Fundamental Groups and Limits of Almost Homogeneous Spaces
aa r X i v : . [ m a t h . M G ] S e p Fundamental Groups and Limits of AlmostHomogeneous Spaces.
Sergio [email protected]
Abstract
We say that the sequence of groups Γ n acts almost transitively on asequence of proper metric spaces X n if for every n , there is an isometricdiscrete cocompact action of Γ n on X n such that the diameters ofquotients X n / Γ n converge to 0 as n → ∞ .In such a case, we prove that if the sequence X n consists of lengthspaces and converges in the pointed Gromov–Hausdorff sense to aproper length space X , then X is isometric to a nilpotent locallycompact group equipped with an invariant length metric.Furthermore, assuming X is either finite dimensional or semilo-cally simply connected, we show that it is a Lie group equipped witha Finsler or sub-Finsler metric, and for large enough n , there are sub-groups Λ n ≤ π ( X n ) and surjective morphisms Λ n → π ( X ). Metric spaces with groups of isometries acting transitively have been studiedextensively (c.f. [4], [5], [6], [21], [24]). In here we will focus on the onesobtained as Gromov–Hausdorff limits of proper length spaces with isometrygroups acting discretely and almost transitively.
Definition 1.
Let X be a metric space and Γ a group acting on X byisometries. We say that Γ acts discretely if for all x ∈ X , r >
0, the set { γ ∈ Γ | d ( γx, x ) < r } is finite. Definition 2.
Let ε > X a metric space, and a group Γ acting on X by isometries. We say that Γ acts ε -transitively on X if for any x, y ∈ X ,there is g ∈ Γ such that d ( gx, y ) < ε . Equivalently, an isometric action is ε -transitive if every orbit intersects every open ball of radius ε .1 efinition 3. We say that a sequence of groups Γ n acts almost transitively on a sequence of metric spaces X n if there is a sequence ε n ∈ R + such thatfor every n , Γ n acts isometrically, ε n -transitively and discretely on X n , and ε n → n → ∞ . In such a case, we say that the sequence X n consists of almost homogeneous spaces . Remark 4.
A sequence of homogeneous length spaces X n is in general nota sequence of almost homogeneous spaces because the group actions are notnecessarily discrete. So the sequences of almost homogeneous spaces do notgeneralize the concept of homogeneous spaces. Let ( X n , p n ) be a sequence of proper almost homogeneous length spaces con-verging in the pointed Gromov–Hausdorff sense to a proper length space( X, p ). Our goal is to classify the spaces X that can arise this way and studythe relationship between those spaces and the sequence X n . Example 5.
This situation arises naturally when we take X n to be a se-quence of Galois covers X n → Y n of a sequence Y n of length spaces with diam ( Y n ) →
0. A priori there is no need for the sequence X n to have alimit, but in the particular case when the spaces Y n are closed Riemannianmanifolds of fixed dimension and Ric ≥ −
1, Gromov’s compactness criterionguarantees the existence of a limit space X up to subsequence (c.f. [19], [20]).In the case when the limit space X is a compact Lie group, Alan Turingshowed that it is necessarily a torus (see [30]). Using this result, TsachikGelander studied the case when X is compact. Theorem 6. [12]. Let X n be a sequence of compact almost homogeneouslength spaces converging in the Gromov–Hausdorff sense to a compact lengthspace X . Then X is a torus, i.e. homeomorphic to a finite or infinite productof circles. Corollary 7.
Let X n be a sequence of compact almost homogeneous lengthspaces converging in the Gromov–Hausdorff sense to a space X homeomor-phic to a closed manifold. Then X is homeomorphic to a finite dimensionaltorus. 2hen X is not compact, the situation is more flexible and allows moredegenerate behaviour. Itai Benjamini, Hilary Finucane and Romain Tesserahave worked on this problem using Pansu’s Theorem to obtain an asmptoticresult. Definition 8.
For a metric space Y and a point q ∈ Y , we say that thepointed metric space ( Y , q ) is a partial asymptotic cone of Y if there is asequence λ k ∈ R + with λ k → λ k Y, q ) → ( Y , q ) in the pointed GH sense.If the convergence ( λ k Y, q ) → ( Y , q ) holds for any sequence λ k →
0, we saythat ( Y , q ) is the asymptotic cone of Y . Theorem 9. (Pansu, [26] ). Let G be a group and S a finite set of gener-ators. If G is of polynomial growth, then the asymptotic cone of the Cayleygraph asociated to ( G, S ) exists and it is a simply connected nilpotent Liegroup N equipped with an invariant sub-Finsler metric. Theorem 10. [2]. Let ( X n , p n ) be a sequence of proper almost homogeneouslength spaces converging in the pointed Gromov–Hausdorff sense to a properlength space ( X, p ). Then the asymptotic cone of X exists and it is a sim-ply connected nilpotent Lie group N equipped with an invariant sub-Finslermetric.In this note we show a similar result classifying the possible limit spaces. Theorem 11.
Let ( X n , p n ) be a sequence of proper almost homogeneouslength spaces converging in the pointed Gromov–Hausdorff sense to a properlength space ( X, p ). Then X is isometric to a locally compact nilpotent groupequipped with a left invariant metric.Our main result concerns when X is semilocally simply connected. Theorem 12.
Let ( X n , p n ) be a sequence of proper almost homogeneouslength spaces converging in the pointed Gromov–Hausdorff sense to a properlength space ( X, p ). If X is semilocally simply connected, then X = N/ Λ,where N is a simply connected nilpotent Lie group equipped with an invariantFinsler or sub-Finsler metric and Λ ≤ N is a central discrete subgroup.For large enough n , there are subgroups Λ n ≤ π ( X n , p n ) and surjectivemorphisms Λ n → Λ = π ( X, p ) . (1)3 emark 13. The hypothesis of X being semilocally simply connected can bereplaced by X having finite topological dimension, because of the followingtheorem (solution to Hilbert’s fifth problem) by Deane Montgomery and LeoZippin. Theorem 14. [24]. Let X be a homogeneous proper length space of fi-nite dimension, then X is homeomorphic to a topological manifold, and itsisometry group is a Lie group. π . In this section we focus our attention on Equation 1, a lower semicontinuityof the fundamental group. It was previously known to hold when X wascompact. Theorem 15. [17]. Let Y n be a sequence of compact length spaces converg-ing to a compact semilocally simply connected length space Y . Then forlarge enough n , there are surjective morphisms π ( Y n ) → π ( Y ) . This property is further studied by Christina Sormani and Guofang Weiin [27], [28], [29]. This lower semicontinuity does not hold when X is notsemilocally simply connected, as one can see from the following example. Example 16.
Let Y k := k S be the re-scaled unit circle with its lengthmetric. Then the sequence X n := Q nk =1 Y k converges in the GH sense to X := Q ∞ k =1 Y k ,where the products are taken using the Pythagoras Theorem. On the otherhand, the fundamental groups satisfy π ( X n ) = Z n , and π ( X ) = Z N , so there is no surjective morphism π ( X n ) → π ( X ) for any n .4heorem 12 states that under a symmetry assumption, we can extendTheorem 15 to the case when the limit is not compact. The following exampleshows that the discreteness hypothesis in Theorem 12 is necessary. Example 17.
