( κ ) metric space. We fix κ ∈ R .We denote by M κ the unique simply connected, complete, -dimensionalRiemannian manifold of constant sectional curvature equal to κ and by D κ the diameter of M κ . So D κ = + ∞ if κ ≤ and D κ = π √ κ if κ > .A metric space X is CAT ( κ ) if any two points at distance less than D κ canbe connected by a geodesic and if the geodesic triangles with perimeter lessthan D κ are thinner than their comparison triangles in the model space M κ .This means the following. For any three points x, y, z ∈ X such that d ( x, y )+ d ( y, z )+ d ( z, x ) < D κ , a geodesic triangle with vertices x, y, z is thechoice of three geodesics [ x, y ] , [ y, z ] and [ x, z ] , denoted by ∆( x, y, z ) . Forany such triangle there exists a unique triangle ∆ κ (¯ x, ¯ y, ¯ z ) in M κ , up to isom-etry, with vertices ¯ x , ¯ y and ¯ z satisfying d (¯ x, ¯ y ) = d ( x, y ) , d (¯ y, ¯ z ) = d ( y, z ) and d (¯ x, ¯ z ) = d ( x, z ) ; such a triangle is called the κ -comparison triangle of ∆( x, y, z ) . The comparison point of p ∈ [ x, y ] is the point ¯ p ∈ [¯ x, ¯ y ] suchthat d ( x, p ) = d (¯ x, ¯ p ) . The triangle ∆( x, y, z ) is thinner than ∆ κ (¯ x, ¯ y, ¯ z ) iffor any couple of points p ∈ [ x, y ] and q ∈ [ x, z ] we have d ( p, q ) ≤ d (¯ p, ¯ q ) .A metric space X is called locally CAT ( κ ) if for any x ∈ X there exists r > such that B ( x, r ) is a CAT ( κ ) metric space. The supremum amongthe radii r < D κ satisfying this property is called the CAT ( κ ) -radius at x and it is denoted by ρ cat ( x ) . The infimum of ρ cat ( x ) among the points x ∈ X is called the CAT ( κ ) -radius of X and it is denoted by ρ cat ( X ) ; therefore, bydefinition, ρ cat ( X ) ≤ D κ .A metric space X is GCBA if there exists a κ such that X is locallyCAT ( κ ) , locally compact, separable and locally geodesically complete. The last property means that any local geodesic in X defined on an interval [ a, b ] can be extended, as a local geodesic, to a bigger interval [ a − ε, b + ε ] .In some case we will write GCBA κ , if we want to emphasize the role of κ .This class of metric spaces is the one studied in [LN19]. A metric spaceis geodesically complete if any local geodesic can be extended, as a localgeodesic, to the whole R . We recall a well known fact: any complete, locallygeodesically complete metric space is geodesically complete.9 tiny ball , according to [LN19], is a metric ball B ( x, r ) such that r < min { , D κ } and B ( x, r ) is compact. We suppose X is a complete, locally geodesically complete, locally CAT ( κ ) ,geodesic metric space. If x, y ∈ X satisfy d ( x, y ) < ρ cat ( x ) then there exists aunique geodesic joining them. Hence for any x ∈ X and < r ≤ R < ρ cat ( x ) it is well defined the contraction map : ϕ Rr : B ( x, R ) → B ( x, r ) by sending a point y ∈ B ( x, R ) to the unique point y ′ along the geodesic [ x, y ] satisfying d ( x, y ′ ) /r = d ( x, y ) /R . Moreover any local geodesic startingat x which is contained in B ( x, ρ cat ( x )) is a geodesic. This fact, togetherwith the locally geodesically completeness and the completeness of X , showsthat the map ϕ Rr is surjective. It is also rR -Lipschitz as stated in [LN19].We skecth here the computation. Lemma 2.1.
Any contraction map is rR -Lipschitz.Proof. By the CAT ( κ ) condition it is enough to prove the thesis on the modelspace M κ . The result is clearly true when κ ≤ , so we can assume κ = 1 .In this case M κ is the standard sphere S . Step 1.
For any x ∈ S and for any ≤ R ≤ π the inverse of the exponentialmap, the logarithmic map log x : B ( x, R ) → B T x S ( O, R ) , is R sin R -Lipschitz.So, for any R in our range we have that the logarithmic map is -Lipschitz.Thus we can conclude that, for any y, z ∈ B ( x, π ) , d ( y, z ) ≤ d (log x ( y ) , log x ( z )) ≤ d ( y, z ) where the first inequality follows by standard comparison results. Step 2.
We fix < r ≤ R ≤ π and y, z ∈ B ( x, R ) . Let y ′ and z ′ be thecontractions of y and z . We observe that the contraction of log x ( y ) , on thetangent space, from the radius R to r coincides with the point log x ( y ′ ) andthe same holds for z ; this contraction map is a dilation of factor rR . Therefore d ( y ′ , z ′ ) ≤ d (log x ( y ′ ) , log x ( z ′ )) = rR d (log x ( y ) , log x ( z )) ≤ rR d ( y, z ) . The natural set of scales where the contraction map is defined is notbounded from above by the CAT ( κ ) -radius but rather from the almost-convexity radius. The almost-convexity radius at a point x ∈ X is defined asthe supremum of the radii r such that for any two geodesics [ x, y ] , [ x, z ] oflength at most r and any t ∈ [0 , it holds: d ( y t , z t ) ≤ td ( y, z ) where y t , z t are respectively the points along [ x, y ] and [ x, z ] satisfying d ( x, y t ) = td ( x, y ) and d ( x, z t ) = td ( x, z ) . The almost-convexity radius at10 does not depend on κ and is denoted by ρ ac ( x ) . Then, by definition, forany point y ∈ B ( x, ρ ac ( x )) there exists a unique geodesic joining x to y (theexistence follows from the assumptions on X ), so the contraction map iswell defined for any < r ≤ R < ρ ac ( x ) . A straightforward modificationof Corollary 8.2.3 of [Pap05] shows that any local geodesic joining x to apoint y at distance d ( x, y ) < ρ ac ( x ) is actually a geodesic. This fact and thegeodesic completeness of X imply again that any contraction map withinthe almost-convexity radius is surjective and rR -Lipschitz, by definition.The (global) almost-convexity radius of the space X , denoted by ρ ac ( X ) , iscorrespondingly defined as the infimum over x of the almost-convexity radiusat x .Clearly, we always have ρ ac ( X ) ≥ ρ cat ( X ) . The inequality can be partiallyreversed when X is proper: indeed, in this case it holds ρ cat ( X ) ≥ min (cid:26) D κ , ρ ac ( X ) (cid:27) , (2)therefore a lower bound on the almost-convexity radius and the knowledgeof the upper bound κ yield a lower bound on the CAT ( κ ) -radius. The proofof (2) follows directly from Corollary II.4.12 of [BH13] once observed thatany two points of X at distance less than ρ ac ( X ) are joined by a uniquegeodesic. We fix a complete, geodesic, GCBA-space X .Given two local geodesics γ , η starting at the same point x ∈ X we canconsider the geodesic triangle ∆( x, γ ( t ) , η ( t )) for any small enough t > .The comparison triangle ∆ κ (¯ x, γ ( t ) , η ( t )) has an angle α t at ¯ x . By theCAT ( κ ) condition, the angle α t is decreasing when t → , see [BH13]. Henceit is possible to define the angle between γ and η at x as lim t → α t : it isdenoted by ∠ x ( γ, η ) and it takes values in [0 , π ] .For any x ∈ X , the space of directions of X at x is defined as Σ x X = { γ local geodesic s.t. γ (0) = x } / ∼ where ∼ is the equivalence relation γ ∼ η if and only if ∠ x ( γ, η ) = 0 .The function ∠ x ( · , · ) defines a distance which makes of Σ x X a compact,geodesically complete, CAT(1) metric space with diameter π (see [LN19]).The tangent cone of X at the point x is the metric space T x X = Σ x X × [0 , + ∞ ) up to the equivalence relation ( v, ∼ ( w, for every v, w ∈ Σ x X .The point corrisponding to t = 0 is called the vertex of the tangent cone,denoted by O . The metric on T x X is given by the following formula: giventwo points V = ( v, t ) and W = ( w, s ) of T x X we define d T ( V, W ) as theunique positive real number satisfying: d T ( V, W ) = t + s − ts cos( ∠ x ( v, w )) . (3)11n other words, T x X is the euclidean cone over Σ x X . With this metric T x X is a proper, geodesically complete, CAT(0) metric space ([LN19]). Remark 2.2.
Let Y = S n − be the euclidean standard sphere of radius .Then the euclidean cone over Y is isometric to R n . For any point x ∈ X the logarithmic map at x is defined as: log x : B ( x, ρ ac ( x )) → T x X, y ([ x, y ] , d ( x, y )) , where [ x, y ] is the unique geodesic from x to y (uniqueness is due to thedefinition of almost-convexity radius).The logarithmic map can be recovered by the contraction maps as follows.First notice that if X is a GCBA-space and λ > , then the space λX isGCBA.Now, let the logarithmic map on the space λX at dil λ ( x ) be denoted by log dil λ ( x ) : B λX ( dil λ ( x ) , λρ ac ( x )) → T dil λ ( x ) ( λX ) . The spaces T dil λ ( x ) ( λX ) and T x X are canonically isometric since the re-spective space of directions are canonically isometric. Let R < ρ ac ( x ) : weconsider a sequence of real numbers r n → , we set λ n = Rr n and we definethe maps g n = log dil λn ( x ) ◦ dil λ n ◦ ϕ Rr n : B X ( x, R ) → T x X where we are using the natural identification T dil λn ( x ) ( λ n X ) ∼ = T x X .By the CAT ( κ ) condition, the map log dil λn ( x ) is (1 + ε n ) -Lipschitz with ε n → , for r n → . So, by Lemma 2.1, the map g n is ε n ) -Lipschitzand for any non-principal ultrafilter ω this sequence defines a ultralimit map g ω between the ultralimit spaces (cp. Proposition A.5 in the Appendix).Since T x X is proper we can apply Proposition A.3 and find that the targetspace of g ω is T x X , i.e. g ω : ω - lim B X ( x, R ) → T x X. Using the definition of the logarithmic map and the natural identification T dil λn ( x ) ( λ n X ) ∼ = T x X as metric spaces, it is straightforward to check that g ω , restricted to the standard isometric copy of B X ( x, R ) in ω - lim B X ( x, R ) given by Proposition A.3, coincides with log x .In general, the logarithmic map of a GCBA space is not injective, dueto the possible branching of geodesics. We summarize its properties in thefollowing lemma: 12 emma 2.3. Let x ∈ X be a point of a complete, geodesic, GCBA space.Then the logarithmic map log x has the following properties:(a) log x ( B ( x, r )) = B ( O, r ) for any r < ρ ac ( x ) ;(b) d ( O, log x ( y )) = d ( x, y ) for any y ∈ B ( x, ρ ac ( x )) ;(c) it is -Lipschitz on B ( x, ρ ac ( x )) .Proof. Let y ∈ B ( x, ρ ac ( x )) . By definition, we have log x ( y ) = ([ x, y ] , d ( x, y )) ,where [ x, y ] is the unique geodesic from x to y . From (3) we immediatelyinfer that d T (log x ( y ) , O ) = d ( y, x ) . This proves (b) and that log x ( B ( x, r )) is included in B ( O, r ) for any r < ρ ac ( x ) . Now let V = ( v, t ) ∈ B ( O, r ) , for r < ρ ac ( x ) . We take a geodesic γ in the class of v . Since X is locally geodesi-cally complete, there exists an extension of γ as a geodesic to the interval [0 , r ] (this follows from the completeness of X and the fact that any localgeodesic is a geodesic if it is contained in a ball of radius smaller than thealmost-convexity radius). Then, using the definition of the logarithmic map,we deduce that log x ( γ ( r )) = V . Now, d ( x, γ ( r )) = r , which concludes theproof of (a). Finally, we have seen that the logarithmic map is obtained asthe restriction of the limit map g ω : ω - lim B X ( x, R ) → T x X to B X ( x, R ) . Itis -Lipschitz for all R ≤ ρ ac , therefore it is -Lipschitz on B ( x, ρ ac ( x )) .The logarithmic map gives a good local approximation of X by the tan-gent cone, as expressed in the following result. Lemma 2.4 ([LN19], Lemma 5.5) . Let x ∈ X be a point of a complete,geodesic, GCBA space. For any ε > there exists δ > such that for all r < δ and for every y , y ∈ B ( x, r ) it holds | d ( y , y ) − d T (log x ( y ) , log x ( y )) | ≤ εr. As a consequence of this fact, Lytchak and Nagano proved that the tan-gent cone at x can be seen as the Gromov-Hausdorff limit of a rescaled tinyball around x . We explicit the proof of this fact because in the following wewill need to write who are the maps realizing the Gromov-Hausdorff approx-imations. Lemma 2.5 ([LN19], Corollary 5.7) . Let x ∈ X be a point of a complete,geodesic, GCBA space. For any sequence λ n → ∞ , consider the sequence of CAT ( κ ) , pointed spaces Y n = ( λ n B ( x, r ) , x ) , for any r < ρ cat ( x ) . Then:(a) Y n → ( T x X, d T , O ) in the pointed Gromov-Hausdorff convergence;(b) the approximating maps f n : Y n → T x X are given by f n = log dil λn ( x ) (using again the natural identification T dil λn ( x ) ( λ n X ) ∼ = T x X )Proof. Fix
R > and any ε > . Let δ be as in Lemma 2.4 and set r n = 1 /λ n .We may assume that r n · R < δ . Then, for all y , y ∈ B Y n ( x, R ) we have y , y ∈ B X ( x, r n R ) and we can apply the Lemma 2.4, which yields13 d ( y , y ) − d T (log x ( y ) , log x ( y )) | ≤ εr n R. We have d Y n ( y , y ) = d ( y ,y ) r n and, by (3) and by the definition of the loga-rithmic map, d T ( f n ( y ) , f n ( y )) = 1 r n d T (log x ( y ) , log x ( y )) . In conclusion, we get | d Y n ( y , y ) − d T ( f n ( y ) , f n ( y )) | ≤ εR. Since this is true for any ε > , the thesis follows from Lemma 2.3.Finally, we observe that this characterization of T x X has another con-sequence. Fix any v ∈ Σ x X , which can be naturally seen as an element of T x X , and take any geodesic γ starting at x defining v : then, for any se-quence r n → we have that the sequence γ ( r n ) ∈ Y n defines v in the limit(indeed, f n ( γ ( r n )) = v for any n ). We recall some fundamental properties of GCBA-spaces proved in [LN19].For any point x ∈ X there exists an integer number k ∈ N such that anysufficiently small ball around x has Hausdorff dimension k . This numberis called the dimension of X at the point x and it is denoted by dim ( x ) .It is possible to show that dim ( x ) is equal to the geometric dimension of thetangent cone to X at x as defined in [Kle99]. The dimension of X is the(possibly infinite) quantity dim ( X ) = sup x ∈ X dim ( x ) ∈ [0 , + ∞ ] .There exists a natural stratification of X into disjoint subsets X k , where X k is the set of points of dimension k , for k ∈ N . In other words, X = F k ∈ N X k .Moreover, the k -dimensional Hausdorff measure H k is locally positive andlocally finite on X k . Hence it is defined a measure on X as µ X = X k ∈ N H k x X k . The measure µ X is locally positive and locally finite: we call it the naturalmeasure of X . Example 2.6. If X is a n -dimensional Riemannian manifold with sectionalcurvatures ≤ κ , then X is a locally geodesically complete, locally compact,separable, locally CAT ( κ ) metric space. In this case µ X is the n -dimensionalHausdorff measure and it coincides with the Riemannian volume measure,up to a multiplicative constant.This stratification of X has good local properties, as shown in [LN19].For any k ∈ N it is possible to define the set of regular points Reg k ( X ) of the k -dimensional part X k of X . We do not present here the definition of regular14oints (they are those points that are ( k, δ ) -strained for a suitable small δ ,according to [LN19], Sec. 11.4). Instead, we recall the main propertiesof the set of k -dimensional and regular k -dimensional points we will need.For every S ⊂ X we will denote S k = S ∩ X k and Reg k ( S ) = S k ∩ Reg k ( X ) .Then: • the set Reg k ( X ) is open in X and dense in X k (Cor. 11.8 of [LN19]); • for any tiny ball B ( x, r ) there exists k such that B ( x, r ) does not containpoints of dimension > k (Corollary 5.4 of [LN19]); • for any tiny ball B ( x, r ) there exists a constant C , only depending onthe maximal number of r -separated points in B ( x, r ) , such that: H k (cid:16) B ( x, r ) k (cid:17) ≤ C · r k (4) H k − (cid:16) ¯ B ( x, r ) k \ Reg k ( B ( x, r )) (cid:17) ≤ C · r k − (5)(Cor.11.8 of [LN19]; see Sec.4 for the definition of r -separated points). We recall here some facts about the behaviour of the natural measures andthe dimension under pointed Gromov-Hausdorff convergence.Consider a proper GCBA-space X and its natural measure µ X = P nk =0 H k x X k ,where n = dim ( X ) is assumed to be finite. The k -dimensional Haus-dorff measure H k restricted to the k -dimensional part is a Radon measure(indeed it is Borel regular and locally finite on the proper metric space X ),so it is µ X . In particular for any open subset U ⊂ X it holds: µ X ( U ) = sup { µ X ( K ) s.t. K is a compact subset of U } . Now suppose to have a sequence of proper GCBA-spaces X n convergingin the pointed Gromov-Hausdorff sense to some (proper) GCBA-space X .Arguing as in the first part of the proof of Theorem 1.5 of [LN19] we deducethat the natural measures µ X n converge in the weak sense to the naturalmeasure of the limit, µ X . This means that for any compact subsets K n ⊂ X n converging to a compact subset K ⊂ X it holds: lim ε → lim inf n → + ∞ µ X n ( B ( K n , ε )) = lim ε → lim sup n → + ∞ µ X n ( B ( K n , ε )) = µ X ( K ) (6)where we denote by B ( K n , ε ) the ε -neighbourhood of K n . As a consequence: Lemma 2.7.