Let Y be S with its standard metric of length 2 π and Z n be S with the round (bi-invariant) metric of constant curvature 1 /n . Let X n be the quotient ( Y × Z n ) / S where S acts on Y × Z n as follows: z ( w, q ) = ( wz − , zq ) : z, w ∈ S , q ∈ S . Then X n is isometric to S equipped with a re-scaled Berger metric. The se-quence X n consists of simply connected homogeneous spaces, but its pointedGromov–Hausdorff limit is S × R , which is not simply connected. One may also study the existence of a partial limit (
X, p ) given a sequenceof proper length spaces ( X n , p n ) with groups Γ n acting discretely and almosttransitively. If the spaces X n satisfy a uniform Ricci curvature-dimensioncondition (they are all CD ( K, N ) spaces) the limit (
X, p ) exists up to subse-quence (see [23]), and by the Bishop–Gromov inequalty, X has finite dimen-sion, so by Remark 13, Theorem 12 holds. Corollary 18.
Let Y n be a sequence of d -dimensional closed Riemannianmanifolds with Ric ≥ −
1, and diam ( Y n ) →
0. Then, up to subsequence, thesequence X n of universal covers, converges to a simply connected nilpontetLie group equipped with an invariant Riemannian metric. Remark 19.
Vitali Kapovitch, Burkhard Wilking, and independently Em-manuel Breuillard, Ben Green, Terence Tao showed ([20] and [7], respec-tively) that under the hypothesis of Corollary 18, the limit is a nilpotent Liegroup equipped with an invariant Riemannian metric, so the only new partof Corollary 18 is the simple connectedness of the limit.Benjamini, Finucane and Tessera also found a sufficient condition for thispartial limit to exist when the spaces X n are graphs. Theorem 20. [2]. Let D n ≤ ∆ n be two sequences going to infnity, and let X n be a sequence of graphs. Assume there are groups Γ n acting discretely5y isomorphisms and transitively on the vertices of X n . If the balls of radius D n satisfy | B ( x, D n ) | = O ( D qn )for some q >
0, then ( X n , d/ ∆ n ) has a subsequence converging in the pointedGH sense to a nilpotent Lie group equipped with a left invariant sub-Finslermetric.It is not known whether Theorem 20 holds if one removes the assumptionthat the groups Γ n act discretely. All subsequent sections are focused on proving Theorems 11 and 12. Sections3 to 8 contain the proof of Theorem 12, assuming the limit space X , ishomeomorphic to a topological manifold (i.e. Theorem 56), while Section 9reduces the proof of Theorem 12 to that case (i.e. Theorem 95) and provesTheorem 11 in the process.We will exploit the ultrafilter techniques from [7] and a holonomy mapsimiliar to the one in [8], [16].In Section 2, we introduce our notation and the standard theory we willuse. Throughout the proof we will use the ultrafilter language. However, inthis outline we will say “for large enough n ” (FLE n ) for simplicity.In Section 3, by repeated applications of the Generalized Margulis Lemmaof Breuillard, Green, and Tao, together with a local uniform doubling con-dition on X , we obtain FLE n that the groups Γ n have subgroups Γ ′ n withbounded index, having normal subgrops H ′ n ⊳ Γ ′ n with the property that the H ′ n orbit of p n is small, and Γ ′ n /H ′ n is nilpotent with nilpotency step boundedby a number independent of n . We replace Γ n by Γ ′ n and restart the proof.In Section 4, we show that X is a nilpotent Lie group with an invaraintFinsler or sub-Finsler metric, and FLE n , the groups Γ n act in an almosttranslational way in arbitrarily large neighborhoods of p n .In Section 5, we make use of the escape norm from [7] to find FLE n ,normal subgroups H n ≤ Γ n such that the H n orbit of p n is small, so X n and X n /H n are GH close, and Z n := Γ n /H n contain large subsets Y n withoutnontrivial subgroups.This allows us to, in Section 6, use the space X as a model (see [7]) forthe ultralimit Y := lim n → α Y n . This implies that FLE n , there are large nicesubsets P n of Z n (precisely, nilprogressions in C -regular form).6n Section 7, we use the Malcev Embedding Theorem to find FLE n ,groups Γ P n isomorphic to lattices in simply connected Lie groups G n , withisometric actions Φ n : Γ P n → Iso ( X n /H n ) . We notice, using an elementary result in algebraic topology, that the kernelsof those actions are isomorphic to quotients of π ( X n , p n ) . Finally, in section 8, we find, FLE n , subgroups of Ker (Φ n ) isomorphic toΛ = π ( X, p ), all of them isomorphic to central discrete subgroups of simplyconnected nilpotent Lie groups, finishing the proof of Theorem 56.On the other hand, in section 9 we specialize the theory of homogeneousproper length spaces developed by Valerii Berestovskii to our setting, showingTheorems 95 and 11.The author would like to thank Vladimir Finkelshtein, Enrico LeDonne,Adriana Ortiz, Anton Petrunin and Burkhard Wilking for helpful commentsand discussions. This research was supported in part by the InternationalCentre for Theoretical Sciences (ICTS) during a visit for participating in theprogram - Probabilistic Methods in Negative Curvature (Code: ICTS/pmnc2019/03).
For any element g in a group G , we will denote by L g the left shift G → G given by L g ( h ) = gh . If G is abelian, we may denote L g by + g . We say that aset A ⊂ G is symmetric if A = A − and e ∈ A . For subsets A , . . . , A k ⊂ G ,we will denote by A . . . A k the set of all products { a . . . a k | a i ∈ A i } ⊂ G. We will denote by A × . . . × A k the set of all sequences { ( a , . . . , a k ) | a i ∈ A i } ⊂ G k . If A i = A for i = 1 , . . . , k , we will also denote A . . . A k by A k , and A × . . . × A k by A × k . 7or H, K subgroups of a group G , we define the commutator subgroup [ H, K ] to be the group generated by the elements [ h, k ] := h − k − hk with h ∈ H, k ∈ K . Define G (0) as G , and G ( j +1) inductively as G ( j +1) := [ G ( j ) , G ].If G ( s ) = { e } for some s ∈ N , we say that G is nilpotent of step ≤ s .For curves β, γ : [0 , → Y , we denote by β : [0 , → Y the curve givenby β ( t ) = β (1 − t ). And if β (0) = γ (1), we denote by β ∗ γ : [0 , → Y thecuncatenation β ∗ γ ( t ) = ( γ (2 t ) if t ≤ / β (2 t −
1) if t ≥ / . If β (0) = γ (1), we say that β ∗ γ is undefined. We call β ∗ γ the concatenation of β and γ . We will write β ≃ γ whenever β and γ are homotopic relative totheir endpoints.If Y is a length space, we say that a curve is an ε -lasso nailed at β (0) if itis of the form β ∗ γ ∗ β , with β (1) = γ (0), and γ a loop contained in a closedball of radius ε . For a pointed space ( Y, y ), we denote its loop space asΩ(
Y, y ) := { γ : [0 , → Y continuous | γ (0) = γ (1) = y } . In a metric space (
Y, d ), we will denote the closed ball of center q ∈ Y andradius r > B Yd ( q, r ) := { y ∈ Y | d ( y, q ) ≤ r } . We will sometimes omit d or Y and write B ( q, r ) if the metric space we areconsidering is clear from the context. Definition 21.
Let
A, B be metric spaces and f, h : A → B . For a subset C ⊂ A , we define the uniform distance between f and h in C as d U ( f, h, C ) := sup c ∈ C d ( f ( c ) , h ( c )) . Definition 22.
Let
A, B be two metric spaces and f : A → B a function.We define the distortion of f as Dis ( f ) := sup a ,a ∈ A | d ( f ( a ) , f ( a )) − d ( a , a ) | . Proposition 23.
Let f, g, h : A → B , f , g : B → B , and C ⊂ A . Then d U ( f, g, C ) ≤ d U ( f, h, C ) + d U ( h, g, C ) ,d U ( f f, g g, C ) ≤ d U ( f, g, C ) + d U ( f , g , g ( C )) + Dis ( f | f ( C ) ∪ g ( C ) ) . In this section we discuss the ultrafilter tools we will use during the proofof Theorem 12, including metric ultralimits and algebraic ultraproducts. Werefer the reader to [1], [2], [7], [17] for proofs and further discussions.
Definition 24.
Let ℘ ( N ) denote the power set of the natural numbers and α : ℘ ( N ) → { , } be a function. We say that α is a non-principal ultrafilter if it satisfies: • α ( N ) = 1. • α ( A ∪ B ) = α ( A ) + α ( B ) for all disjoint A, B ⊂ N . • α ( F ) = 0 for all finite F ⊂ N .Using Zorn’s Lemma it is not hard to show that nonprincipal ultrafiltersexist. We will choose one ( α ) and fix it for the rest of these notes. Sets A ⊂ N with α ( A ) = 1 are called α -large. For a property P : N → { , } , if α ( P − (1)) = 1 we will often say “ P holds for n sufficiently close to α ”, or“ P holds for n sufficiently α -large”. Definition 25.
Let A n be a sequence of sets. In the product A ′ := ∞ Y n =1 A n , we say that two sequences { a n } , { a ′ n } are α -equivalent if α ( { n | a n = a ′ n } ) = 1 . The set A ′ modulo this equivalence relation is called the algebraic ultraproduct of the sets A n and is denoted by A α := lim n → α A n . A n = R for each n , an element in A α is called a non-standard realnumber . For x = { x n } , y = { y n } nonstandard real numbers, we say that x ≤ y if α ( { n ∈ N | x n ≤ y n } ) = 1. Definition 26.
Let x n be a sequence in a metric space X . We say that thesequence ultraconverges to a point x ∞ if for every ε > α ( { n | d ( x n , x ∞ ) < ε } ) = 1 . The point x ∞ is called the ultralimit of the sequence and we write x n α −→ x ∞ ,or lim n → α x n = x ∞ . Since α is fixed, the ultralimit of a sequence only depends on the topologyof X and not on the particular metric. It is easy to show that if a sequence x n has an ultralimit, then it is unique. Furthermore, if X is compact, thenany sequence in X ultraconverges. Definition 27.
Let ( X n , p n ) be a sequence of pointed metric spaces. Let X ′ α be the set of sequences x n ∈ X n such thatsup n ∈ N d ( x n , p n ) < ∞ . Equip X ′ α with the pseudometric d ( { x n } , { y n } ) := lim n → α d ( x n , y n ) . Let X α be the metric space corresponding to the pseudometric space X ′ α .The pointed metric space ( X α , p α ), where p α is the class of the sequence p n ,is called the metric ultralimit of the sequence ( X n , p n ). It is straightforwardto show that X α is always a complete metric space. Remark 28.
If a sequence of proper metric spaces ( X n , p n ) converges in thepointed GH sense to a proper metric space ( X, p ), then ( X α , p α ) and ( X, p )are isometric. Conversely, if the sequence ( X n , p n ) is precompact in thepointed GH topology, then there is a subsequence that converges to ( X α , p α )in the pointed GH sense. Remark 29.
To define an algebraic ultraproduct A α = lim n → α A n or ametric ultraproduct X α = lim n → α X n , we don’t require the sets A n or X n tobe defined for every n , but only for all n in an α -large set.10 .4 Approximate Isometries, Equivariant Convergence. Definition 30.
We say that f : A → B is an ε -approximation between A and a subset C ⊂ B if Dis ( f ) < ε , and the Hausdorff distance between f ( A )and C is less than ε . Lemma 31.
Let (
Y, q ) be a proper length space and φ n : Y → Y a sequenceof maps satisfying sup n →∞ d ( φ n q, q ) < ∞ . Assume that for every
R > Dis ( φ n | B ( q,R ) ) →
0, and consider the map φ α : Y → Y given by φ α ( y ) := lim n → α φ n ( y ) . Then φ α is an isometry, and for all R > n → α d U ( φ n , φ α , B ( q, R )) = 0 . We will call φ α the ultralimit of the sequence φ n .If we further assume that for every R , φ n | B ( q,R ) are δ n -approximationsbetween B ( q, R ) and B ( φ n ( q ) , R ) for numbers δ n α −→
0, then φ α is surjective. Definition 32.
Let ( X n , p n ), ( Y n , q n ) be two sequences of pointed metricspaces and let φ n , ψ n : X n → Y n be two sequences of maps. We say that φ n is ultraequivalent to ψ n if for every R > n → α d U ( φ n , ψ n , B ( p n , R )) = 0.Let ( X n , p n ) be a sequence of pointed proper metric spaces, converging inthe pointed GH sense to a proper length space ( X, p ). By definition of GHconvergence, there are maps f n : X n → X and h n : X → X n with f n ( p n ) = p and the property that for all ε, R >
0, there is M ∈ N so that for all n > M ,there exists r ≥ R such that f n | B ( p n ,r ) is an ε -approximation between B ( p n , r )and B ( p, r ), and d U ( h n ◦ f n , id, B ( p n , R )) < ε .11et Γ n act by isometries on X n . We say that a sequence g n ∈ Γ n ultra-converges to a map g : X → X if the sequence of maps ( f n ◦ g n ◦ h n ) isultraequivalent to the constant sequence g .We say that a sequence g n ∈ Γ n is stable ifsup n ∈ N d ( g n ( p n ) , p n ) < ∞ . The set Γ α of stable sequences modulo ultraequivalence is called the equiv-ariant ultralimit of the sequence Γ n . Lemma 33.
The map Φ : Γ α → Iso ( X ) that sends a sequence to its ul-tralimit is well defined (doesn’t depend on the representative in the equiva-lence class), and it is injective (that is, if two stable sequences have the sameequivariant ultralimit, then the sequences are ultraequivalent). Furthermore,if the sequence Γ n acts almost transitively on the sequence X n , then Γ α actstransitively on X . Remark 34.
The set Γ α has two equivalent group structures: The one ob-tained by pulling back the group structure in Iso ( X ) through Φ, a ∗ b := Φ − (Φ( a ) ◦ Φ( b )) . And the one given by { g n } ∗ { g ′ n } := { g n ◦ g ′ n } .Let ( Y, q ) be a proper pointed metric space. When we equip the isome-try group
Iso ( Y ) with the compact-open topology, the following family is acompact neighbourhood basis of the identity: U YR,ε := { g ∈ Iso ( Y ) | sup y ∈ B ( q,R ) d ( g ( y ) , y ) < ε } with R, ε ∈ R . This gives Iso ( Y ) the topology of a locally compact Hausdorffgroup. Remark 35.