Let X n be a sequence of proper, GCBA-spaces converging in thepointed Gromov-Hausdorff sense to a proper, GCBA-space X . Let x n ∈ X n be a sequence of points converging to x ∈ X . Then, for any R > it holds: µ X ( B ( x, R )) ≤ lim sup n → + ∞ µ X n ( B ( x n , R )) . (7)15 roof. The natural measure µ X is Radon and any compact subset containedin B ( x, R ) is contained in B ( x, R − η ) for some η > , therefore µ X ( B ( x, R )) = sup η> µ X ( B ( x, R − η )) . On the other hand, for any η > we have by (6) µ X ( B ( x, R − η )) ≤ lim sup n → + ∞ µ X n ( B ( x n , R − η )) ≤ lim sup n → + ∞ µ X n ( B ( x n , R )) . The equality in (7) would follow from a uniform estimate on the volumesof the annulii of a given thickness. Indeed this is the case when the metricspaces satisfy a uniform doubling condition, as we will see in Section 5.We end this preliminary section recalling some facts about the stability ofthe dimension under Gromov-Hausdorff convergence. In [LN19] (Def. 5.12),Lytchak and Nagano introduce the notion of standard setting of convergence .This means considering a sequence of tiny balls B ( x n , r ) ⊂ B ( x n , r ) in a sequence of GCBA-spaces X n , satisfying the following assumptions: • the closed balls B ( x n , r ) have uniformly bounded r -covering number(i.e. ∃ C such that the ball B ( x n , r ) can be covered by C closed ballsof radius r with centers in B ( x n , r ) for all n , cp. Sec.4); • the balls B ( x n , r ) converge to a compact ball B ( x, r ) of a GCBA-space X in the Gromov-Hausdorff sense; • the closures B ( x n , r ) converge to the closure B ( x, r ) of a tiny ball in X .We then have: Lemma 2.8 (Lemma 11.5 & Lemma 11.7 of [LN19]) . Let B ( x n , r ) be a sequence of tiny balls in the standard setting of conver-gence.Let y n ∈ B ( x n , r ) be a sequence converging to y ∈ B ( x, r ) . Then:(a) dim ( y ) ≥ lim sup n → + ∞ dim ( y n ) ;(b) if y is k -regular then dim ( y ) = dim ( y n ) for all n large enough. For non-compact spaces, the following general result is known:
Lemma 2.9 (Lemma 2.1 of [Nag18]) . Let ( X n , x n ) be a sequence of pointed, proper, geodesically complete CAT ( κ )spaces converging to some ( X, x ) in the pointed Gromov-Hausdorff sense.Then, dim ( X ) ≤ lim inf n → + ∞ dim ( X n ) . Estimate of volume of balls from below
We fix again a complete, geodesic, GCBA-space X .From (4) & (5) it follows that there exists an upper bound for the measureof any tiny ball B ( x, r ) ; moreover, one can find a uniform upper boundof the measure of all balls, independently of the center x , provided that X satisfies a uniform packing condition at some scale (see Theorem 4.9 inSection 4 for a precise statement). It is less clear if there exists a lowerbound on the measure, and in particular if this lower bound depends onlyon some universal constant. Indeed, in general the µ X -volume of balls of agiven radius is not uniformly bounded below independently of the space X .For instance, consider the balls of radius inside R n : when n grows, themeasure of these balls tends to . The next theorem shows that, if thedimension is bounded from above, then there is a uniform bound from belowto the measure of balls of a given (sufficiently small) radius: Theorem 3.1.
Let X be a complete, geodesic, GCBA metric space.If dim ( X ) ≤ n then for any x ∈ X and any r < min { , ρ ac ( x ) } it holds µ X ( B ( x, r )) ≥ c n · r n , where c n is a constant only depending on n . The proof of this fact is based on ideas most of which are already presentin [LN19]. First of all we have:
Proposition 3.2.
Let X be a complete, geodesic, GCBA metric space and x ∈ X be a point of dimension n . Then, there exists a -Lipschitz, surjectivemap P : T x X → R n such that:(a) P ( O ) = 0 ;(b) P ( B ( O, r )) = B (0 , r ) for any r > ;(c) d T ( V, O ) = d R n ( P ( V ) , for any V ∈ T x X .Proof. As the point x has dimension n , then the geometric dimension of T x X is n . This implies that Σ x X is a space of dimension n − satisfyingthe assumptions of Proposition 11.3 of [LN19]. So, there exists a -Lipschitzsurjective map P ′ : Σ x X → S n − . We extend the map P ′ to a map P overthe tangent cones by sending the point V = ( v, t ) to the point ( P ′ ( v ) , t ) .It is immediate to check that P is surjective and that P (0) = 0 .Moreover, the tangent cone over S n − is R n , as said in Example 2.2; therefore,the equality P ( B ( O, R )) = B (0 , R ) follows directly from (3). Always by (3),we have d T ( V, O ) = d R n ( P ( V ) , for any V ∈ C x X . Finally, the -Lipschitzproperty of P follows from the same property of P ′ and from the propertiesof the cosine function. 17ombining this result with the properties of the logarithmic map ex-plained in Section 2.3, we deduce the following : Proposition 3.3.
Let X be a complete, geodesic, GCBA metric space and x ∈ X be a point of dimension n . Then, there exists a -Lipschitz, surjectivemap Ψ x : B ( x, ρ ac ( x )) → R n such that(a) Ψ x ( x ) = 0 ;(b) Ψ x ( B ( x, r )) = B (0 , r ) for any < r < ρ ac ( x ) ;(c) d ( x, y ) = d (0 , Ψ x ( y )) for any y ∈ B ( x, ρ ac ( x )) .Proof. Define Ψ x = P ◦ log x , where P is the map of the previous propositionand log x is the logarithmic map at x . Then Ψ satisfies the thesis.Using the map Ψ x we can transport metric and measure properties from R n to X . We denote by ω n the H n -volume of the ball of radius of R n . Corollary 3.4.
Let X be a complete, geodesic, GCBA metric space and x ∈ X be a point of dimension n . Then H n ( B ( x, r )) ≥ n ω n r n for any < r < ρ ac ( x ) .Proof. It follows directly from the properties of the map Ψ x and the be-haviour of the Hausdorff measure under Lipschitz maps. Proof of Theorem 3.1.
We fix x ∈ X , < r < min { , ρ ac ( x ) } and ε = r n .We call d the dimension of x . We look for the biggest ball around x ofHausdorff dimension exactly d . In order to do that we define r = sup { ρ > s.t. HD ( B ( x, ρ )) = d } . (where HD denotes the Hausdorff dimension). Notice that HD ( B ( x, ρ )) is monotone increasing in ρ . If r ≥ r we stop and we redefine r = r .Otherwise, there exists a point x such that d ( x, x ) ≤ r + ε and the di-mension of x is d > d , by definition of r . Now we look for the biggestball around x of Hausdorff dimension d . We define r = sup { ρ > s.t. HD ( B ( x , ρ )) = d } . Arguing as before, if r + ε + r ≥ r we stop the algorithm and we redefine r as r = r + ε + r . Otherwise we can find again a point x such that d ( x , x ) ≤ r + ε and whose dimension is d > d . We continue the algorithmuntil r + ε + . . . + r k = r . It happens in at most n steps. At the end wehave points x = x , x , . . . x k with k ≤ n such that d ( x i , x j ) ≤ r j + ε ,18 + ε + . . . + r k = r and such that the dimension of x j is d j , with d i > d j if i > j . We observe that the d j -dimensional parts of the balls B ( x j , r j ) ,denoted by B d j ( x j , r j ) , are disjoint and contained in B ( x, r ) , by construction.Moreover the open ball B ( x j , r j ) has no point of dimension greater than d j .So µ X ( B ( x, r )) = n X k =0 H k x B k ( x, r ) ≥ X j H d j ( B d j ( x j , r j )) . The last step is to estimate the last term of the sum. Since k ≤ n and r + ε + . . . + r k = r then r + . . . + r k = r − ( k − ε ≥ r . Hence there existsan index j such that r j ≥ r n . By definition, any point of the ball B ( x j , r j ) is of dimension ≤ d j . Hence by the properties of the Hausdorff measure weget H d j ( B d j ( x j , r j )) = H d j ( B ( x j , r j )) ≥ d j ω d j r d j j ≥ c n r n , where the first inequality follows directly from the previous corollary, andthe last one holds since r ≤ . So we can choose c n = (cid:18) n (cid:19) n min k =0 ,...,n ω k that is a constant depending only on n . This concludes the proof. Let Y ⊂ X be any subset of a metric space:– a subset S of Y is called r -dense if ∀ y ∈ Y ∃ z ∈ S such that d ( y, z ) ≤ r ;– a subset S of Y is called r -separated if ∀ y, z ∈ S it holds d ( y, z ) > r .The r -packing number of Y is the maximal cardinality of a r -separatedsubset of Y and is denoted by Pack ( Y, r ) . The r -covering number of Y is theminimal cardinality of a r -dense subset of Y and is denoted by Cov ( Y, r ) .These two quantities are classically related by the following relations:Pack ( Y, r ) ≤ Cov ( Y, r ) ≤ Pack ( Y, r ) . (8)On a given space X , the numbers Pack ( B ( x, R ) , r ) and Cov ( B ( x, R ) , r ) , for < r ≤ R , depend in general on the chosen point x . We are interestedin the case where these numbers can be bounded independently of x ∈ X .Therefore, consider the functionsPack ( R, r ) = sup x ∈ X Pack ( B ( x, R ) , r ) , Cov ( R, r ) = sup x ∈ X Cov ( B ( x, R ) , r ) called, respectively, the packing and covering functions of X . They takevalues on [0 , + ∞ ] ; moreover, as an immediate consequence of (8) we havePack ( R, r ) ≤ Cov ( R, r ) ≤ Pack ( R, r ) . (9)19 efinition 4.1. Let X be a metric space and let C , P , r > .We say that X is P -packed at scale r if Pack (3 r , r ) ≤ P , that is everyball of radius r contains no more than P points that are r -separated.Analogously, we say that X is C -covered at scale r if Cov (3 r , r ) ≤ C ,i.e. every ball of radius r can be covered by at most C balls of radius r .The next theorem affirms that the packing functions can be well con-trolled for complete, locally CAT ( κ ) -spaces which are locally geodesicallycomplete (notice that no local compactness is assumed, since it will follow from the packing condition): Theorem 4.2.
Let X be a complete, locally CAT ( κ ) , locally geodesicallycomplete, geodesic metric space with ρ ac ( X ) > . Suppose that X satisfies Pack (cid:16) r , r (cid:17) ≤ P for < r < ρ ac ( X ) / . Then X is proper and geodesically complete; so, it is a GCBA metric space.Moreover, for any < r ≤ R it holds: Pack ( R, r ) ≤ P (1 + P ) Rr − , if r ≤ r ; Pack ( R, r ) ≤ P (1 + P ) Rr − , if r > r . We want to remark that, in general, a control of the packing function atsome fixed scale does not imply any control at smaller scales, as shown inthe following example.
Example 4.3.
Let D n ⊂ R n be the closed Euclidean disk of radius .Let X n be the space obtained gluing a Euclidean ray [0 , + ∞ ) to a pointof the boundary of D n . Fix r = 1 . Any r -separated subset S of X n con-tains at most one point of D n . Hence Pack (3 r , r ) ≤ , in other words X n is -packed at scale for every n . However at smaller scales, for exampleat scale r = , we can easily show that Pack (3 r, r ) → + ∞ when n → + ∞ .Notice that the spaces X n in this example are complete and CAT(0) butthey fail to be geodesically complete .We also remark that a packing condition to scales bigger than the almost-convexity radius does not propagate to smaller scales: Example 4.4.
Let X n be the graph with one vertex and n loops of length .For any n , we glue an half-line to the vertex obtaining a complete, GCBA,length metric space Y n . As in Example 4.3 it is easy to show that at bigscales the spaces Y n satisfy a uniform packing condition, while at small scalesthey do not.The proof of Theorem 4.2 is based on some preliminary lemmas.20 emma 4.5. Let X be a space satisfying the assumptions of Theorem 4.2.Then, X is P -packed at scale r for any r ≤ r .Proof. We fix x ∈ X and r ≤ r . We take a r -separated subset { x , . . . , x N } of B ( x, r ) . We consider the contraction map ϕ r r which is surjective and rr -Lipschitz. For any i we fix a preimage y i of x i under ϕ r r . We have r < d ( x i , x j ) ≤ rr d ( y i , y j ) for any i = j . This means that the set { y , . . . , y N } is r -separated in B ( x, r ) , hence N ≤ P . Corollary 4.6.
Let X be as in Theorem 4.2. Then X is locally compact.Proof. We fix a point x ∈ X . The ball B ( x, r ) is complete since it is closedand X is complete. Moreover for any ε > the maximal cardinality of a ε -separated subset of B ( x, r ) is finite, hence this ball is totally bounded.We can conclude it is compact.As a consequence, since X is a locally compact, complete, geodesic metricspace, then by Hopf-Rinow theorem it is proper. Moreover since it is com-plete and locally geodesically complete then it is also geodesically complete.This proves the first assertion of Theorem 4.2.We will now prove that the P -packing condition at every scale r ≤ r impliesthe announced estimate of Pack ( R, r ) for every R . First, we show: Lemma 4.7.