If we equip
Iso ( X ) with the topology described above, it iseasy to check that the image Φ(Γ α ) is a closed subgroup. Remark 36.
The topologies on Γ α induced by the two following neighbor-hood bases of the identity are equivalent. The one obtained by the ultralimit:ˆ U αR,ε := {{ g n } ∈ Γ α | α ( { n | g n ∈ U X n R,ε } ) = 1 } . The topology inherited from Φ: U αR,ε := Φ − ( U XR,ε ).12 .5 Ultraconvergence of Polynomials.
Definition 37.
Let Q n : R ℓ → R m be a sequence of polynomials of boundeddegree. We say that the sequence converges well to a polynomial Q : R ℓ → R m if the sequence Q n is ultraequivalent to the constant sequence Q . Equiv-alently, the sequence Q n converges well to Q if the sequences of coefficientsof Q n ultraconverge to the corresponding coefficients of Q .Let [ · , · ] n : R m × R m → R m be a sequence of Lie algebra structures in R m . We say that the sequence converges well to a Lie algebra structure[ · , · ] : R m × R m → R m if for each i, j, k ∈ { , . . . , m } , c ki,j ( n ) α −→ c ki,j , where [ e i , e j ] n = m X k =1 c ki,j ( n ) e k , [ e i , e j ] = m X k =1 c ki,j e k . Lemma 38.
Let Q n : R r × R r → R r be a sequence of polynomial groupstructures in R r of bounded degree. Assume Q n converges well to a polyno-mial group structure Q : R r × R r → R r . Then the corresponding sequenceof Lie algebra structures on R r converges well to the Lie algebra structure of Q . Proof.
This follows from the fact that the coefficients c ki,j ( n ) depend contin-uously on the second derivatives of Q n , which by hypothesis, ultraconvergeto the corresponding second derivatives of Q . Lemma 39.
For each d ∈ N , there is N ∈ N such that the followingholds. Let I N := {− , . . . , − N , , N , . . . , } , and assume we have polynomials Q n , Q : R ℓ → R m of degree ≤ d such that Q n ( x ) α −→ Q ( x ) for all x ∈ ( I N ) × ℓ .Then Q n converges well to Q . Proof.
Working on each coordinate, we may assume that m = 1. We proceedby induction on ℓ , the case ℓ = 1 being elementary Lagrange interpolation.Name the variables x , . . . , x ℓ . Since R [ x , . . . , x ℓ ] = ( R [ x ])[ x , . . . , x ℓ ] , we can consider the polynomials Q n , Q as polynomials ˜ Q n , ˜ Q in the variables x , . . . , x ℓ with coefficients in R [ x ]. 13f Q n ( x ) α −→ Q ( x ) for all x ∈ ( I N ) × ℓ , we would have ˜ Q n ( q, x ′ ) α −→ ˜ Q ( q, x ′ )for all q ∈ I N and x ′ ∈ ( I N ) × ( ℓ − . By the induction hypothesis, if N waslarge enough, depending on d , the coefficients of ˜ Q n , which are polynomialsin R [ x ], ultraconverge to the coefficients of ˜ Q whenever x ∈ I N . By thecase ℓ = 1, if N was large enough, the coefficients of Q n ultraconverge tothe coefficients of Q . Definition 40.
For x ∈ R r , we define its support as supp ( x , . . . , x r ) := { i ∈ { , . . . , r }| x i = 0 } . For x, y ∈ R r , we say that x (cid:22) y if i ≤ j for every i ∈ supp ( x ), j ∈ supp ( y ).We say that a polynomial group structure Q : R r × R r → R r is quasilinear if Q ( x, y ) = Q ( x,
0) + Q (0 , y ) = x + y whenever x (cid:22) y .Note that for any quasilinear group structure in R r , the coordinate axesare one-parameter subgroups, and the exponential mapexp : T R r = R r → R r is the identity when restricted to such axes. Lemma 41.
Consider quasilinear polynomial nilpotent group structures Q n , Q : R r × R r → R r of bounded degree. Let log n , log : R r → R r = T R r denote the logarithm maps for the group structures Q n and Q , respectively.Assume the sequence Q n converges well to Q , and a sequence x n ∈ R r ultra-converges to a point x ∈ R r . Thenlim n → α log n ( x n ) = log( x ) . (2) Proof.
Let x n = ( x n, , . . . , x n,r ). By quasilinearity, we havelog n ( x n ) = log n ( x n, e + . . . + x n,r e r )= log n (( x n, e ) . . . ( x n,r e r ))= log n (exp n (log n ( x n, e )) . . . exp n (log n ( x n,r e r )))= log n (exp n ( x n, e ) . . . exp n ( x n,r e r ))By the Baker–Campbell–Hausdorff formula, the last expression is a polyno-mial in the variables x n, , . . . , x n,r with coefficients depending continuouslyon the coefficients c ki,j ( n ). By Lemma 38, Equation 2 is established.14 .6 Constructing Covering Spaces. Let ε > ε -transitively by isometries ona proper length space Z . Lemma 42.
For any z ∈ Z , { g ∈ Γ || gz − z | < ε } generates Γ. Proof.
Let Γ ′ be the subgroup generated by { g ∈ Γ || gz − z | < ε } andassume Γ ′ = Γ. Let M = inf { d ( gz, z ) | g ∈ Γ \ Γ ′ } and let g ∈ Γ \ Γ ′ suchthat d ( g z, z ) = M . Take a minimizing geodesic c : [0 , M ] → Z from g z to z parametrized by arc length. By ε -transitivity, there is g ∈ Γ suchthat d ( g z, c (3 ε/ < ε . It follows that d ( g z, z ) < M , and g ∈ Γ ′ . Also, d ( g − g z, z ) = d ( g z, g z ) < ε and g g − ∈ Γ ′ , contradicting that g ∈ Γ \ Γ ′ .Let z ∈ Z , ρ > ε , and set B := B ( z , ρ ) ,S := { g ∈ Γ || gz − z | ≤ ρ } . Let ˜Γ be the abstract group generated by S , with relations s = s s in ˜Γ, whenever s, s , s ∈ S and s = s s in Γ.Then there is a canonical map Φ : ˜Γ → Γ that extends the embedding
S ֒ → Γ. It is surjective by Lemma 42. Equip ˜Γ with the discrete topology,and consider the topological space˜ Z := (˜Γ × B ) / ∼ ,where ∼ is the minimal equivalence relation such that( gs, x ) ∼ ( g, Φ( s ) x ) if s ∈ S , x, Φ( s ) x ∈ B .There is a well defined contnuous map Ψ : ˜ Z → Z given byΨ( g, x ) := Φ( g )( x ) . Definition 43.
Let ε >
0. We say that a covering map ˜ Y → Y is ε -wide iffor every y ∈ Y , the ball B Y ( y, ε ) is an evenly covered neighborhood of y . Theorem 44. (Monodromy).
The map Ψ is a regular ρ/ Ker (Φ). 15 roof Sketch:
To prove Theorem 44, one has to observe that for every z ∈ B Z ( z , ρ/ z ∈ Ψ − ( z ) has a unique representative of the form( g, z ), with g ∈ Ker
Φ. From there, it is not hard to prove that B Z ( z , ρ/ z . Then the result follows from thefact that Ψ is compatible with the actions of ˜Γ on ˜ Z and Z . Lemma 45.
Let Y be a length space, y ∈ Y , ε >
0, and ∆ a group. Thereis a regular ε -wide covering map ˜ Y → Y with Galois group ∆ if and only ifthere is a surjective map π ( Y, y ) → ∆ whose kernel contains all the classescontaining ε -lassos nailed at y .Lemma 45 follows from the standard construction of covering spaces (see[9] Chapter II, Sections VI-IX). Theorem 46.
Let
A, B be proper length spaces with d GH ( A, B ) ≤ ε/ , and ˜ B → B a regular ε -wide covering map with Galois group ∆. Then thereis a regular covering map ˜ A → A with Galois group ∆. Proof.
There is an ε/ f : A → B . Let p ∈ A and q = f ( a ).Consider π ( B, q ) → ∆ the map from Lemma 45 and let H be its Kernel.Construct a map f ♯ : Ω( A, p ) → Ω( B, q )as follows. For γ ∈ Ω( A, p ), by absolute continuity, there is a partition0 = t < . . . < t k = 1such that γ ([ t i − , t i ]) is contained in a ball of radius ε/
100 for each i ∈{ , . . . , k } . Let σ i be a minimizing path from f ( t i − ) to f ( t i ). Define f ♯ ( γ ) tobe the concatenation σ k ∗ . . . ∗ σ . Define ∧ w ∧ to be the minimal equivalencerelation in Ω( A, p ) so that β ∧ w ∧ γ whenever d U ( β, γ, [0 , ≤ ε . Also define ≡ to be the minimal equivalence relation in Ω( B, q ) so that β ≡ γ whenever d U ( β, γ, [0 , ≤ ε . One needs to verify that 16 β ≃ γ ⇒ β ∧ w ∧ γ ⇒ f ♯ ( β ) ≡ f ♯ ( γ ) ⇒ ([ f ♯ ( β )]) − [ f ♯ ( γ )] ∈ H. • f ♯ ( β ∗ γ ) ≡ f ♯ ( β ) ∗ f ♯ ( γ ) . • For every γ ∈ Ω( B, q ), there is β ∈ Ω( A, p ) such that f ♯ ( β ) ≡ γ . • For any ε/ p , its image under f ♯ is ≡ -equivalent toan ε -lasso centered at q .Each item being an easy exercise in topology. We have a well defined surjec-tive map f ∗ : π ( A, p ) → π ( B, q ) /H = ∆such that every class containing an ε/ ε/ A → A with Galois group ∆. Definition 47.
We say that a continuous map Y → Z between path con-nected topological spaces has no content or has trivial content if the inducedmap π ( Y ) → π ( Z ) is trivial. Otherwise, we say that the map has nontrivialcontent . Lemma 48.
Let G be connected nilpotent Lie group and K a compactsubgroup such that its connected component K is nontrivial. Then theinclusion K → G has nontrivial content. Proof.
Since K is a connected compact nilpotent Lie group, it is homeo-morphic to a torus, and π ( K ) is nontrivial. By the long exact homotopysequence of the fibration G → G/K , we have the exact sequence π ( G/K ) → π ( K ) → π ( G )Since compact subgroups of connected nilpotent Lie groups are central, G/K is a connected nilpotent Lie group, and is therefore aspherical (see [11], Sec-tion 1.2). This means that π ( G/K ) = 0, and the map π ( K ) → π ( G ) isnontrivial. Lemma 49.
Let G , ˜ G be connected Lie groups such that ˜ G is a discreteextension of G (i.e. there is a surjective continuous morphism f : ˜ G → G with discrete kernel). Assume G and ˜ G are equipped with invariant length17etrics such that f is a local isometry. Let δ > B G ( e, δ )contains no nontrivial subgroups, and the inclusion B G ( e, δ ) → G has nocontent. Then B ˜ G ( e, δ ) contains no nontrivial subgroups and the inclusion B ˜ G ( e, δ ) → ˜ G has no content. Proof.
If there is a group H ⊂ B ˜ G ( p, δ ), its image f ( H ) is a subgroup of G contained in B G ( e, δ ), so f ( H ) = { e } ≤ G . If there is a nontrivial element h ∈ H \{ e } , we can take a shortest path γ : [0 , → ˜ G from e to h . Theprojection f ◦ γ would be a noncontractible loop in G contained in B G ( e, δ ),contradicting the hypothesis that B G ( e, δ ) → G has no content. Also, wecan consider the commutative diagram B ˜ G ( e, δ ) ˜ GB G ( e, δ ) G f Since f is a covering map, the right vertical arrow induces an injectivemap at the level of fundamental groups, and the bottom horizontal arrowhas no content by hypothesis. Therefore the top horizontal arrow has trivialcontent as well. Lemma 50.
Let Y be a proper length space, and γ : [0 , → Y be acontinuous curve. Then for every ε > β : [0 , → Y with β (0) = γ (0), β (1) = γ (1), and d U ( γ, β, [0 , ≤ ε. Proof.
Let k > | x − x | ≤ /k implies d ( γ ( x ) , γ ( x )) ≤ ε/ x , x ∈ [0 , β : [0 , → Y to be such that β | [ j/k, ( j +1) /k ] is a constant speed minimizing curve between γ ( j/k ) and γ (( j + 1) /k ) for j = 0 , , . . . , k −
1. It is easy to see that β satisfies the desired properties. Lemma 51.
Let f : Y → Z be a continuous map between proper lengthspaces. Assume that Y is semilocally simply connected, and the composition B ( y, r ) → Y → Z has nontrivial content for some y ∈ Y , r >
0. Thenthere is a loop β : [0 , → Y based at y of length ≤ r such that f ◦ β isnoncontractible in Z . 18 roof. Since Y is semilocally simply connected, there is ε > β , β : [0 , → B ( y, r ) with d U ( β , β , [0 , ≤ ε , arehomotopic to each other in Y . By hypothesis, there is a loop γ : [0 , → Y based at y whose image is contained in B ( y, r ) and such that f ◦ γ isnoncontractible in Z . By Lemma 50, there is a Lipschitz curve γ : [0 , → B ( y, (1 . r ) such that f ◦ γ is noncontractible in Z . Let L be the Lipschitzconstant of γ , and pick ℓ ∈ N with ℓ ≥ L/r . For each j = 0 , . . . , ℓ , set σ j to be a minimizing path from y to γ ( j/ℓ ), and define β j as the concatenation σ j +1 ∗ γ | [ j/ℓ, ( j +1) /ℓ ] ∗ σ j . Since γ is homotopic to the concatenation β ℓ − ∗ . . . ∗ β ,at least one of the curves β j is satisfies that f ◦ β j is noncontractible in Z ,and all of them have length ≤ r . Corollary 52.
Let Y be a proper semilocally simply connected length space.Assume the inclusion B ( y, r ) → Y has nontrivial content for some y ∈ Y , r >
0. Then there is a noncontractible loop based at y of length ≤ r . Proof.
Apply Lemma 51 with f = id Y . Definition 53.
Let f : Y → Y be a map between proper length spaces.We say that f is a metric submersion if for every y ∈ Y , and r >
0, we have f ( B Y ( y, r )) = B Y ( f ( y ) , r ) Lemma 54.
Let Y be a proper length space, and Γ ≤ Iso ( Y ) a closedsubgroup. Then the quotient map f : Y → Y /
Γ is a metric submersion.
Proof.