Let X be a geodesic metric space that is P -packed at scale r .Then, for any R ≥ r , it holds: Pack ( R, r ) ≤ P (1 + P ) Rr − . Proof.
We prove the thesis by induction on k , where k is the smallest integersuch that R ≤ r + kr . The case k = 0 clearly holds as for R = 3 r wehave Pack ( R, r ) ≤ P ≤ P (1 + P ) . Let now k ≥ and R ≥ r such that R ≤ r + kr . We consider the sphere S ( x, R − r ) of points at distanceexactly R − r from x . We observe that R − r ≤ r + ( k − r , so byinduction we can find a r -separated subset y , . . . , y n of S ( x, R − r ) ofmaximal cardinality, where n ≤ P (1 + P ) R − r r − . Moreover n [ i =1 B ( y i , r ) ⊃ A ( x, R − r , R ) . Indeed for any y ∈ A ( x, R − r , R ) we take a geodesic [ x, y ] and we call y ′ thepoint on the geodesic [ x, y ] at distance R − r from x . Then y ∈ B ( y ′ , r ) .21oreover there exists y i such that d ( y ′ , y i ) ≤ r , because of the maximalityof the set { y , . . . , y n } . Hence d ( y, y i ) ≤ r . Therefore we get:Pack ( B ( x, R ) , r ) ≤ Pack ( B ( x, R − r ) , r ) + Pack ( A ( x, R − r , R ) , r ) ≤ Pack ( B ( x, R − r ) , r ) + n X i =1 Pack ( B ( y i , r ) , r ) . Since Pack ( B ( y i , r ) , r ) ≤ P , we obtainPack ( B ( x, R ) , r ) ≤ Pack ( B ( x, R − r ) , r ) + P · n ≤ Pack ( B ( x, R − r ) , r ) + P · Pack ( B ( x, R − r ) , r ) ≤ (1 + P ) P (1 + P ) R − r r − = P (1 + P ) Rr − . We can now prove Theorem 4.2.
Proof of Theorem 4.2.
We have already shown that X is proper and geodesi-cally complete, and that it is P -packed at every scale < r ≤ r . Therefore,for these values of r , Lemma 4.7 yieldsPack ( R, r ) ≤ P (1 + P ) Rr − ∀ R ≥ r ; but this also holds for R ≤ r , since then Pack ( R, r ) ≤ Pack (3 r, r ) .On the other hand, if r ≥ r the thesis follows directly from Lemma 4.7.Indeed, when R ≥ r then Pack ( R, r ) ≤ Pack ( R, r ) and Lemma 4.7 con-cludes. If R < r we getPack ( R, r ) ≤ Pack ( R, r ) ≤ Pack (3 r , r ) ≤ P and P (1 + P ) Rr − ≥ P . We can read this result in terms of the covering functions instead of thepacking functions using (9).
Corollary 4.8.
Let X be a complete, locally CAT ( κ ) , locally geodesicallycomplete, geodesic metric space with ρ ac ( X ) > . Suppose that X satisfies Cov (cid:16) r , r (cid:17) ≤ C for r < ρ ac ( X ) / . Then for any < r ≤ R it holds: Cov ( R, r ) ≤ C (1 + C ) Rr − , if r ≤ r ; Cov ( R, r ) ≤ C (1 + C ) Rr − , if r > r . roof. By (9) we have that X satisfies Pack (3 r , r ) ≤ C . Hence we canapply the previous proposition to get:Cov ( R, r ) ≤ Pack (cid:16)
R, r (cid:17) ≤ C (1 + C ) Rr − , if r ≤ r Cov ( R, r ) ≤ C (1 + C ) Rr − , if r > r . We are ready to characterize the packing condition in terms of dimensionand measure of a GCBA metric space.
Theorem 4.9.
Let X be a complete, geodesic GCBA κ metric space with ρ ac ( X ) ≥ ρ > . The following facts are equivalent.a) There exist P > and < r < ρ such that Pack (3 r , r ) ≤ P ;b) There exist n , V , R > such that dim ( X ) ≤ n and µ X ( B ( x, R )) ≤ V for any x ∈ X ;c) There exists a measure µ on X and there exist two functions c ( r ) , C ( r ) such that for any x ∈ X and for any < r < ρ : < c ( r ) ≤ µ ( B ( x, r )) ≤ C ( r ) < + ∞ . Moreover the set of constants ( n , V , R , ρ , κ ) can be expressed only in termsof the set of constants ( P , r , ρ , κ ) and viceversa.Finally, if any of the above conditions holds then the natural measure µ X satisfies condition (c), and X is proper and geodesically complete.Proof. Assume first that X satisfies Pack (3 r , r ) ≤ P . First of all it followsthat the dimension of X is bounded. Indeed, we fix any point x ∈ X and wedenote by n its dimension. We consider the map Ψ x : B ( x, r ) → R n givenby Proposition 3.3. Let x , . . . , x k be a r -separated subset of B R n (0 , r ) .Since Ψ x is surjective we can take preimages y i of x i under Ψ x . Moreover d ( y i , x ) = d (Ψ x ( y i ) , , hence y i ∈ B ( x, r ) . As Ψ x is -Lipschitz the set { y , . . . , y k } is a r -separated subset of B ( x, r ) . Then k ≤ Pack (2 r , r ≤ Pack (3 r , r ≤ P by Theorem 4.2. But it is easy to show that k ≥ n . Therefore n ≤ P is the bound on the dimension we were looking for. We observe that thisbound is expressed only in terms of P . We fix now x ∈ X and any R > .Let r = min { , R, r , D κ } . We take a covering of B ( x, R ) with balls ofradius r . By Theorem 4.2 it is possible to do that with k balls, where k canbe estimated in the following way: k = Cov ( B ( x, R ) , r ) ≤ Pack (cid:18) B ( x, R ) , r (cid:19) ≤ P (1 + P ) Rr − .
23e call y , . . . , y k the centers of these balls. By Theorem 4.2 the space X isproper, then from the choice of r we get that B ( y i , r ) is a tiny ball for any i , as follows from (2). Moreover the maximal number of r -separated pointsinside B ( y i , r ) is bounded by Pack (10 r, r ) ≤ P (1 + P ) , as follows againby Theorem 4.2. Hence by (4) we have H j ( B ( y i , r ) j ) ≤ C ( P ) r j , where C ( P ) is a constant depending only on P . Therefore, using the factthat the dimension of X is bounded above by n = P and r ≤ , we get: µ X ( B ( y i , r )) = n X j =0 H j ( B ( y i , r ) j ) ≤ P · C ( P ) for any i . Finally, µ X ( B ( x, R )) ≤ P (1 + P ) Rr − · P · C ( P ) = V ( P , r , R, κ ) . (10)This shows that for any x ∈ X and any R we can find the desired uniformbound on the volume of the ball B ( x, R ) . This ends the proof of the impli-cation (a) ⇒ (b). Moreover this part of the proof, together with Theorem3.1, shows that if (a) holds then the measure µ X is a measure that satisfiescondition (c) of the theorem.Assume now that has dimension bounded above by n and that the vol-ume of the balls of radius R are uniformly bounded above by V . We set r = min { R , , ρ } . The claim is that X satisfies Pack (3 r , r ) ≤ P forsome P depending only on V , R , n and ρ . We consider the ball of radius R centered at a point x ∈ X . We take a r -separated subset of B (cid:0) x, R (cid:1) and we suppose its cardinality is bigger than some k . It means that thereare k points y , . . . , y k ∈ B (cid:0) x, R (cid:1) such that d ( y i , y j ) > r for any i = j .Hence the balls centered at y i of radius r are pairwise disjoint and satisfy B (cid:0) y i , r (cid:1) ⊂ B (cid:0) x, R + r (cid:1) ⊂ B ( x, R ) , since R + r ≤ R + R < R .We can apply Theorem 3.1 to get µ X (cid:0) B ( y i , r ) (cid:1) ≥ c n (cid:0) r (cid:1) n for any i .Thus V ≥ µ X ( B ( x, R )) ≥ k X i =1 µ X (cid:18) B (cid:18) y i , r (cid:19)(cid:19) ≥ k · c n (cid:18) r (cid:19) n , then k ≤ n V c n r n = 2 n V c n · min (cid:26) , (cid:18) ρ (cid:19) n , (cid:18) R (cid:19) n (cid:27) = P . It means that Pack ( B ( x, R ) , r ) ≤ P . Since R ≥ r we can concludethat Pack (3 r , r ) ≤ P that is what claimed.24inally, assume that there exists a measure µ such that for any x ∈ X andfor any < r < ρ it holds < c ( r ) ≤ µ ( B ( x, r )) ≤ C ( r ) < + ∞ . We take any r < ρ and we fix any point x ∈ X . Let k be the max-imal cardinality of a r -separated subset of B ( x, r ) . Then, arguing asbefore, we can find k disjoint balls of radius r contained in B ( x, r ) . Since C (4 r ) ≥ µ ( B ( x, r )) ≥ k · c ( r ) then k ≤ C (4 r ) c ( r ) = P . This shows that X satisfies (a) with these choices of r and P . In this section X will be a complete, geodesic GCBA-space.We say that X is purely n -dimensional if dim ( x ) = n for any x ∈ X .Moreover, we say that a measure µ on X is : • D-doubling up to the scale t at x ∈ X if there exists a constant D > such that for any < t ′ ≤ t it holds µ ( B ( x, t ′ )) µ ( B ( x, t ′ )) ≤ D ; • D -doubling up to scale t if it is D -doubling up to scale t at any point x ∈ X (for a uniform doubling constant D ).When uniformity of the constant and of the scale is not an issue, we willsimply say that µ is locally doubling on X : that is, if for any x ∈ X thereexist t x > and D x > such that µ is D x -doubling up to scale t x at anypoint of B ( x, t x ) . Remark 5.1.
Notice that any metric measured space ( X, µ ) satisfying a D -doubling condition up to scale t is P -packed at scale r = t for P = D (provided that the measure gives positive mass to the balls of positive radius).Actually, let x ∈ X and take any r -separated subset { y , . . . , y k } of B ( x, r ) .So, the balls B ( y i , r ) are pairwise disjoint. From the doubling property weget: µ X ( B ( x, r )) ≥ k X i =1 µ ( B ( y i , r / ≥ k X i =1 D µ ( B ( y i , r )) and since B ( y i , r ) ⊃ B ( x, r ) we deduce that k ≤ D .The next result characterizes GCBA-spaces whose natural measure islocally doubling: Theorem 5.2.
Let X be a proper, geodesic GCBA metric space. Suppose µ X is locally doubling: then X is purely n -dimensional for some n .
25e begin the proof of Theorem 5.2 with the following two preliminary results.
Lemma 5.3.
Let X be a proper, geodesic GCBA metric space and x ∈ X .Let v ∈ Σ x X , and assume that every point of B (( v, , ε ) is a k -regular pointof T x X , for some ε > . Then, there exists r > such that all points of theset A v,ǫ ( r ) = { y ∈ X s.t. d T (log x ( y ) , ( v, d ( x, y ))) ≤ εd ( x, y ) } ∩ B ( x, r ) have dimension k . We recall that, since T x X is a GCBA-space and since the set of k -regularpoints is open in T x X , if ( v, is k -regular point in T x X then it is alwayspossible to find ε satisfying the assumptions of the lemma. Proof.
Suppose the thesis is false. Then, there exists a sequence of points y n of dimension different from k at distance r n → from x such that d T (log x ( y n ) , ( v, r n )) ≤ εr n . We consider rescaled tiny balls Y n = r n B ( x, r ) as in Lemma 2.5, togetherwith the approximating maps f n ; so, for all n we have: d T ( f n ( y n ) , ( v, ≤ ε. Moreover, we are in the standard setting of convergence. Indeed, the GCBA-space X is geodesic and complete, so the contraction maps ϕ Rr are well-defined for any R < ρ cat ( x ) , and they are surjective and rR -Lipschitz; there-fore, by applying the same proof as in Lemma 4.5, we conclude that therescaled balls are uniformly packed (the other properties follow from the dis-cussion in Section 2). Moreover, the sequence y n ∈ Y n converges to somepoint y ∞ ∈ B (( v, , ε ) . So, y ∞ is k -regular by assumption. But, by Lemma2.8, the points y n must be k -dimensional for n large enough, which is acontradiction. Lemma 5.4.
Let v ∈ Σ x X and let γ be a geodesic starting at x defining v .For any < ε < we have, for all r > small enough: B (cid:18) γ (cid:16) r (cid:17) , εr (cid:19) ⊂ A v,ε ( r ) Proof.
Actually, as the logarithm map is -Lipschitz we have d T (log x ( y ) , ( v, d ( x, y ))) ≤ d T (cid:16) log x ( y ) , log x ( γ (cid:0) r (cid:1) ) (cid:17) + d T (cid:16)(cid:0) v, r (cid:1) , ( v, d ( x, y )) (cid:17) ≤ d (cid:16) y, γ (cid:0) r (cid:1)(cid:17) + (cid:12)(cid:12)(cid:12) r − d ( x, y ) (cid:12)(cid:12)(cid:12) ≤ d (cid:16) y, γ (cid:0) r (cid:1)(cid:17) ≤ εr ≤ εd ( x, y ) since d ( x, y ) ≥ r − εr . On the other hand, if y ∈ B ( γ ( r ) , εr ) we have d ( x, y ) ≤ r + εr < r , so the ball B ( γ ( r ) , εr ) is included in A v,ε ( r ) .26 roof of Theorem 5.2. Let us suppose X is not pure dimensional. We takea point x ∈ X of minimal dimension d . Then, we have by assumption r = sup { ρ > s.t. HD ( B ( x , ρ )) = d } < + ∞ . We can find a point x ∈ X with dimension d > d such that d ( x , x ) = r .Indeed, for any n we can find a point x n such that d ( x , x n ) < r + n anddim ( x n ) > d . The sequence of points x n converge, as the space is proper,to a point x at distance exactly r from x . Assume that dim ( x ) = d : then,there would exist a small radius ρ such that the Hausdorff dimensions of B ( x, ρ ) is exactly d . But x n belongs to B ( x, ρ ) for n ≫ , and any openball around x n has Hausdorff dimension strictly greater than d ; thereforeHD ( B ( x, ρ )) > d , a contradiction.Now, the tangent cone T x X at x has dimension d . Hence, there exists apoint v ∈ Σ x X and ε > such that any point of the ball B (( v, , ε ) isregular and of dimension d . We take any geodesic γ starting at x and defin-ing v , and we set y r = γ ( r ) . Applying the two lemmas above we havethat, for all r small enough, any point of the ball B ( y r , εr ) is d -dimensional.Since X satisfies a doubling condition around x , we know by Remark 5.1that a ball B ( x, r ) is P -packed, for some r , P depending on x . So, byTheorem 4.2 and by the properties of the natural measure recalled in Section2.4, there exists a constant C , only depending on r and P , such that forall sufficiently small r we have: µ X (cid:16) B (cid:16) y r , εr (cid:17)(cid:17) ≤ C · (cid:16) εr (cid:17) d . Consider now the ball B ( y r , r ) : notice that there exists a ball of radiusat least r contained in B ( y r , r ) ∩ B ( x , r ) , so made only of d -dimensionalpoints.In particular by Corollary (3.4) we have µ X ( B ( y r , r )) ≥ c d ( r ) d , where c d is a constant depending only on d . Thus µ X ( B ( y r , r )) µ X ( B ( y r , εr )) ≥ C ′ r d − d , where C ′ is a constant that does not depend on r . Since this is true forany r small enough and d < d , this inequality contradicts the doublingassumption at y r , when r goes to .As a consequence of what proved in Section 4 we obtain the following:27 orollary 5.5. Let X be a complete, geodesic GCBA κ metric space with ρ ac ( X ) ≥ ρ > . The following facts are equivalent:(a) there exist D > and t > such that the natural measure µ X is D -doubling up to scale t ;(b) X is purely dimensional and there exist P > and < r < ρ / suchthat Pack (3 r , r ) ≤ P ;(c) there exist n , V , R > such that X is purely n -dimensional and µ X ( B ( x, R )) ≤ V for any x ∈ X .Moreover each of the three sets of constants ( D , t , ρ , κ ) , ( P , r , ρ , κ ) , ( n , V , R , ρ , κ ) can be expressed in terms of the others.Finally if the conditions hold then X is proper and geodesically complete.Proof of Corollary 5.5. The implication (a) ⇒ (b) follows from Theorem 5.2 and from Remark5.1together with Theorem 4.2.Assume now X purely n -dimensional and Pack (3 r , r ) ≤ P . We recallthat by Theorem 4.9 n can be bounded from above in terms of P . We fix t < min { , R, r , D κ } as in the proof of Theorem 4.9. By Theorem4.2 we know X is proper, so it is easy to check that ρ cat ( X ) ≥ t by (2).Therefore by Theorem 3.1 we have µ X ( B ( x, t )) ≥ c n t n = c ( P ) t n for any t ≤ t . Moreover, by the same estimate used in the proof of Theorem4.9, and using the fact that µ X is just the n -dimensional Hausdorff measure,we get µ X ( B ( x, t )) ≤ P (1 + P ) · P · C ( P ) t n for any t ≤ t . Hence µ X ( B ( x, t )) µ X ( B ( x, t )) ≤ P (1 + P ) · P · C ( P ) c ( P ) = D which shows the implication (b) ⇒ (a).The equivalence between (b) and (c) is proved in Theorem 4.9.Finally, the doubling condition also implies the uniform continuity of thenatural measure of annuli: Lemma 5.6.
Let X be a complete, geodesic, GCBA κ -space which is D -doubling up to scale t and satisfies ρ ac ( X ) ≥ ρ . There exists β > ,only depending on D , such that for every R > and for every positive ε < min (cid:8) t R , (cid:9) it holds : µ X ( A ( x, R, (1 − ε ) R )) ≤ (cid:18) max (cid:26) Rt , (cid:27)(cid:19) β · ε β · µ X ( B ( x, R )) . roof. The proof is exactly the same as in Proposition 11.5.3 of [HKST15],with a minor modification due to the fact that we assume that µ X is doublingonly up to scale t . Actually, arguing as in the first part of the proof ofProposition 11.5.3 of [HKST15] one deduces that µ X ( A ( x, R, R − t )) ≤ D · µ X ( A ( x, R − t, R − t )) (11)for all x ∈ X and all positive t ≤ min (cid:8) t , R (cid:9) := t R . From (11), we deducethat for all t ≤ t R it holds µ X ( A ( x, R, R − t )) ≤ D (cid:16) µ X ( B ( x, R )) − µ X ( A ( x, R, R − t )) (cid:17) hence µ X ( A ( x, R, R − t )) ≤ (cid:18) D D (cid:19) · µ X ( B ( x, R )) Setting t m = · m one then shows by induction as in [HKST15] that µ X (cid:16) A ( x, R, (1 − t m ) R ) (cid:17) ≤ (cid:18) D D (cid:19) m +1 − m · µ X ( B ( x, R )) for all m ≥ m = l log ( R t R ) m . Our claim then follows for ε ≤ min (cid:8) t R , (cid:9) choosing β = log (cid:16) D D (cid:17) . Indeed for every such ε we choose the uniqueinteger m ≥ m such that t m +1 ≤ ε ≤ t m . Therefore we have µ X ( A ( x, R, (1 − ε ) R )) ≤ µ X ( A ( x, R, (1 − t m ) R )) ≤ (cid:18) D D (cid:19) m +1 − m · µ X ( B ( x, R )) . Using the fact that m + 1 ≥ − log ε we get µ X ( A ( x, R, (1 − ε ) R )) ≤ (2 · m ) β · ε β · µ X ( B ( x, R )) . Since m ≤ log (cid:16) R t R (cid:17) + 1 the thesis follows.As a consequence, we deduce that for D -doubling GCBA-spaces the mea-sure of balls is continuous under the Gromov-Hausdorff convergence, whichsharpens Lemma 2.7: Corollary 5.7.
Let X n be a sequence of geodesic, GCBA κ -spaces which are D -doubling up to scale t and satisfying ρ ac ( X ) ≥ ρ . Assume that the X n converge in the pointed Gromov-Hausdorff sense to some GCBA-space X and let x n ∈ X n be a sequence of points converging to x ∈ X . Then forany R ≥ it holds µ X ( B ( x, R )) = lim n → + ∞ µ X n ( B ( x n , R )) . roof. By Remark 5.1 and Theorem 4.2 the space X is P -packed at somescale r ≤ ρ / for P , r only depending on D , t , ρ and κ . By Theorem4.9, precisely by (10), the balls of radius R in X have uniformly boundedvolume, that is µ X ( B ( x, R )) ≤ C ( R ) for a universal function C ( R ) only depending on D , t , ρ and R .By the above Corollary, for all R > and ε > there exists δ > ,depending only on D , t and R such that for any x n ∈ X n it holds µ X n ( A ( x n , R + δ, R )) ≤ ε. The proof then follows directly from (6).
The aim of this section is to study properties that are stable under Gromov-Hausdorff convergence and the relations between ultralimits and Gromov-Hausdorff convergence.Throughout the section, we fix P , r , ρ > with r < ρ / and κ ∈ R .We denote by GCBA κ pack ( P , r ; ρ ) the class of complete, geodesic GCBA κ metric spaces X with ρ ac ( X ) ≥ ρ and Pack (3 r , r ) ≤ P . Then, we havethe following result which is strictly related to Gromov’s PrecompactnessTheorem, see [Gro81]: Theorem 6.1.
The class
GCBA κ pack ( P , r ; ρ ) is closed under ultralimitsand compact under pointed Gromov-Hausdorff convergence.Proof. Any space X ∈ GCBA κ pack ( P , r ; ρ ) is proper by Theorem 4.2,geodesic and geodesically complete. Consider any sequence ( X n , x n ) of ele-ments of GCBA κ pack ( P , r ; ρ ) and any non-principal ultrafilter ω .For any n we have ρ cat ( X n ) ≥ min { D κ , ρ } = ρ ′ > from (2). Then,by Corollary A.10 we have that X ω is a complete, locally geodesically com-plete, locally CAT ( κ ) , geodesic metric space with again ρ cat ( X ω ) ≥ ρ ′ .We want to prove now that Pack (3 r , r ) ≤ P holds on X ω . We fix apoint y = ( y n ) ∈ X ω : by Lemma A.8, we have B ( y, r ) = ω - lim B ( y n , r ) .Let z i = ( z in ) , i = 1 , . . . , N be a r -separated subset of B ( y, r ) , that is d ( z i , z j ) > r for all i = j . For any couple i = j we have d ( z in , z jn ) > r , ω -a.s.Since there are a finite number of couples, d ( z in , z jn ) > r for any i = j , ω -a.s.Moreover the points z in belong to B ( y n , r ) for any i . So, ω -a.s., there is a r -separated subset of B ( y n , r ) of cardinality N . Therefore N ≤ P andin particular Pack (3 r , r ) ≤ P on X ω . We can now apply again Theorem4.2 to conclude that X ω is proper, hence a GCBA κ metric space.To finish the first part of the proof we need to show that ρ ac ( X ω ) ≥ ρ .This is the object of the following: 30 roposition 6.2. Let ( X n , x n ) be GCBA κ -spaces converging to ( X, x ) withrespect to the pointed Gromov-Hausdorff topology. Then: ρ ac ( X ) ≥ lim sup n →∞ ρ ac ( X n ) We postpone the proof of this proposition, to end the proof of Theorem 6.1.In order to prove the compactness under pointed Gromov-Hausdorff conver-gence we take a sequence of spaces ( X n , x n ) ∈ GCBA κ pack ( P , r ; ρ ) and wefix any non-principal ultrafilter ω . Let ( X ω , x ω ) ∈ GCBA κ pack ( P , r ; ρ ) bethe ultralimit. Since the limit is proper we can apply Proposition A.11 to finda subsequence ( X n k , x n k ) that converges in the pointed Gromov-Hausdorffsense to ( X ω , x ω ) , showing the compactness part of the statement. Proof of Proposition 6.2 . Assume that ρ ac ( X n ) ≥ ρ > for infinitely many n .Take any non-principal ultrafilter ω : since by definition X is proper, then byProposition A.11 we have X = ω - lim X n . If ρ ≤ D κ we have ρ cat ( X n ) ≥ ρ for all n , so by Corollary A.10 we conclude immediately that ρ ac ( X ω ) ≥ ρ cat ( X ω ) ≥ ρ .Assume now that ρ > D κ ; in particular, as before we deduce ρ cat ( X ω ) = D κ .The strategy is the following: we claim that for any y = ( y n ) ∈ X ω and forany point z = ( z n ) at distance < ρ from y there exists a unique geodesicjoining y to z . In particular this geodesic must coincide with the ultralimitof the geodesics [ y n , z n ] of length < ρ . If this is true, then for any two points z = ( z n ) , w = ( w n ) of X ω at distance < ρ from y and any t ∈ [0 , we get d ( z t , w t ) = ω - lim d (( z n ) t , ( w n ) t ) ≤ ω - lim 2 td ( z n , w n ) = 2 td ( z, w ) which implies that ρ ac ( y ) ≥ ρ for any y ∈ X ω .So, suppose our claim is not true: that is, assume that there exists a point y = ( y n ) ∈ X ω , a radius ρ ∈ ( D κ , ρ ) such that any point at distance < ρ from y is joined to y by a unique geodesic, while for arbitrarily small values ǫ > there exist two different geodesics γ ε , γ ′ ε joining y to the same point z ε = ( z ε,n ) with d ( y, z ε ) = ρ + ε .We consider the points w ε = γ ε ( ρ − ε ) , w ′ ε = γ ′ ε ( ρ − ε ) and set ℓ = d ( w ε , w ′ ε ) .We observe we have ℓ ≤ ε and ℓ > since the ball of radius D κ around z ε is CAT ( κ ) by assumption, so uniquely geodesic. Similarly, we consider thepoints u ε = γ ε ( ρ + ε − D κ ) , u ′ ε = γ ′ ε ( ρ + ε − D κ ) and we set L = d ( u ε , u ′ ε ) .Our first step is to prove that L = d ( u ε , u ′ ε ) ≥ D κ · ℓ ε =: δ. (12)So, suppose by contradiction that (12) does not hold. First of all we remarkthat δ ≤ D κ , since ℓ ≤ ε . Then, as the ball B ( z ε , D κ ) is CAT ( κ ) , we canconsider the κ -comparison triangle ∆ κ ( z ε , u ε , u ′ ε ) . As usual we denote by31 ε , w ′ ε the comparison points of w ε and w ′ ε , respectively. By definition theedges of ∆ κ ( z ε , u ε , u ′ ε ) have length D κ , D κ , L . We consider another triangle ∆( Z, V, V ′ ) on M κ with edges [ Z, V ] , [ Z, V ′ ] , [ V, V ′ ] of length respectively D κ , D κ , δ . We denote by W, W ′ the points along [ Z, V ] and [ Z, V ′ ] at distance ε from Z . Since the contraction map ϕ Rr towards Z is rR -Lipschitz and d ( W, Z ) = d ( W ′ , Z ) = 2 ε we deduce d ( W, W ′ ) ≤ · ε ( D κ / d ( V, V ′ ) = 8 εD κ δ = ℓ . Since we are assuming by contradiction that
L < δ , we have by comparisonthat d ( w ε , w ′ ε ) < d ( W, W ′ ) . So, applying the CAT ( κ ) condition, we obtain ℓ = d ( w ε , w ′ ε ) ≤ d ( w ε , w ′ ε ) < d ( W, W ′ ) ≤ ℓ a contradiction. Therefore (12) holds.Now, by assumption there exists a unique geodesic from y to any pointin B ( y, ρ ) . Since d ( y, w ε ) < ρ by construction, if w ε = ( w ε,n ) then theultralimit of the geodesics γ ε,n = [ y n , w ε,n ] is the unique geodesic joining y to w ε , that is γ ε = ω - lim γ ε,n . Analogously, if w ′ ε = ( w ′ ε,n ) , we have γ ′ ε = ω - lim γ ε,n where γ ′ ε,n = [ y n , w ′ ε,n ] . Applying the contraction propertyon X n from R = ρ − ε to r = ρ + ε − D κ / we get L = d ( u ε , u ′ ε ) = ω - lim d (cid:0) γ ε,n ( ρ + ε − D κ / , γ ′ ε,n ( ρ + ε − D κ / (cid:1) ≤ ω - lim 2( ρ + ε − D κ / ρ − ε · d ( w ε,n , w ′ ε,n )= 2( ρ + ε − D κ / ρ − ε · ℓ. (13)As ρ > D κ , combining (12) and (13) gives a contradiction for ǫ → .We have therefore proved that ρ ac ( X ) ≥ ρ . This implies the upper semi-continuity of the almost-convexity radius since we can apply the same argu-ment to any subsequence. Remark 6.3.
In particular, for any sequence of metric spaces X n inGCBA κ pack ( P , r ; ρ ) and for any non-principal ultrafilter ω the ultralimit X ω is a proper space. Notice that, in general, the ultralimit of a sequence ofproper spaces is not proper, even if the spaces are really mild.For instance, let ( X n , x n ) = ( R n , and ω be any non-principal ultrafilter.Then X ω is isometric to ℓ ( R ) , the spaces of sequences { a n } of real numberssuch that P a n < + ∞ . This is a non-proper space of infinite dimension.The compactness of a class of proper metric spaces C is hard to achievesince properness and dimension are in general not stable under limits.In the following theorem we characterize the classes of proper, GCBA κ ,geodesic metric spaces with almost-convexity radius uniformly bounded frombelow that are precompact and compact under pointed Gromov-Hausdorffconvergence: 32 heorem 6.4. Let C be a class of proper, GCBA κ , geodesic metric spaces X with ρ ac ( X ) ≥ ρ > . Then, C is precompact under the pointed Gromov-Hausdorff convergence if and only if there exist P , r > such that C ⊂
GCBA κ pack ( P , r ; ρ ) . Moreover, C is compact if and only if is precompact and closed under ultralim-its. We stress the “only if” part in the above statement: for GCBA κ spaces, auniform packing assumption at some fixed scale is a necessary and sufficientcondition in order to have precompactness (we recall that, in the generalGromov’s Precompactness Theorem, one needs to have a uniform control ofthe packing function at every scale in order to achieve precompactness). Proof of Theorem 6.4.