Let y ∈ Y , r >
0, and z ∈ Y /
Γ with d Y/ Γ ( f ( y ) , z ) ≤ r . Since Γ isclosed, the orbits are closed, and since Y is proper, there is z ∈ f − ( z ) with d ( y, z ) = d ( f ( y ) , z ) ≤ r . This proves f ( B Y ( y, r )) ⊂ B Y/ Γ ( f ( y ) , r ). Theother contention is immediate from the definition of the metric in Y / Γ. Lemma 55.
Let f : Y → Y be a metric submersion between proper lengthspaces, γ : [0 , → Y a Lipschitz curve, and q ∈ f − ( γ (0)). Then there is acurve ˜ γ : [0 , → Y with f ◦ ˜ γ = γ, ˜ γ (0) = q , and length (˜ γ ) = length ( γ ). Proof.
For each j ∈ N , let D j := { , j , . . . , j − j , } , and define h j : D j → Y as follows: Let h j (0) = q , and inductively, let h j ( x + 1 / j ) be a point in f − ( γ ( x + 1 / j )), such that d Y ( h j ( x + 1 / j ) , h j ( x )) = d Y ( γ ( x + 1 / j ) , γ ( x )) , for x ∈ D j \{ } . Using Cantor’s diagonal argument, we can find a subsequence of h j that con-verges for every dyadic rational. Since the maps h j are uniformly Lipschitz,we can extend this map to a Lipschitz map ˜ γ : [0 , → Y . It is easy to checkthat ˜ γ satisfies the desired properties.19 Virtual Nilpotency of Γ n . As stated in the Summary, Sections 3 to 8 focus on proving the followingtheorem, which is a slightly weakened version of Theorem 12.
Theorem 56.
Let ( X n , p n ) be a sequence of proper length spaces convergingin the pointed Gromov–Hausdorff sense to a proper length space ( X, p ). As-sume there is a sequence of groups Γ n acting discretely and almost transitivelyon X n . If X is homeomorphic to a topological manifold, then X = N/ Λ,where N is a simply connected nilpotent Lie group equipped with an invari-ant Finsler or sub-Finsler metric and Λ ≤ N is a central discrete subgroup.For large enough n , there are subgroups Λ n ≤ π ( X n , p n ) and surjectivemorphisms Λ n → Λ = π ( X ) . First, by Lemma 33, the isometry group of X acts transitively. So thefollowing theorem by Berestovskii tells us that its metric is either Finsler orsub-Finsler. Theorem 57. [4], [5]. Let Y be a proper length space whose isometry groupacts transitively. If Y is homeomorphic to a topological manifold, then itsmetric is given by a Finsler or a sub-Finsler structure.Knowing this, we can start extracting information from the groups Γ n . Lemma 58.
Let Y be a manifold equipped with a Finsler or sub-Finslermetric. Assume the group Γ acts on Y transitively by isometries. Thenthere is a constant K > q ∈ Y , r ∈ (0 , γ , . . . , γ m ∈ Γ, m ≤ K such that B ( q, r ) ⊂ m [ j =1 B ( γ j q, r ) . Proof.
By [25], the Hausdorff dimension d of Y is an integer, and the cor-responding Hausdorff measure µ on Y is positive on open sets and finite oncompact sets. Furthermore, there is a constant N > s ∈ (0 , µ ( B ( q, s )) µ ( B ( q, s )) ≤ N . γ , . . . , γ m ∈ Γ such that γ q, . . . , γ m q ∈ B ( q, r )and the family of balls (cid:8) B (cid:0) γ j q, r (cid:1)(cid:9) mj =1 is disjoint. Clearly, m ≤ N , and if the balls { B ( γ j q, r ) } mj =1 do not cover B ( q, r ), there is q ′ ∈ B ( q, r ) withmin j =1 ,...,m d ( q ′ , γ j q ) > r. Choosing γ m +1 sending q to q ′ contradicts the maximality of γ , . . . , γ m .Now our goal is to apply a Generalized Margulis Lemma by Breuillard,Green and Tao to Γ n to find large virtually nilpotent subgroups. Fix δ < A n ( δ ) := { g ∈ Γ n | d ( gp n , p n ) < (0 . δ } . If we have good enough GH approximations between B X n ( p n ,
10) and B X ( p, B X n ( p n , (2 . δ ) can be covered by K translates of B X n ( p n , (0 . δ ),where K comes from applying Lemmma 58 to X . This implies that ( ˇ A n ( δ )) can be covered by K translates of ˇ A n ( δ ). Definition 59.
Let A be a finite symmetric subset of a group and K > A is a K -approximate group if there is a symmetric set W ⊂ A such that | W | ≤ K , A ⊂ W A .Fix M ∈ N . Then by Lemma 42, for large enough n there are symmetricgenerating sets ˇ S n ⊂ Γ n with ˇ S Mn ⊂ ˇ A n ( δ ). Theorem 60. (Margulis Lemma, [7] ) For each
K >
1, there exists M ∈ N such that the following holds. Let A be a K -approximate group in a group G generated by a finite symmetric set S with S M ⊂ A . Then there existsubgroups H ⊳ G ≤ G such that: • [ G : G ] = O K (1). • G /H is nilpotent of nilpotency step O K (1). • A ∩ G generates G and contains H .21 emark 61. Theorem 60 represents one of the most important breakthroughsin modern mathematics. It shows that one can, after quotienting by a finitegroup, control the index and step of nilpotency in Gromov’s theorem ongroups of polynomial growth ([15]). Its proof idea is, on the surface, thesame as Gromov’s original proof. However, the techniques are much morerefined so that it took 30 years to achieve this improvement.Applying Theorem 60 with K = K to ˇ A n ( δ ), we get that for large enough n , there are subgroups ˆ H n ( δ ) ⊳ G n ( δ ) ≤ Γ n such that • [Γ n : G n ( δ )] ≤ I for some I ∈ N independent of n and δ . • G n ( δ ) / ˆ H n ( δ ) is nilpotent of nilpotency step ≤ s for some s ∈ N inde-pendent of n and δ . • ˆ H n ( δ ) ⊂ { g ∈ Γ n | d ( gp n , p n ) < δ } . For each n , let δ n ≥ H ′ n ⊳ Γ ′ n ≤ Γ n such that • [Γ n : Γ ′ n ] ≤ I . • Γ ′ n /H ′ n is nilpotent of nilpotency step ≤ s . • H ′ n ⊂ { g ∈ Γ n | d ( gp n , p n ) ≤ δ n } . By the previous analysis, δ n → n → ∞ . Since [Γ n : Γ ′ n ] ≤ I , thediameters diam ( X n / Γ ′ n ) go to 0 as n → ∞ , so the groups Γ ′ n also act almosttransitively. Remark 62.
The conclussion of Theorem 12 does not involve Γ n , and thegroups Γ ′ n also act almost trnasitively on the sequence X n . This implies thatwe can replace the groups Γ n by the groups Γ ′ n and assume that every g ∈ Γ ( s ) n satisfies d ( gp n , p n ) ≤ δ n . Let Γ α be the equivariant ultralimit of the sequence Γ n . In this section we usetechniques similar to the discrete ones in [8], [16] to identify X with Γ α andshow that for large n , the action of Γ n is almost translational near p n ∈ X n .22 emma 63. Let Γ be a group that acts ε -transitively by isometries on apointed metric space ( Y, q ). Let H ⊳ G be a normal subgroup such that forevery h ∈ H , d ( hq , q ) ≤ ε . Then H ⊂ { g ∈ G | d ( gq, q ) ≤ ε + ε for all q ∈ Y } . Proof.