Let C be a class of proper, GCBA κ , geodesic spaces X with ρ ac ( X ) ≥ ρ > . Let us prove the first equivalence stated in 6.4.So, assume that it is precompact in the pointed Gromov-Hausdorff sense,i.e. the closure C is compact under pointed Gromov-Hausdorff convergence.Suppose C is not contained in GCBA κ pack ( P , r ; ρ ) for any choice of P and r . Hence there exists r < ρ such that for any n there is a space X n ∈ C and a point x n ∈ X n with a set of r -separated points inside B ( x n , r ) of cardinality at least n . By assumption, there exists a subse-quence, denoted again ( X n , x n ) , converging in the pointed Gromov-Hausdorffsense to ( X, x ) . The space X is proper, see Section 2. Fix now any non-principal ultrafilter ω . Then ( X ω , x ω ) is isometric to ( X, x ) by Proposi-tion A.11. We are going to prove that inside B ( x, r ) there are infinitelymany points that are at distance at least r one from the other: there-fore, X cannot be proper and this is a contradiction. For any n we de-note the set of r -separated points of cardinality n inside B ( x n , r ) by { z n , . . . , z nn } . Then, for any fixed k ∈ N , we consider the admissible sequence z k = ( z kn ) ∈ X ω (notice that z kn is defined only for n ≥ k , but this sufficesto define a point z k in the ultralimit). Clearly, z k ∈ B ( x ω , r ) for all k .Moreover if k = l then d ( z kn , z ln ) > r for all n , hence d ( z k , z l ) ≥ r .This shows that C is a subclass of GCBA κ pack ( P , r ; ρ ) for some P and r .Viceversa, if C ⊂
GCBA κ pack ( P , r ; ρ ) then its closure C is contained in thecompact space GCBA κ pack ( P , r ; ρ ) by Theorem 6.1, so C is compact.Let us show now the second equivalence. Suppose that C is precompactand closed under ultralimits. Applying the same proof of the second part ofTheorem 6.1 we get that C is compact under pointed Gromov-Hausdorff con-vergence. Viceversa, if C is compact under Gromov-Hausdorff convergencethen it is contained in GCBA κ pack ( P , r ; ρ ) for some P , r . In particularfor any non-principal ultrafilter ω and any sequence of spaces ( X n , x n ) ∈ C we have that X ω is a proper metric space. By Proposition A.11 there existsa subsequence that converges in the pointed Gromov-Hausdorff sense to X ω ,hence X ω ∈ C since C is compact. 33s a consequence of Theorem 6.4 and of the estimates on volumes andpacking proved in Sections 3 & 4, we deduce that the dimension is almost stable under pointed Gromov-Hausdorff convergence, in the following sense: Proposition 6.5.
Let ( X n , x n ) be a sequence of GCBA κ -spaces with almostconvexity radius ρ ac ( X n ) ≥ ρ > , converging to ( X, x ) in the pointed Gromov-Hausdorff sense. Let X max n be the maximal dimensional subspace of X .Then, dim ( X ) ≤ lim inf n → + ∞ dim ( X n ) and the equality dim ( X ) = lim n → + ∞ dim ( X n ) holds if and only if the dis-tance d ( x n , X max n ) stays uniformly bounded when n → ∞ .Proof. As the spaces ( X n , x n ) converge to ( X, x ) , they form a precompactfamily and so they belong to GCBA κ pack ( P , r , ρ ) , for some constants P and r , by Theorem 6.4. Let us first show that we always havedim ( X ) ≤ lim inf n → + ∞ dim ( X n ) (14)Actually, consider a subsequence, we we still denote ( X n ) , whose dimen-sions equal the limit inferior, denoted d . Now suppose that there exists apoint y ∈ X with dim ( y ) = d > d . We may assume that y is d -regular,since Reg d ( X ) is dense in X d . The point y is the limit of a sequence ofpoints y n ∈ X n and for any r > the volume of the ball B ( y, r ) is big-ger than or equal to the limit of the volumes of the balls B ( y n , r ) , by (6).By Theorem 3.1 we have for all n : µ X (cid:16) B (cid:16) y n , r (cid:17)(cid:17) ≥ c d · (cid:16) r (cid:17) d where c d is a constant depending only on d . Moreover, since y is d -regular,then for any r small enough the ball B ( y, r ) contains only d -dimensionalpoints. We conclude by (4) & (5) that µ X ( B ( y, r )) ≤ C · r d , where C is a constant depending only on y and not on r . Therefore, as d < d , we have a contradiction if r is small enough, and (14) is proved.Assume now that d ( x n , X max n ) < D for all n . Since the almost convexityradius is bounded below by ρ both for X n and for X , also the CAT ( κ ) -radiusis bounded below by (2). So we can consider tiny balls B ( y n , r ) centered atregular points y n of maximal dimension, all with the same radius r , suchthat the closed ball B ( y n , r ) converge to some ball B ( y, r ) of X andsatisfy the condition Pack ( P , r ) for some constant P for all n , by 4.2.We are then in the standard setting of convergence, which implies by 2.8that dim ( X ) ≥ dim ( y ) ≥ lim sup n →∞ dim ( y n ) = lim sup n →∞ dim ( X n ) . Conversely, if we assume that dim ( X ) = lim n → + ∞ dim ( X n ) , then in partic-ular dim ( X n ) is constant for n ≫ and equal to d = dim ( X ) . Consider a34egular point y = ( y n ) ∈ X of dimension d : then, the points y n are admis-sible by definition (that is, d ( x n , y n ) stays uniformly bounded); moreover, aswe can choose as before uniformly packed tiny balls with B ( y n , r ) con-verging to B ( y, r ) , then the points y n necessarily belong to X max n , againby Lemma 2.8 (b). Example 6.6.
Let ( X, x ) ∈ GCBA κ pack ( P , r , ρ ) be any space. We con-sider the space Y obtained by gluing the half-line [0 , + ∞ ) to X at the point x .Clearly Y belongs to GCBA κ pack ( P ′ , r ′ , ρ ′ ) . The pointed Gromov-Hausdorfflimit of the sequence ( Y, n ) , where n ∈ [0 , + ∞ ) , is the real line. This isan example where the maximal dimension part escapes to infinity and thedimension is not preserved.We are going now to explore some variations of Theorem 6.4.We fix constants κ ∈ R and P , r , V , R , D , t , ρ , n > , with r ≤ ρ / ,and consider the following classes of complete, geodesic GCBA κ spaces X :– the class GCBA κ pack ( P , r ; ρ , n =0 ) of spaces which are P -packed at scale r , with almost-convexity radius ρ ac ( X ) ≥ ρ and dimension equal to n ;– the class GCBA κ pack ( P , r ; ρ , n pure ) of spaces P -packed at scale r ,with almost-convexity radius ρ ac ( X ) ≥ ρ and of pure dimension n ;– the classes GCBA κ vol ( V , R ; ρ , n =0 ) , GCBA κ vol ( V , R ; ρ , n pure ) of thosesatisfying µ X ( B ( x, R )) ≤ V , ρ ac ( X ) ≥ ρ and which have,respectively, dimension equal to n and pure dimension n ;– the class GCBA κ doub ( D , t ; ρ ) of spaces D -doubled up to scale t ,with ρ ac ( X ) ≥ ρ .Then: Corollary 6.7.
All the above classes are compact with respect to the pointedGromov-Hausdorff convergence.Proof.
By Theorem 4.9 and Corollary 5.5, the above are all subclasses ofGCBA κ pack ( P , r ; ρ ) , for suitable P and r . By the compactness Theorem6.4, the proof then reduces to show that the additional conditions on thedimension, on the measure of balls of given radius or on the doubling constantare stable under Gromov-Hausdorff limits. By Lemma 2.7, if a sequence X n in GCBA κ vol ( V , R ; ρ , n =0 ) converges to X , then µ X ( B ( y, R )) ≤ V for any y ∈ X . On the other hand, from Corollary 5.7 it follows that the doublingcondition is preserved to the limit. The stability of the dimension is provedin Proposition 6.5. To conclude, we need to show that pure-dimensionality isstable under Gromov-Hausdorff limits: this is the object of the Propositionwhich follows. Proposition 6.8.
Let ( X n , x n ) be a sequence of GCBA κ -spaces with al-most convexity radius ρ ac ( X n ) ≥ ρ > , converging to ( X, x ) in the pointedGromov-Hausdorff sense. Assume that X n is pure-dimensional for all n :then, X is pure-dimensional of dimension dim ( X ) = lim n → + ∞ dim ( X n ) . roof. The spaces ( X n , x n ) form a precompact family and so, by Theorem6.4, they belong to GCBA κ pack ( P , r , ρ ) , for suitable constants P and r .Then, by Theorem 4.9 the numbers dim ( X n ) belong to the finite set [0 , n ] .Suppose to have two integers d = d and two infinite subsequences X n i , X n i such that dim ( X n i ) = d for any i and dim ( X n i ) = d for any i . Weconsider the sequences x n i and x n i : for any r > the volumes of the ballsof radius r around these points converge to the volume of the ball of radius r around x , by Corollary 5.7 and Corollary 5.5. By (4), (5) and Theorem3.1 we have C r d ≤ µ X ni ( B ( x n i , r )) ≤ Cr d , C r d ≤ µ X ni ( B ( x n i , r )) ≤ Cr d , where C is a constant depending only on P and r . Since this is true for anyarbitrarily small r , we deduce that d = d . Therefore lim n → + ∞ dim ( X n ) exists and we denote it by d . We again apply the same estimate as beforeto conclude that for any y ∈ X and for any small r > we have C r d ≤ µ X ( B ( y, r )) ≤ Cr d . Therefore the dimension of y is d , which concludes the proof.Finally, we can specialize these theorems to subclasses of compact spaces.Clearly, the subclasses of the above classes made of spaces with diameter lessthan or equal to some constant ∆ will be compact with respect to the usualGromov-Hausdorff distance. We state here just two particularly interestingcases, which are reminiscent of the classical finiteness theorems of Rieman-nian geometry. Consider the classes:GCBA κ vol ( V ; ρ , n =0 ) , GCBA κ vol ( V ; ρ , n pure ) of complete, geodesic GCBA κ with total measure µ ( X ) ≤ V , almost con-vexity radius ρ ac ( X ) ≥ ρ and which are, respectively, n -dimensional andpurely n -dimensional. Corollary 6.9.
The classes
GCBA κ vol ( V ; ρ , n =0 ) and GCBA κ vol ( V ; ρ , n pure ) are compact under Gromov-Hausdorff convergence and contain only finitelymany homotopy types.Proof. First, we show that the diameter is uniformly bounded in both classes.Actually, consider X ∈ GCBA κ vol ( V ; ρ , n =0 ) and take any two points y, y ′ ∈ X such that d ( y, y ′ ) = ∆ > ρ := min { ρ , } . Let γ be a geodesic joining y to y ′ .Along γ we take points at distance ρ one from the other: they are at least ∆ ρ − , and the balls of radius ρ around these points are disjoint. Then, byTheorem 3.1 we get 36 ≥ µ X ( X ) ≥ c n n ρ n (cid:18) ∆ ρ − (cid:19) so the diameter of X is bounded from above in terms of n , ρ and V only.Let R such un upper bound. Then these classes are included inGCBA κ vol ( V , R ; ρ , n ) , whose compactness we have just proved. The con-clusion follows from Propositions 6.5 and 6.8.Finally, notice that any element of both classes has local geometric con-tractibility function LGC ( r ) = r for r ≤ ρ (see [Pet90] for the definition).Moreover the covering dimension of any space in both classes coincides withthe Hausdorff dimension, so it is uniformly bounded from above. We canthen apply Corollary B of [Pet90] to conclude that there are only finitelymany homotopy types inside any of the two classes. M κ -complexes Beyond Riemannian manifolds with uniform upper bounds on the sectionalcurvature and injectivity radius bounded below, an important class of GCBA κ spaces is provided by M κ -complexes with bounded geometry , in a sense weare going to explain. We will prove that the metric spaces in this class areuniformly packed and we will show that this class is compact under pointedGromov-Hausdorff convergence. Other finiteness results will be presented. M κ -complexes First of all we recall briefly the definitions and the properties of the class ofsimplicial complexes we are interested in. A κ -simplex S is the convex setgenerated by n + 1 points v , . . . , v n of M κn in general position, where M κn is the unique n -dimensional space-form with constant sectional curvature κ .If κ > the points v , . . . , v n are required to belong to an open emisphere.We say that S has dimension n . Each v i is called a vertex . A d-dimensionalface T of S is the convex hull of a subset { v i , . . . , v i d } of ( d + 1) vertices.The interior of S , denoted ˙ S , is defined as S minus the union of its lowerdimensional faces; the boundary ∂S is the union of its codimension faces.Let Λ be any set and E = F λ ∈ Λ S λ , where any S λ is a κ -simplex. Let ∼ bean equivalence relation on E satisfying:(i) for any λ ∈ Λ the projection map p : S λ → E/ ∼ is injective;(ii) for any λ, λ ′ ∈ Λ such that p ( S λ ) ∩ p ( S λ ′ ) = ∅ there exists an isometry h λ,λ ′ from a face T ⊂ S λ onto a face T ′ ⊂ S λ ′ such that p ( x ) = p ( x ′ ) ,for x ∈ S λ and x ′ ∈ S λ ′ , if and only if x ′ = h λ,λ ′ ( x ) .The quotient space K = E/ ∼ is called a M κ -simplicial complex or simply M κ -complex ; the set E is the total space . A subset S ⊂ K is called an m -simplex of K if it is the image under p of an m -dimensional face of some S λ ;37ts interior and its boundary are, respectively, the image under p of the inte-rior and the boundary of S λ . The support of a point x ∈ K , denoted supp ( x ) ,is the unique simplex containing x in its interior (notice that supp ( v ) = v when v is a vertex).The open star around a vertex v is the union of the interior of all sim-plices having v as a vertex.Metrically, K is equipped with the quotient pseudometric. By Lemma I.7.5of [BH13] the pseudometric can be expressed using strings. A m -string in K from x to y is a sequence Σ = ( x , . . . , x m ) of points of K such that x = x , y = x m and for each i = 0 , . . . , m − there exists a simplex S i containing x i and x i +1 . Moreover, a m -string Σ = ( x , . . . , x m ) from x to y is taut if • there is no simplex containing { x i − , x i , x i +1 } ; • if x i − , x i ∈ S i and x i , x i +1 ∈ S i +1 then the concatenation of the seg-ments [ x i − , x i ] and [ x i , x i +1 ] is geodesic in the subcomplex S i ∪ S i +1 .The length of Σ is defined as: ℓ (Σ) = m − X i =0 d S i ( x i , x i +1 ) where d S denotes the standard M κ -metric on a geodesic simplex S of M κ .Then, any string can be identified to a path in K , and the natural quotientpseudometric on K coincides with the following ([BH13], Lemma I.7.21): d K ( x, y ) = inf { ℓ (Σ) s.t. Σ is a taut string from x to y } . Moreover, for any x ∈ K one can define the number ε ( x ) = inf S simplex of Kx ∈ S inf T face of Sx / ∈ T d S ( x, T ) (15)which has the following fundamental property: Lemma 7.1 (Lemma I.7.9 and Corollary I.7.10 of [BH13]) . If ε ( x ) > for any x and K is connected then d K is a metric and ( K, d K ) isa length space. Moreover if y ∈ K satisfies d K ( x, y ) < ε ( x ) then any simplex S containing y contains also x and d K ( x, y ) = d S ( x, y ) . For any vertex v ∈ K it is possible to define the link Lk ( v, K ) of K at v asfollows. We fix any λ ∈ Λ such that v = p ( v λ ) , where v λ is a vertex of S λ .The set of unit vectors w of T v λ M κn such that the geodesic starting in direc-tion w stays inside S λ for a small time is a geodesic simplex of M n − = S n − ,denoted Lk ( v λ , S λ ) . Consider the equivalence relation on the disjoint union F p ( S λ ) ∋ v S λ , given by w λ ∼ w λ ′ if and only if p ( S λ ) ∩ p ( S λ ′ ) = ∅ and38 dh λ,λ ′ ) v λ ( w λ ) = w ′ λ : the link Lk ( v, K ) is the quotient space under thisequivalence relation. It is clearly a M -complex.We introduce now the class of simplicial complexes we are interested in.We say that K has valency at most N if for all v ∈ K the number of simpliceshaving v as a vertex is bounded above by N . Notice that if the valency is atmost N , then the maximal dimension of a simplex of K is at most N too.We say that a simplex S has size bounded by R > if it contains a ball ofradius R and it is contained in a ball of radius R ; accordingly, we say thesimplicial complex K has size bounded by R if all the simplices S λ defining K have size bounded by R . Lemma 7.2.