Let h ∈ H , q ∈ Y . Then by ε -transitivity, there is g ∈ G such that d ( gq, q ) ≤ ε . Since H is normal, ghg − ∈ H , so d ( hq, q ) = d ( hg − gq, g − gq )= d ( ghg − ( gq ) , gq ) ≤ d ( ghg − ( gq ) , ghg − ( q )) + d ( ghg − ( q ) , q ) + d ( q , gq ) ≤ ε + ε . Lemma 64. Γ α is a nilpotent Lie group. Proof.
Since X is a locally compact homogeneous metric space of finite di-mension, by Theorem 14, Iso ( X ) is a Lie group. Since Γ α is a closed subgroupof Iso ( X ), it is also a Lie group.Since Γ n acts ε n -transitively on X n , Γ α acts transitively on X . By Remark62, every g ∈ Γ ( s ) α leaves p invariant. Then by Lemma 63, Γ ( s ) α is trivial.To show that Γ α acts freely on X , we will require the following result byEnrico LeDonne and Alessandro Ottazzi. Lemma 65. [22]. Let G be a nilpotent Lie group equipped with a leftinvariant Finsler or sub-Finsler metric, and f : G → G an isometry. Then f is smooth, and uniquely defined by its first order data at the identity ( f ( e ), df ( e )). Theorem 66. Γ α is connected and acts freely on X . Proof.
Let Γ be the connected component of the identity in Γ α . Since X isconnected, Γ acts transitively on X . For any point q ∈ X , its stabilizerΓ q := { g ∈ Γ | gq = q } is a compact subgroup. Since compact subgroups of connected nilpotent Liegroups are central, and all stabilizers are conjugate, all stabilizers coincide.23n the other hand, the action of Γ is faithful, so the stabilizers are trivial.Therefore we can identify X with Γ and equip X with a nilpotent Lie groupstructure with an invariant Finsler or sub-Finsler metric (see [4]). By Lemma65, the compact subgroup Γ ′ p := { g ∈ Γ α | gp = p } consists of diffeomorphisms, hence there is an inner product h , i in T p X invariant under Γ ′ p . Then for each g ∈ Γ ′ p there is an h , i -orthonormal basis a , b , . . . , a k , b k , c , . . . , c k , d , . . . , d k ∈ T p X and angles θ , . . . , θ k ∈ S \{ } such that d p g ( a j ) = cos θ j a j + sin θ j b j ,d p g ( b j ) = − sin θ j a j + cos θ j b j ,d p g ( c j ) = − c j ,d p g ( d j ) = d j . Assume by contradiction that there exists g ∈ Γ ′ p \{ id } , then again by Lemma65, d p g = id , so k + k >
0. Let d denote the left invariant Riemannianmetric in X given by h , i and consider d U the uniform distance with respectto d . Assume k >
0, then by the Baker–Cambell–Hausdorff formula, forevery ε > δ > d U ( L exp( δa ) , exp ◦ (+ δa ) ◦ exp − , B d ( p, k + k + k ) δ )) < εδ and d U ( g, exp ◦ d p g ◦ exp − , B d ( p, k + k + k ) δ )) < εδ. Since s ≤ k + k + k , by Lemma 23, there is C > s commutators satisfy d U ([ . . . [ L exp( δa ) , g ] , . . . ] , g ] , exp ◦ [ . . . [+ δa , d p g ] , . . . ] , d p g ] ◦ exp − , B d ( p, δ )) < Cεδ. However, by direct computation, d (exp ◦ [ . . . [+ δa , d p g ] , . . . ] , d p g ] ◦ exp − ( p ) , p ) = δ | θ − | s + o ( δ )24o, as δ →
0, [ . . . [ L exp( δa ) , g ] , . . . ] , g ]( p ) = p, contradicting the fact that every step s commutator is trivial. The case k > c instead of a .Note that we have proved the following result. Lemma 67.
Let Y be a proper length space, and Γ ≤ Iso ( Y ) be a closednilpotent subgroup acting transitively which is also a Lie group. Then Γ isconnected and acts freely.By the above Lemmas, the identification Γ α → X given by g → gp is ahomeomorphism. Therefore, X = Γ α = N/ Λ, where N is the universal coverof Γ α equipped with an invariant Finsler or sub-Finsler metric and Λ ≤ N isa discrete central subgroup.We have sequences ε n → R n → ∞ , and maps f n : X n → X, h n : X → X n such that f n ( p n ) = p , f n is an ε n -approximation between B X n ( p n , R n ) and B X ( p, R n ), and d U ( f n ◦ h n , id, B ( p, R n )) < ε n . Definition 68.
Let
R > to be chosen later (Section 8) and defineΘ n := { g ∈ Γ n | d ( gp n , p n ) < R } , Θ ′ n := { g ∈ Γ n | d ( gp n , p n ) < R/ } . Define the translation maps t : Θ n → X and ˆ t : Θ n → Iso ( X ) as t ( g ) := f n ( g ( h n ( p ))) , ˆ t ( g ) := L t ( g ) . The following two lemmas show that the maps ˆ t are approximate mor-phisms from Θ n to Iso ( X ). Lemma 69.
For any R ′ > g n ∈ Θ n ,lim n → α d U ( f n ◦ g n ◦ h n , ˆ t ( g n ) , B ( p, R ′ )) = 0 . roof. By contradiction, assume there is ε >
0, and a sequence g n ∈ Θ n ( R )with lim n → α d U ( f n ◦ g n ◦ h n , L t ( g n ) , B ( p, R ′ )) ≥ ε. By Lemma 66, the equivariant ultralimit g α of g n coincides with L lim n → α t ( g n ) = lim n → α L t ( g n ) . Therefore, by Lemma 31,lim n → α d U ( f n ◦ g n ◦ h n , lim n → α L t ( g n ) , B ( p, R ′ )) = 0 , but also by Lemma 31,lim n → α d U ( L t ( g n ) , lim n → α L t ( g n ) , B ( p, R ′ )) = 0 , conntradicting the triangle inequality of the uniform distance (Proposition23) for α -large enough n . Lemma 70.
For any R ′ > g n , g ′ n ∈ Θ ′ n , we havelim n → α d U (ˆ t ( g n g ′ n ) , ˆ t ( g n )ˆ t ( g ′ n ) , B ( p, R ′ )) = 0 . Proof.
By Proposition 23, d U (ˆ t ( g n g ′ n ) , ˆ t ( g n )ˆ t ( g ′ n ) , B ( p, R ′ ))is bounded above by d U (ˆ t ( g n g ′ n ) , f n ◦ g n ◦ g ′ n ◦ h n , B ( p, R ′ ))+ d U (( f n ◦ g n ) ◦ ( g ′ n ◦ h n ) , ( f n ◦ g n ) ◦ ( h n ◦ f n ) ◦ ( g ′ n ◦ h n ) , B ( p, R ′ ))+ d U (( f n ◦ g n ◦ h n ) ◦ ( f n ◦ g ′ n ◦ h n ) , ˆ t ( g n )ˆ t ( g ′ n ) , B ( p, R ′ )) . Also by Proposition 23, the second summand goes to 0 as n → α , and thethird summand is bounded above by d U ( f n ◦ g ′ n ◦ h n , ˆ t ( g ′ n ) , B ( p, R ′ ))+ d U ( f n ◦ g n ◦ h n , ˆ t ( g n ) , B ( p, R ′ + 2 R ))By Lemma 69, all those summands go to 0 as n → α . Corollary 71.