Let S be a M κ -simplex of dimension n and size bounded by R .Then any face of S of dimension d has size bounded by n − d R .Proof. We prove the lemma by induction on the dimension n . If n = 0 , there is nothing to prove. Assume now that the bounds hold for all faces of M κ -simplices of dimension ≤ n − , and consider a n -dimensional M κ -simplex S = Conv ( v , . . . , v n ) of size bounded by R . Let S ′ = Conv ( v , . . . , v n − ) be the face of S opposite to v n , and identify M κn − with the κ -model spacecontaining S ′ . It is clear that S ′ is contained in a ball B M κn − ( x, R ) of M κn − .On the other hand, let B M κn ( x, R ) be the ball of M κn which is contained in S .Call ψ : S → S ′ the map sending every point z of S to the intersection of theextension of the geodesic [ v n , z ] after z with S ′ , and let y = ψ ( x ) ; moreover,let ϕ be the contraction map centered at v n sending y to x . Notice that ψ ◦ ϕ ( z ) = z for all z ∈ M κn − . The map ϕ is at most -Lipschitz, so anypoint of B M κn ( y, R ) is sent to B M κn ( x, R ) under ϕ . Therefore, B M κn − (cid:0) y, R (cid:1) = B (cid:0) y, R (cid:1) ∩ M κn − ⊂ ψ (cid:0) B M κn ( x, R ) (cid:1) ⊂ S ′ which proves the induction step. Proposition 7.3.
The class of n -dimensional κ -simplices of size boundedby R and having a fixed point o as a vertex is compact under the Hausdorffdistance on M κn . Moreover, under this convergence, any face of the limitspace is limit of faces of the simplices in the sequence. Finally, the sameclass is closed under ultralimits.Proof. We take a sequence of simplices S l as in the assumption. We de-note by v l = o, v l . . . , v ln the vertices of S l . All the sequences ( v li ) are con-tained in a compact subset of M κn , so up to subsequence they converge to v i for all i = 0 , . . . , n , in particular v = o . Then, the ε -neighbourhoodConv ( v , . . . , v n ) ε of Conv ( v , . . . , v n ) is a convex subset of M κn which defi-nitely contains v l = o, v l . . . , v ln , hence Conv ( v , . . . , v n ) ε ⊃ Conv ( v l = o, v l . . . , v ln ) .Analogously, Conv ( v , . . . , v n ) ⊂ Conv ( v l = o, v l . . . , v ln ) ε definitely, hence39onv ( v , . . . , v n ) → Conv ( v l = o, v l . . . , v ln ) for the Hausdorff distance.Similarly, any face of S is limit of corresponding faces of S l . We now claimthat v , . . . , v n are in general position. If not, then there are three vertices,say v , v , v , belonging to the same -dimensional space. This means thefaces Conv ( v l , v l , v l ) tend to a 1-dimensional face, therefore thay cannothave size bounded below uniformly, which contradicts Lemma 7.2. There-fore S is a n -dimensional simplex. Moreover it is clear it is contained in aball of radius R and it contains a ball of radius R . Fix now any non-principalultrafilter ω and a sequence S l as above. Each S l is proper and the sequenceconverges in the Gromov-Hausdorff sense to the proper space S . Then byProposition A.11 we get that the ultralimit S ω is isometric to S .Clearly the same conclusion holds for the class of simplices of dimensionat most n and size bounded by R since it is the finite union of compactclasses. From this compactness result we get useful uniform estimates. Lemma 7.4.
Let K be a M κ -complex of size bounded by R and dim ( K ) ≤ n . Then there exists a constant ε ( R, n ) > depending only on R and n such that for all vertices v, w of K it holds ε ( v ) > ε ( R, n ) and d K ( v, w ) ≥ ε ( R, n ) .Proof. The class of simplices with size bounded by n − d R and dimensionexactly d is compact with respect to the Hausdorff distance of M κd by 7.3.Moreover the map Conv ( v , . . . , v d ) d M κd ( v , Conv ( v , . . . , v d )) is contin-uous with respect to the Hausdorff distance and it is positive. Therefore itattains a global minimum ε d > . Setting ε ( R, n ) = min d =0 ,...,n ε d , we have ε ( v ) ≥ ε ( R, n ) > for every vertex v ∈ K . Therefore, every two vertices v, w of K satisfy d K ( v, w ) ≥ ε ( R, n ) (or, by Lemma 7.1, there would exista simplex S of K such that d K ( v, w ) = d S ( v, w ) < ε ( R, n ) , a contradic-tion). Lemma 7.5.
Let S be a κ -simplex of size bounded by R and dim ( S ) ≤ n .Let ∂T ε denote the ε -neighbourhood of the boundary of any face T of S .For any positive τ there exists ε ( R, n, τ ) > such that for all faces T of S ,for all x ∈ T \ ∂T τ and all the faces T ′ of S which do not contain x it holds: d ( x, T ′ ) ≥ ε ( R, n, τ ) Moreover, for any integer d ≥ there exist η d = η d ( R, n ) , ε d = ε d ( R, n ) > ,where ε = ε ( R, n ) is the function given by Lemma 7.4 and η = ε n +1) ,satisfying the following conditions:(a) for all d -dimensional faces T of S , for every x ∈ T \ ∂T η d − and everyface T ′ of S not containing x it holds: d ( x, T ′ ) ≥ ε d ;(b) η k + η k +1 + · · · η m ≤ ε k , for all ≤ k ≤ m ≤ n . roof. The proof follows same arguments of Lemma 7.4. Indeed it is suffi-cient to consider the positive, lower semicontinuous map h ( S ) = min T face of S inf x ∈ T \ ∂T τ min T ′ face of Sx / ∈ T ′ d ( x, T ′ ) on the compact set of κ -simplices of size bounded by R and dimension atmost n , and take as ε ( R, n, τ ) its positive minimum.To prove the second part of the Lemma, we define ε ( R, n ) as ε ( R, n, η ) ,where this is the number given by the first statement with τ = η . Then,we choose < η = min { ε n +1) , ε n +1) } and again we define ε > as ε ( R, n, η ) .We can continue choosing < η = min { ε n +1) , ε n +1) , ε n +1) } and so on.This process produces the announced ε i , η i , which clearly satisfy (b).As a consequence, we get the following useful estimates (the second ofwhich is similar to Lemma I.7.54 of [BH13]): Lemma 7.6.
Let K be a M κ - complex of size bounded by R and dim ( K ) ≤ n .For all τ > there exists ǫ ( R, n, τ ) > with the following property: for all x ∈ K whose support is S satisfying d S ( x, ∂S ) ≥ τ we have ε ( x ) ≥ ε ( R, n, τ ) .In particular, if K is connected then ( K, d K ) is a length metric space.Proof. Let x ∈ K . Any simplex containing x must contain supp ( x ) as a face.It is then enough to apply the first claim of Lemma 7.5 to get the estimateon ε ( x ) . The second part follows immediately from Lemma 7.1. Lemma 7.7.
Let K be a M κ - complex of size bounded by R and dim ( K ) ≤ n .Then, there exists δ = δ ( R, n ) > depending only on R and n such that:(a) if two simplices S, S ′ of K are at distance ≤ δ ( R, N ) , they share a face;(b) moreover, for every x ∈ K the ball B ( x, δ ) is contained in the open starof some vertex;(c) finally, for every x ∈ K there exists y ∈ K such that B ( x, δ ) ⊂ B ( y, ε ( y )4 ) (where ε ( y ) is the function defined in (15) ).Proof. We start proving (c). Consider the numbers ε d , η d given by Lemma7.5.The claim is that δ = min d =0 ,...,n η d satisfies the thesis of (c). Actually,take any x ∈ K and consider the d -dimensional simplex S = supp ( x ) .There are two possibilities: either x ∈ S \ ∂S η d − or there exists a point y ∈ ∂S such that d ( x, y ) ≤ η d − . In the first case we observe that anysimplex S ′ containing x must have S has a face and by Lemma 7.5 we canconclude that ε ( x ) ≥ ε d . Hence, in this case B ( x, δ ) ⊂ B ( x, ε d ) ⊂ B ( x, η ( x )4 ) as follows by Lemma 7.5.(b). Otherwise, let S = supp ( y ) and call ≤ d ≤ − its the dimension. Arguing as before, we find that either η ( y ) ≥ ε d ,or there exists again a point y whose support S has dimension ≤ d < d such that d ( y , y ) ≤ η d − . In the first case we have B ( x, δ ) ⊂ B ( y , η d − + η d ) ⊂ B (cid:16) y , ε d (cid:17) ⊂ B (cid:18) y , ε ( y )4 (cid:19) , otherwise we continue the procedure inductively. Then either at some stepwe have the thesis, or we find a vertex v of K such that d ( x, v ) ≤ η d − + η d − + . . . + η ≤ ε . Therefore B ( x, δ ) ⊂ B (cid:16) v, ε ( R,n )4 (cid:17) ⊂ B (cid:16) v, ε ( v )4 (cid:17) , which proves (c).In order to prove (b) we fix x ∈ K and we find the corresponding y given by(c).Then for all point z ∈ B ( x, δ ) we can apply Lemma 7.1 and find that anysimplex S containing z must contain also y . This means that any such S hasthe vertices of supp ( y ) as vertices. This concludes the proof of (b).Finally, the proof of (a) is an easy consequence: suppose to have two points x and x ′ , belonging to two simplices S, S ′ respectively, such that d ( x, x ′ ) ≤ δ ;then, they belong to the open star of a same vertex by (b). In particular S and S ′ share a vertex.Another straightforward application of compactness and continuity yieldsthe following, whose proof is omitted: Lemma 7.8.
Let K be a M κ - complex of size bounded by R and dim ( K ) ≤ n .Then, there exists R ′ = R ′ ( R, n ) depending only on R and n such that forevery vertex v of K the M -complex Lk ( v, K ) has size bounded by R ′ . We start now considering M κ -complexes with bounded size and valency: Proposition 7.9.
Let K be a connected M κ -complex of size bounded by R and valency at most N . Then K is locally finite (i.e. for all x ∈ K there area finite number of simplices containing x ) and ( K, d K ) is a proper, geodesicmetric space.Proof. Any simplex S containing a point x must have supp ( x ) as a face;in particular, if v is a vertex of supp ( x ) , then it is also a vertex of S . Sothe number of simplices containing x is bounded by the number of simplicescontaining v , which is bounded by N by assumption. By Lemma 7.6 we knowthat ( K, d K ) is a length metric space. Finally, by Lemma 7.7 for all y ∈ K the ball B ( y, δ ) belongs to the open star of a vertex, which is the union of afinite number of simplices, hence K is locally compact and complete. Then,as K is a complete, locally compact, length metric space, it is proper andgeodesic by Hopf-Rinow’s Theorem. 42he following is the analogue of Theorem I.7.28 of [BH13]: Proposition 7.10.
Let K be a connected M κ -complex of size bounded by R and valency at most N . Then for any ℓ > there exists m = m ( ℓ, R, N ) depending only on ℓ, R and N such that any m -taut string of length ≤ ℓ satisfies m ≤ m .Proof. We use the same proof of Theorem I.7.28 of [BH13] (which is for M κ -complexes of finite shape ), proceeding by induction on the dimension of K .The first step is to prove that if a m -string Σ is included in the open star ofa vertex v , then m is bounded by a function m ′ ( ℓ, R, N ) . This is clear with m ′ = 3 if the geodesic associated to Σ passes through v , otherwise it followsby the inductive hypothesis by projecting radially Σ to Lk ( v, K ) (which haslower dimension), using Lemma 7.8.Now, if the bound stated in the proposition did not hold, there would existtout m -strings Σ i in M κ -complexes K i with length ≤ ℓ and arbitrary large m .Then, there would exist also tout m ′ -substrings Σ ′ i of the Σ i , with m ′ >m ′ ( ℓ, R, N ) , included in some ball ¯ B ( x i , δ ) ⊂ K i , for δ = δ ( R, N ) definedin Lemma 7.7. By the same Lemma, Σ ′ i would be included in the openstar of some vertex, which by step one implies that m ′ ≤ m ′ ( ℓ, R, N ) , acontradiction. Corollary 7.11.
Let K be a connected M κ -complex of size bounded by R and valency at most N . Let x, y ∈ K such that d K ( x, y ) ≤ ℓ . Then, thereexists a geodesic joining x to y realized as the concatenation of at most m ( ℓ, R, N ) geodesic segments, each contained in a simplex of K .Proof. Immediate from the fact that K is a geodesic space (by 7.9), thecharacterization of d K in terms of tout strings and Proposition 7.10.In order to establish if a M κ -complex is a locally CAT ( κ ) space we use thefollowing improvement of a well-known criteria. We recall that the injectivityradius of a complex K , denoted ρ inj ( K ) , is defined as the supremum of the r ≥ such that any two points of K that are at distance at most r are joinedby a unique geodesic. Proposition 7.12.
Let K be a connected M κ -complex of size bounded by R and valency at most N . The following facts are equivalent:(a) ( K, d K ) is locally CAT ( κ ) ;(b) K satisfies the link condition, i.e. the link at any vertex is CAT (1) ;(c) ( K, d K ) is locally uniquely geodesic;(d) ( K, d K ) has positive injectivity radius;(e) ρ inj ( K ) ≥ δ ( R, N ) , where δ ( R, N ) is the function defined in Lemma 7.7. oreover if K satisfies one of the equivalent conditions above, then for any x ∈ K the ball B ( x, δ ( R, N )) is a CAT ( κ ) space, i.e. the CAT ( κ ) -radius of K is at least δ ( R, N ) . The equivalences between (a), (b) and (c) are quite standard. The equiv-alence of these conditions with (d) is known for simplicial complexes withfinite shapes, see [BH13]. The main point of Proposition 7.12 is that the lastequivalence continues to hold in our setting, and moreover we can boundfrom below the injectivity radius of K in terms of R and N only. Proof of Proposition 7.12.
The equivalence between (a) and (b) follows fromTheorem II.5.2 and Remark II.5.3 of [BH13], while (a) ⇒ (c) is straightfor-ward. The implication (c) ⇒ (e) follows as in Proposition I.7.55 of [BH13].Actually, by Proposition 7.9 we have ε ( x ) > for every x ∈ K , so the ball B ( x, ε ( x )2 ) is isometric to the open ball B ( O, ε ( x )2 ) of the κ -cone C κ ( Lk ( v, K )) centred at the cone point O (cp. Theorem I.7.39 in [BH13]). Moreoverby assumption a neighbourhood of O of the cone C κ ( Lk ( v, K )) is uniquelygeodesic, which implies that the whole C κ ( Lk ( v, K )) is uniquely geodesic(cp. Corollary I.5.11, [BH13]), and this in turns implies that B ( x, ε ( x )2 ) is.By Lemma 7.7(c), we conclude that the injectivity radius is bounded belowby δ ( R, N ) (recall that the dimension of K is bounded above by N ). Theimplication (e) ⇒ (d) is obvious, while (c) ⇒ (b) follows exactly as in The-orem II.5.4 of [BH13]. Finally, the last remark follows from Theorem I.7.39&Theorem II.3.14 of [BH13] together with Lemma 7.7(c).We recall that a locally compact, locally CAT ( κ ) M κ -complex is locallygeodesically complete if and only if it has no free faces (see II.5.9 and II.5.10of [BH13] for the definition of having free faces and the proof of this fact).We can finally show that the class of metric spaces we are studying in thissection is uniformly packed. Proposition 7.13.
Let K be a connected M κ -complex without free faces,of size bounded by R , valency at most N and positive injectivity radius.Then, K is a proper, geodesic GCBA κ -space with ρ cat ( K ) ≥ ρ and sat-isfying Pack (3 r , r ) ≤ P , for constants ρ , P , r depending only on R, N and κ , and r ≤ ρ / .Proof. By the proof of Proposition 7.10 we know that K is proper andgeodesic. Moreover since the injectivity radius is positive then K is locallyCAT ( κ ) , and by Proposition 7.12 the CAT ( κ ) -radius is at least ρ = δ ( N, R ) .Since K has no free faces then it is locally geodesically complete. This showsthat K is also a GCBA κ -space. We remark that clearly H k ( K ) = 0 if k > N since the projection map from a simplex to K is -Lipschitz; this showsthat there are no points of dimension greater than N , i.e. dim ( K ) ≤ N .We now use Lemma 7.7 to estimate the number of simplices intersecting a44all around any point x ∈ K . Any simplex S which intersect B ( x, δ ( R, N )) intersects the open star around some vertex v , by Lemma 7.7.(b). Therefore v must be a vertex of S . If follows that the number of simplices intersecting B ( x, δ ( R, N )) is bounded by N . Therefore, for any x ∈ K we have µ K ( B ( x, δ ( R, N ))) ≤ N X d =0 N · H d ( B M κd ( o, δ ( R, N ))) ≤ V , where V depends just on R, N and κ (here is o is any point of M κd ).The conclusion follows from Theorem 4.9. M κ -complexes The aim of this section is to provide compactness and finiteness results forsimplicial complexes. We denote by M κ ( R, N ) the class of M κ -complexeswithour free faces, of size bounded by R , valency at most N and positiveinjectivity radius. Theorem 7.14.
The class M κ ( R, N ) is compact under pointed Gromov-Hausdorff convergence. By Proposition 7.13 there exist P , r , ρ such that any K ∈ M κ ( R, N ) belongs to GCBA κ pack ( P , r ; ρ ) . So, by Theorem 6.4, the class M κ ( R, N ) is precompact, and it is compact if and only if it is closed under ultralimits.We are going now to show that M κ ( R, N ) is closed under ultralimits.We fix a non-principal ultrafilter ω and we take any sequence ( K n , o n ) in M κ ( R, N ) . We denote by K ω the ultralimit of this sequence. Our aimis to prove that K ω is isometric to a M κ -complex ˆ K ω satisfying the sameconditions as the K n ’s. Step 1: construction of the simplicial complex ˆ K ω .Let us start definining who are the simplices of ˆ K ω . Let ( x n ) be any ad-missible sequence of points, with x n ∈ K n , and consider the unique sim-plex supp ( x n ) of K n containing x n in its interior: we define S ( x n ) = ω - lim supp ( x n ) .The metric space S ( x n ) is a κ -complex with size bounded by R by 7.3.Notice that, a priori, if y n is another sequence defining the same point as x n in K ω , then S ( y n ) might be different from S ( x n ) .Now we define ˆ K ω as follows. Let p n : S → K n denote the projection of anysimplex of the total space of K n to K n . The total space of ˆ K ω will be G ( x n ) admissible S ( x n ) where ( x n ) is any admissible sequence of points in K n , and the equivalencerelation is: if z ω = ω - lim z n ∈ S ( x n ) and z ′ ω = ω - lim z ′ n ∈ S ( x ′ n ) (i.e. ( z n ) , ( z ′ n ) respectively in supp ( x n ) and supp ( x ′ n ) ),we say that z ω ∼ z ′ ω if and only if ω - lim d K n ( p n ( z n ) , p n ( z ′ n )) = 0 . That is,we compare the points z n and z ′ n in the common space K n where they live.For simplicity we will abbreviate d K n ( p n ( z n ) , p n ( z ′ n )) with d K n ( z n , z ′ n ) .First of all we need to check that the relation is well defined: given otheradmissible sequences ( w n ) , ( w ′ n ) with w n ∈ supp ( x n ) and w ′ n ∈ supp ( x ′ n ) such that z ω = ω - lim w n and z ′ ω = ω - lim w ′ n , we have d K n ( w n , w ′ n ) ≤ d supp ( x n ) ( w n , z n ) + d K n ( z n , z ′ n ) + d supp ( x ′ n ) ( z ′ n , w ′ n ) hence ω - lim d K n ( w n , w ′ n ) = 0 . Once proved it is well defined it is easy toshow it is an equivalence relation. We call ˆ K ω the quotient space, and denote p ω : S ( x n ) → ˆ K ω the projections. Step 2: ˆ K ω satisfies axiom (i) of M κ -complexes. We fix an admissible sequence ( x n ) and the corresponding simplex S ( x n ) .We need to prove that the map p ω : S ( x n ) → ˆ K ω is injective. For this, con-sider points z ω = ω - lim z n and z ′ ω = ω - lim z ′ n in S ( x n ) ,with z n , z ′ n ∈ supp ( x n ) for all n ; then there exists ε > such that ω - lim d supp ( x n ) ( z n , z ′ n ) > ε > . In particular d supp ( x n ) ( z n , z ′ n ) > ε ω -a.e.(n). Now, for any point z of a M κ -complex define dim ( z ) as the di-mension of supp ( z ) . The strategy to prove the injectivity is by induction on d = max { ω - lim dim ( z n ) , ω - lim dim ( z ′ n ) } . Observe that if ω - lim dim ( z n ) = k then we have dim ( z n ) = k ω -a.e. ( n ) because the possible dimensions belong to a finite set. For d = 0 , we havethat z n , z ′ n are both vertices of supp ( x n ) , ω -a.e. ( n ) . If p ω is not injectivethen for every ε > we have d K n ( z n , z ′ n ) ≤ ε ω -a.e. ( n ) . By Lemma 7.4 weknow that if d K n ( z n , z ′ n ) ≤ ε ( R, N ) then z n = z ′ n as points of supp ( x n ) .We consider now the inductive step. We denote by T n , T ′ n the faces of S n containing z n and z ′ n in their interior, respectively. We suppose there exists τ > such that for ω -a.e. ( n ) it holds z n ∈ T n \ ( ∂T n ) τ . By Lemma 7.6 wehave ε ( z n ) ≥ ε ( R, N, τ ) ω -a.e. ( n ) , and similarly for z ′ n . Once again this factimplies the injectivity. Consider now the case where for all τ > the set { n ∈ N s.t. d ( z n , ∂T n ) ≤ τ and d ( z ′ n , ∂T ′ n ) ≤ τ } belongs to ω . Therefore ω - lim d ( z n , ∂T n ) = ω - lim d ( z ′ n , ∂T ′ n ) = 0 . Thismeans that z ω belongs to ∂T ω and z ′ ω belongs to ∂T ′ ω , by Proposition 7.3.Hence z ω = ω - lim w n and z ′ ω = ω - lim w ′ n , where w n and w ′ n belong to a lowerdimensional face of T n and T ′ n respectively. We then apply the inductiveassumption to get the thesis. Step 3: ˆ K ω satisfies axiom (ii) of M κ -complexes. Consider two simplices S ( x n ) , S ( x ′ n ) and suppose p ω ( S ( x n ) ) ∩ p ω ( S ( x ′ n ) ) = ∅ .This means that for any ε > there exist y ω = ω - lim y n and y ′ ω = ω - lim y ′ n y n ∈ supp ( x n ) and y ′ n ∈ supp ( x ′ n ) such that d K n ( y n , y ′ n ) < ε , ω -a.e. ( n ) .If ε < δ ( R, N ) then by Lemma 7.7.(a) we know that supp ( x n ) and supp ( x ′ n ) share a face in K n . Let then T n ⊂ supp ( x n ) and T ′ n ⊂ supp ( x ′ n ) such facesand h n : T n → T ′ n an isometry such that p n ( z ) = p n ( z ′ ) for z ∈ T n , z ′ ∈ T ′ n if and only if z ′ = h n ( z ) . By assumption this holds ω -a.e. ( n ) . By Proposition7.3it is easy to see that the metric spaces T ω = ω - lim T n and T ′ ω = ω - lim T ′ n are,respectively, faces of S ( x n ) and S ( x ′ n ) . Moreover the sequence of maps ( h n ) defines a limit map h ω : T ω → T ′ ω which is an isometry, by Proposition A.5.It remains to show that p ω ( z ω ) = p ω ( z ′ ω ) , for z ω ∈ T ω and z ′ ω ∈ T ′ ω , if andonly if h ω ( z ω ) = z ′ ω . But given z ω = ω - lim z n and z ′ ω = ω - lim z ′ n with z n ∈ supp ( x n ) , z ′ n ∈ supp ( x ′ n ) we have p ω ( z ω ) = p ω ( z ′ ω ) by definition if andonly if ω - lim d K n ( p n ( z n ) , p n ( z ′ n )) = 0 . This happens if and only if for any ε > the inequality d K n ( p n ( z n ) , p n ( z ′ n )) < ε holds ω -a.e. ( n ) . This means that d K n ( p n ( h n ( z n )) , p n ( z ′ n )) < ε holds ω -a.e. ( n ) , in particular p ω ( h ω ( z ω )) = p ω ( z ′ ω ) . By the injectivity of the pro-jection map p ω we then obtain h ω ( z ω ) = z ′ ω , which is the thesis. Step 4: ˆ K ω belongs to M κ ( R, N ) . It is clear that ˆ K ω has size bounded by R by construction.We want to show it has valency at most N . Fix a vertex v of ˆ K ω and pa-rameterize by α ∈ A the set of simplices S ( x n ( α )) of ˆ K ω having v as a vertex.For any fixed α ∈ A there is a vertex v n ( α ) of supp ( x n ( α )) such that thesequence ( v n ( α )) converges ω -a.e. ( n ) to v , by Proposition 7.3. In particularfor all α, α ′ ∈ A we get d K n ( v n ( α ) , v n ( α ′ )) < ε ( R, N ) ω -a.e. ( n ) , and then v n ( α ) = v n ( α ′ ) by Lemma 7.4.Let now S ( x n ( α )) = S ( x n ( α ′ )) be distinct elements of ˆ K ω , for α, α ′ ∈ A . Then,there exist a vertex of the first simplex u = ω - lim u n , with u n ∈ supp ( x n ( α )) ,which does not belong to the second one. So, d K n ( u n , supp ( x n ( α ′ ))) > ω -a.e. ( n ) , hence supp ( x n ( α )) = supp ( x n ( α ′ )) ω -a.e. ( n ) . Therefore, if ˆ K ω has m different simplices S ( x n ( α )) sharing the vertex v , there also exist m differ-ent simplices supp ( x n ( α )) of K n sharing the same vertex v n ( α ) , ω -a.e. ( n ) .This contradicts our assumptions if m > N .Finally, the fact that ˆ K ω has positive injectivity radius and has not free faceswill follow from the last step, where we prove that ˆ K ω and K ω are isometric.In fact, K ω is geodesically complete and locally CAT ( κ ) , as ultralimit ofcomplete, geodesically complete, locally CAT ( κ ) spaces with CAT ( κ ) -radiusuniform bounded below; hence, K ω (and in turns ˆ K ω ) has positive injectivityradius and no free faces, by Proposition 7.12 and II.5.9&II.5.10 of [BH13]. Step 5: ˆ K ω is isometric to K ω . We define a map
Φ : K ω → ˆ K ω as follows. Let y ω = ω - lim y n the ω -limitof be an admissible sequence ( y n ) of K n . Any y n belongs to supp ( y n ) : we47ill denote by ( y n ) supp ( y n ) the point, in the ultralimit of the sequence of sim-plices supp ( y n ) , which is defined by the admissible sequence of points ( y n ) .We then define Φ as Φ( y ω ) = p ω (( y n ) supp ( y n ) ) . It is easy to see it is well defined and surjective.It remains to prove it is an isometry. Let y n , z n ∈ K n define admissiblesequences. So, the distances d K n ( y n , z n ) are uniformly bounded by someconstant L . Therefore by Proposition 7.10 for any n there exists a geodesicbetween y n and z n which is the concatenation of at most m ( L, R, N ) seg-ments, each of them contained in a simplex. Since the number of segments isuniformly bounded, we can define a path in ˆ K ω which is the concatenationof geodesic segments, each contained in a simplex of ˆ K ω , and whose lengthis the limit of the lengths of the segments in K n . This shows that d ˆ K ω ( p ω (( y n ) supp ( y n ) ) , p ω (( z n ) supp ( z n ) ) ≤ ω - lim d K n ( y n , z n ) . In order to prove the other inequality, we fix two points y = p ω (( y n ) supp ( y n ) ) and z = p ω (( z n ) supp ( z n ) ) of ˆ K ω . Notice that from the inequality above wededuce that ˆ K ω is path-connected. Hence, by Proposition 7.10, we know thatthere exists a geodesic between y and z which is the concatenation of at most m ( ℓ, R, N ) geodesic segments, each of them contained in a simplex, where ℓ = d ˆ K ω ( x, y ) . These segments cross finitely many simplices, each of whichcan be seen as the ω -limit of a sequence of simplices in K n . Since the numberis finite we can see the union of these simplices of ˆ K ω as the ultralimit of theunion of the corresponding simplices in K n . We can therfore approximatethe geodesic in ˆ K ω with paths in K n between y n and z n , whose total lengthtend to ℓ . So d ˆ K ω ( p ω (( y n ) supp ( y n ) ) , p ω (( z n ) supp ( z n ) ) ≥ ω - lim d K n ( y n , z n ) . which ends the proof of Theorem 7.14.We can specialize this compactness teorem to other families of M κ -complexes, as done for GCBA κ pack ( P , r ; ρ ) . Namely, consider:– the subclass M κ ( R, N ; ∆) ⊂ M κ ( R, N ) of complexes with diameter ≤ ∆ ;– the class M κ ( R, V, n ) of M κ -complexes without free faces, with size boundedby R , total volume ≤ V , dimension bounded above by n and positive injec-tivity radius. Remark 7.15.
We should specify the measure on the complexes K of theclass M κ ( R ; V, n ) under consideration. Any such space is stratified in sub-spaces of different dimension, so it is naural to consider the measure which The notation stresses the fact that we see ( y n ) supp ( y n ) as limit of points in the abstractsimplices supp ( y n ) (not in K n ). Namely, ( y n ) supp ( y n ) belongs to the total space of ˆ K ω ,while y ω ∈ K ω .
48s the sum over k = 0 , . . . , n of the k -dimensional Hausdorff measure on each k -dimensional part. This clearly coincides with the natural measure µ K of K seen as GCBA-space. Corollary 7.16.
For any choice of R , n , V , N and ∆ , the above classesare compact under Gromov-Hausdorff convergence and contain only finitelymany simplicial complexes up to simplicial homeomorphisms.Proof. The compactness of M κ ( R, N ; ∆) is clear from the one of M κ ( R, N ) .Moreover, by Proposition 7.13 we know that any K ∈ M κ ( R, N ; ∆) satis-fies the condition Pack (3 r , r ) ≤ P for constants P , r only depending on R and N . Moreover, by Lemma 7.4 we know that any two vertices of K are η ( R ) -separated: in particular, the number of vertices of K is boundedabove by Pack ( ∆2 , η ( R )) which is a number depending only on R, N, κ and ∆ .Since the valency is bounded and the total number of vertices is bounded,we have only finitely many possible simplicial complexes up to simplicialhomeomorphisms.On the other hand, it is straightforward to show that any K ∈ M κ ( R ; V, n ) has valency bounded from above by a function depending only on R, V, n and κ , because any simplex of locally maximal dimension contributes to the totalvolume with a quantity greater than a universal function v ( R, n, κ ) > .This also shows also that the total number of simplices of K is uniformlybounded in terms of R, V and n , hence the combinatorial finiteness of M κ ( R ; V, n ) . Moreover, since any simplex has uniformly bounded size, alsothe diameters of complexes in this class are uniformly bounded. There-fore, M κ ( R ; V, n ) ⊂ M κ ( R, N ) for a suitable N and, as the class is made ofcompact metric spaces, it is actually precompact under (unpointed) Gromov-Hausdorff convergence. It remains to show that M κ ( R ; V, n ) is closed.By the proof of Theorem 7.14 it is clear that the upper bound on the dimen-sion of the simplices is preserved under limits. The stability of the upperbound on the total volume is proved as for the class GCBA κ vol ( V , R ; ρ , n =0 ) in Corollary 6.7.Finally, we want to point out that the assumptions on size and diameterin the above compactness results are essential: Examples 7.17.
Non-compact families of M κ -complexes. (1) Let X n be a wedge of n circles of radius . The family of M -complexes { X n } has uniformly bounded size and uniformly bounded diameter, butthe valency is not bounded. Notice that this family is neither finite noruniformly packed. In particular, it is not precompact.492) Let X n be obtained from a circle of radius , then choosing n equidistantpoints on the circle and gluing n circles of radius to them. The X n ’sadmit M -complex structures with uniformly bounded valency and uni-formly bounded diameter, but the size of the simplices is not bounded.Again, this family is neither finite nor uniformly packed, hence not pre-compact. A Ultralimits An ultrafilter on N is a subset ω of P ( N ) such that:1) ∅ / ∈ ω ;2) if A, B ∈ ω then A ∩ B ∈ ω ;3) if A ∈ ω and A ⊂ B then B ∈ ω ;4) for any A ⊂ N then either A ∈ ω or A c ∈ ω .We recall that there is a one-to-one correspondance between the ultrafilters ω on N and the finitely-additive measures defined on the whole P ( N ) withvalues on { , } such that ω ( N ) = 1 . Indeed given an ultrafilter ω we definethe measure ω ( A ) = 1 if and only if A ∈ ω ; conversely, given a measure ω as before we define the ultrafilter as the set ω = { A ⊂ N s.t. ω ( A ) = 1 } (it is easy to show it actually is an ultrafilter). In the following, ω will denoteboth an ultrafilter and the measure that it defines. Therefore we will writethat a property P ( n ) holds ω -a.s. when the set { n ∈ N s.t. P ( n ) } ∈ ω .There is an easy example of ultrafilter: fix n ∈ N and consider the set ω ofsubsets of N containing n . An ultrafilter of this type is called principal .The interesting ultrafilters are the non-principal ones; it turns out thatan ultrafilter is non-principal if and only if it does not contain any finiteset. The existence of non-principal ultrafilters follows from Zorn’s lemma.The interest on non-principal ultrafilters is due to the fact that they candefine a notion of limit of a bounded sequence of real numbers: Lemma A.1.
Let a n ∈ [ a, b ] be a bounded sequence of real numbers.Let ω be a non-principal ultrafilter. Then, there exists a unique point x in [ a, b ] such that for all η > the set { n ∈ N s.t. | a n − x | < η } belongs to ω .The real number x is said the ω -limit of the sequence ( a n ) and it is denotedby x = ω - lim a n . Moreover, if a n and b n are two bounded sequence of realnumbers, it holds:(a) ω - lim( a n + b n ) = ω - lim a n + ω - lim b n ;(b) if λ ∈ R then ω - lim( λa n ) = λ · ω - lim a n ; c) if a n ≤ b n then ω - lim a n ≤ ω - lim b n ;(d) if a = ω - lim a n and f is continuous at a , then ω - lim f ( a n )= f ( ω - lim a n ) . (The proof of the main part can be found in [DK18], Lemma 7.23, whileproperties (a)-(d) are trivial.)The ultralimit of unbounded sequences of real numbers can be defined inthe following way. Given an unbounded sequence of real numbers a n thefollowing mutually exclusive situations can occur: • there exists L > such that a n ∈ [ − L, L ] for ω -a.e. n .In this case the ultralimit of ( a n ) can be defined using Lemma A.1. • for any L > the set { n ∈ N s.t. a n ≥ L } belongs to ω .In this case we set ω - lim a n = + ∞ . • for any L < the set { n ∈ N s.t. a n ≤ − L } belongs to ω .In this case we set ω - lim a n = −∞ .We remark that the limit depends strongly on the non-principal ultrafilter ω .The ultralimit of a sequence of metric spaces is defined as follows. Definition A.2.
Let ( X n , x n ) be a sequence of pointed metric spaces and ω be a non-principal ultrafilter. We set: X = { ( y n ) : y n ∈ X n and ∃ L > s.t. d ( y n , x n ) ≤ L for any n } . and, for ( y n ) , ( z n ) ∈ X , we define the distance as: d (( y n ) , ( z n )) = ω - lim d ( y n , z n ) . The space X ω = ( X, d ) / d =0 is a metric space and it is called the ω -limitof the sequence of spaces ( X n , x n ) . The fact that ( X, d ) is a metric spacefollows immediately from the properties of the ultralimit of a sequence ofreal numbers and from the fact that d n is a distance for any n . In generalthe limit depends on the non-principal ultrafilter ω and on the basepoints.A basic example is provided by the ultralimit of a constant sequence. Proposition A.3.
Let ( X, x ) be a metric space and ω a non-principal ultra-filter.Consider the constant sequence ( X, x ) and the corresponding ultralimit ( X ω , x ω ) ,where x ω is the constant sequence of points ( x ) . Then(a) The map ι : ( X, x ) → ( X ω , x ω ) that sends y to the constant sequence ( y n = y ) is an isometric embedding;(b) if X is proper then ι is surjective, and ( X ω , x ω ) is isometric to ( X, x ) . roof. The first part is obvious by the definitions. If X is proper and ( y n ) is an admissible sequence defining a point of X ω , then it is contained in aclosed ball of X , that is compact. By Lemma 7.23 of [DK18] we find y ∈ X such that for all ε > the set { n ∈ N s.t. d ( y, y n ) < ε } belongs to ω . Therefore it is clear that the constant sequence ( y n = y ) defines the same point as the sequence ( y n ) in X ω , which proves (b).An interesting consequence of the definition is that the ultralimit ofpointed metric spaces is always complete (the proof is given in [DK18],Proposition 7.44): Proposition A.4.
Let ( X n , x n ) be a sequence of pointed metric spaces andlet ω be a non-principal ultrafilter. Then X ω is a complete metric space. Once defined the limit of pointed metric spaces it is useful to definelimit of maps. We take two sequences of pointed metric spaces ( X n , x n ) and ( Y n , y n ) . A sequence of maps f n : X n → Y n is said admissible if there exists M ∈ R such that d ( f n ( x n ) , y n ) ≤ M for any n ∈ N . In general an admissiblesequence of maps does not define a limit map, but it is the case if the mapsare equi-Lipschitz. A sequence of maps f n : X n → Y n is equi-Lipschitz ifthere exists λ ≥ such that f n is λ -Lipschitz for any n . Proposition A.5.
Let ( X n , x n ) , ( Y n , y n ) be two sequences of pointed metricspaces. Let f n : X n → Y n be an admissible sequence of equi-Lipschitz maps.Let ω be a non-principal ultrafilter. Let X ω and Y ω be the ω -limits of ( X n , x n ) and ( Y n , y n ) respectively. Define f = f ω : X ω → Y ω as f (( z n )) = ( f n ( z n )) .Then:a) f is well defined;b) f is Lipschitz with the same constants of the sequence f n .In particular if for any n the map f n is an isometry then f is an isometry,while if f n is an isometric embedding for any n then f is again an isometricembedding. The map f = f ω is called the ω -limit of the sequence of maps f n and wedenote it by f ω = ω - lim f n . The proof in case of isometric embeddings isgiven in [DK18], Lemma 7.47; the general case is analogous.This result can be applied to the special case of geodesic segments,since they are isometric embeddings of an interval into a metric space X .However, we first need to explain what is the ultralimit of a sequence ofintervals: 52 emma A.6. Let I n = [ a n , b n ] ⊂ R be a sequence of intervals containing (possibly with a n = −∞ or b n = + ∞ ). Let ω be a non-principal ultra-filter. Then ω - lim( I n , is isometric to I , where I = [ ω - lim a n , ω - lim b n ] (possibly with a = −∞ or b = + ∞ ) contains 0.Proof. We define a map from I ω to I as follows. Given an admissible sequence ( x n ) such that x n ∈ I n then x n is ω -a.s. bounded, so it is defined ω - lim x n by Lemma A.1. We define the map as ( x n ) ω - lim x n . It is easy to checkit is surjective. Moreover it is an isometry, indeed: | ω - lim x n − ω - lim y n | = ω - lim | x n − y n | = d (( x n ) , ( y n )) . In particular, the limit of geodesic segments is a geodesic segment.
Lemma A.7.
Let ( X n , x n ) be a sequence of pointed metric spaces and let ω be a non-principal ultrafilter. Let X ω be the ultralimit of ( X n , x n ) , andlet z = ( z n ) , w = ( w n ) ∈ X ω . Suppose that for all n there exists a geodesic γ n : [0 , d ( z n , w n )] → X n joining z n and w n : then, there exists a geodesicjoining z and w in X ω . In particular, if X n is a geodesic space for all n ,then the ultralimit X ω is a geodesic space.Proof. We denote by I n the interval [0 , d ( z n , w n )] . Since z and w belongsto X ω then the distance between them is uniformly bounded. Hence fromthe previous lemma it follows that the ultralimit of the spaces ( I n , is I ω = [0 , ω - lim d ( z n , w n )] = [0 , d ( z, w )] . The maps γ n define an admissiblesequence of isometric embedding, so in particular they define a limit isometricembedding γ ω : I ω → X . So γ ω is a geodesic and clearly γ ω (0) = ( γ n (0)) =( z n ) = z and γ ω ( d ( z, w )) = w .In order to prove stability results for classes of metric spaces we also needto establish the convergence of balls under ultralimits: Lemma A.8.
Let ( X n , x n ) be a sequence of geodesic metric spaces and ω bea non-principal ultrafilter. Let X ω be the ultralimit of the sequence ( X n , x n ) .Let y = ( y n ) be a point of X ω . Then for any R ≥ it holds B ( y, R ) = ω - lim B ( y n , R ) . Proof.
First of all ω - lim B ( y n , R ) ⊂ B ( y, R ) . Indeed z = ( z n ) belongs to ω - lim B ( y n , R ) if and only if d ( z n , y n ) ≤ R for all n . Then d ( z, y ) ≤ R ,i.e. z ∈ B ( y, R ) . The next step is to show that the set ω - lim B ( y n , R ) is closed. We take a sequence z k = ( z kn ) of points of ω - lim B ( y n , R ) thatconverges to some point z = ( z n ) of X ω . This implies that d ( y, z ) ≤ R .We consider a geodesic segment of X n between y n and z n and we denote by w n the point along this geodesic at distance exactly R from y n , if it exists.Otherwise z n ∈ B ( y n , R ) and in this case we set w n = z n . We observe that w = ( w n ) ∈ ω - lim B ( y n , R ) by definition. We claim that w = z . In order to53rove that we fix ε > . Then ω -a.s. d ( y n , z n ) < R + ε . This implies that d ( y n , w n ) < ε . Since it holds ω -a.s. then d ( w, z ) < ε . From the arbitrarinessof ε the claim is proved. The last step is to show that the open ball B ( y, R ) is contained in ω - lim B ( y n , R ) . Indeed, given z = ( z n ) ∈ B ( y, R ) , thenthere exists ε > such that d ( z, y ) < R − ε . The set of indices n such that d ( z n , y n ) < d ( z, y ) + ε < R belongs to ω , hence z ∈ ω - lim B ( y n , R ) . Since X ω is geodesic and in any length space the closed ball is the closure of theopen ball the proof is concluded.In general, even if every space X n is uniquely geodesic, the ultralimit X ω may be not uniquely geodesic. This is because, in general, it is nottrue that all the geodesics of X ω are limit of sequences of geodesics of X n .The fact that the geodesics of X ω are actually limit of geodesics of the spaces X n is true when all the X n are CAT ( κ ) . We recall the following fact whichis well known (see [BH13] or [DK18] for instance): Proposition A.9.
Let ( X n , x n ) be a sequence of CAT ( κ ) pointed metricspaces and ω be a non-principal ultrafilter. Then any geodesic of length < D κ in X ω is limit of a sequence of geodesics of X n . As a consequence X ω is a CAT ( κ ) metric space. The main result of the appendix is the following stability property forthe CAT ( κ ) -radius: Corollary A.10.
Let ( X n , x n ) be a sequence of complete, locally geodesicallycomplete, locally CAT ( κ ) , geodesic metric spaces with ρ cat ( X n ) ≥ ρ > .Let ω be a non-principal ultrafilter. Then X ω is a complete, locally geodesi-cally complete, locally CAT ( κ ) , geodesic metric space with ρ cat ( X ω ) ≥ ρ .Proof. Let y = ( y n ) be a point of X ω . For any r < ρ and for any n theball B ( y n , r ) is a CAT ( κ ) metric space. Moreover by Lemma A.8 we havethat B ( y, r ) is the ultralimit of a sequence of CAT ( κ ) metric spaces, henceit is CAT ( κ ) by Proposition A.9. This shows that X ω is locally CAT ( κ ) and ρ cat ( X ω ) ≥ ρ by the arbitrariness of r . Moreover X ω is geodesic byCorollary A.7. We fix now a geodesic segment γ of X ω defined on [ a, b ] .We look at the ball B ( γ ( a ) , ρ ) which is CAT ( κ ) and we take a sequenceof points z n such that ( z n ) = γ ( a ) . The subsegment of γ inside this ball,defined on [ a, a + ρ ) is the limit of a sequence of geodesics γ n inside thecorresponding balls B ( z n , ρ ) , by Proposition A.9. Each γ n can be extendedto a geodesic segment ˜ γ n on the interval ( a − ρ , a + ρ ) since each X n islocally geodesically complete and complete. The ultralimit of the maps ˜ γ n isa geodesic segment defined on [ a − ρ , a + ρ ] which extends γ . We can do thesame around γ ( b ) . This proves that X ω is locally geodesically complete.We conclude the appendix recalling the relations between ultralimits andpointed Gromov-Hausdorff convergence, which we will use in Section 6:54 roposition A.11 (see [Jan17]) . Let ( X n , x n ) be a sequence of proper,length metric spaces and ω be a non-principal ultrafilter. Then:(a) if the ultralimit ( X ω , x ω ) is proper then it is the limit of a convergentsubsequence in the pointed Gromov-Hausdorff sense;(b) reciprocally, if ( X n , x n ) converges to ( X, x ) in the pointed Gromov-Hausdorffsense then for any non-principal ultrafilter ω the ultralimit X ω is isomet-ric to ( X, x ) (we recall that, in this case, ( X, x ) is proper by definitionof Gromov-Hausdorff convergence). eferences [BCGS] Gérard Besson, Gilles Courtois, Sylvestre Gallot, and AndreaSambusetti. Bishop-gromov inequality generalized. In prepara-tion .[BCGS17] Gérard Besson, Gilles Courtois, Sylvestre Gallot, and An-drea Sambusetti. Curvature-free margulis lemma for gromov-hyperbolic spaces. arXiv preprint arXiv:1712.08386 , 2017.[BGT11] Emmanuel Breuillard, Ben Green, and Terence Tao. The struc-ture of approximate groups.
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