For any R ′ > g n , g ′ n ∈ Θ ′ n , wehave, lim n → α ˆ t ( g n g ′ n ) = lim n → α ˆ t ( g n )ˆ t ( g ′ n ) = lim n → α ˆ t ( g n ) lim n → α ˆ t ( g ′ n ) . Getting Rid of the Torsion.
Now we want to identify and get rid of the torsion elements of Γ n . Unfortu-nately, for g ∈ Θ n , t ( g ) being close to p , may not imply that high powers of g do not “escape” Θ n . That is because the definition of t : Θ n → X containsan “error” coming from the fact that f n is not an actual isometry. To dealwith the torsion elements, we will need the escape norm from [7].Since X is locally simply connected, there is ε ∈ (0 ,
1) such that everyloop of length ≤ ε is nullhomotopic. Let B be a small open convexsymmetric set in the Lie algebra n of Γ α such thatexp( B ) ⊂ B ( p, ε ) . Define sets A n and S n asˆ A n := { g ∈ Θ n | t ( g ) ∈ exp( B ) } , ˆ S n := { g ∈ Θ n | t ( g ) ∈ exp( B/ K ) } ,A n := ˆ A n ∪ ˆ A − n ,S n := ˆ S n ∪ ˆ S − n , where K ∈ N was obtained from Lemma 58. Definition 72. [7]. Let A be a K -approximate group of a group G . We saythat A is a strong K -approximate group if there is a symmetric set S ⊂ A satisfying the following: • ( { asa − | a ∈ A , s ∈ S } ) K ⊂ A . • If g, g , . . . , g ∈ A , then g ∈ A . • If g, g , . . . , g K ∈ A , then g ∈ S .By the Baker–Campbell–Hausdorff formula, Lemma 69, and Lemma 70,we see that if B was chosen small enough, for n sufficiently close to α , { asa − | a ∈ A n , s ∈ S n } ⊂ S n , and all three conditions of a strong approximate group hold. Lemma 73.
For n sufficiently close to α , the set A n , thanks to S n , is astrong K -approximate group of Γ n . 27 efinition 74. Let A be a subset of a group G . For g ∈ G , we define theescape norm as k g k A := inf (cid:26) m + 1 | id, g, g , . . . , g m ∈ A (cid:27) .In strong approximate groups, the escape norm satisfies really nice prop-erties. Theorem 75.
Let A be a strong K -approximate group. Then for g , g , . . . , g n ∈ A , we have • k g k A = k g − k A . • k g k k A ≤ | k |k g k A • k g g g − k A ≤ k g k A . • k g g · · · g n k A ≤ O K (1) P ni =1 k g i k A . • k [ g , g ] k A ≤ O K (1) k g k A k g k A . Proof.
The first three properties are immediate. We refer the reader to ([7],Section 8) for a proof of the other two properties.Lemma 73 and Theorem 75 imply that for n sufficiently close to α , H n := { g ∈ A n |k g k A n = 0 } is a subgroup of Γ n normalized by A n . By Lemma 42, for n sufficiently closeto α , H n is normal in Γ n and we can form the quotient Z n := Γ n /H n . Remark 76.
For any sequence h n ∈ H n ,lim n → α t ( h n ) = p. Therefore lim n → α d GH ( X n /H n , X n ) = 0 . What is special about the groups Z n is that they don’t have small sub-groups. Let Y n := ˆ π ( A n ), where ˆ π : Γ n → Z n is the standard quotientmap. For [ g ] ∈ Y n \{ e Z n } , we have k g k A n = 0, and therefore g M is not in A n ⊃ A n H n for some M >
0. This implies that ˆ π ( g M ) = [ g ] M does notbelong to Y n . In other words, every nonidentity element in Y n eventually“escapes” from Y n .We still have the map 28 : ˆ π (Θ n ) → X = Γ α given by t ([ g ]) := t ( g ).Of course, to make this map well defined, we have to choose one representa-tive from each class in ˆ π (Θ n ). However, different choices of representativesonly change the value of t by an error which goes to 0 as n → α . Remark 77.
For any pair of sequences g n , g ′ n ∈ Θ n with ˆ π ( g n ) = ˆ π ( g ′ n ) for α -large enough n , there is a sequence w n ∈ H n such that g n = g ′ n w n for α -large enough n , and by Corollary 71,lim n → α ˆ t ( g n ) = lim n → α ˆ t ( g ′ n )ˆ t ( w n ) = lim n → α ˆ t ( g ′ n ) . Remark 78.
By our choice of B , for α -large enough n , and g ∈ Y n , t ( g ) ∈ (exp( B )) ⊂ B X ( p, ε ) ∼ = B N ( id, ε ) , and we can think think of t as a map from Y n to N . We will denote thismap by t N : Y n → N . We refer to ([7], Appendix B) for an introduction to local groups and mul-tiplicative sets. Local K -approximate groups are defined identically as K -approximate groups (Definition 59), but we replace the word group by mul-tiplicative set. Definition 79.
Let A m be a sequence of subsets of multiplicative sets G m .If there is a K > A m are local K -approximate groups for m sufficiently close to α , we say that the algebraic ultraproduct A = lim m → α A m is an ultra approximate group . If for α -large enough m , the approximategroups A m do not contain nontrivial subgroups, we say that A is a NSS (nosmall subgroups) ultra aproximate group.For subsets A ′ m ⊂ A m with the property ( A ′ m ) ⊂ A m , we say that thealgebraic ultraproduct A ′ = Q m → α A ′ m is a sub-ultra approximate group if itis an ultra approximate group, and there is a constant C ∈ N such that A m can be covered by C many translates of A ′ m for m sufficiently close to α .29et Y be the algebraic ultraproduct Q n → α Y n . Consider the map˜ t : Y → X given by the metric ultralimit˜ t ( { g n } ) := lim n → α t ( g n ).Corollary 71 implies that ˜ t is a homomorphism, but moreover, it is a goodmodel . Definition 80. [18], [7]. Let A = Q m → α A m be an ultra approximate group.A good Lie model for A is a connected local Lie group L , together with amorphism σ : A → L satisfying: • There is an open neighborhood U ⊂ L of the identity with U ⊂ σ ( A )and σ − ( U ) ⊂ A . • σ ( A ) is precompact. • For F ⊂ U ⊂ U with F compact and U open, there is an algebraicultraproduct A ′ = Q m → α A ′ m of finite sets A ′ m ⊂ A m with σ − ( F ) ⊂ A ′ ⊂ σ − ( U ). Definition 81.
Let B be a local group, u , u , . . . , u r ∈ B , and N , N , . . . , N r ∈ R + . The set P ( u , . . . , u r ; N , . . . , N r ) is defined as the set of words in the u i ’s and their inverses such that the number of appearances of u i and u − i is not more than N i . We say that P ( u , . . . , u r ; N , . . . , N r ) is well defined ifevery word in it is well defined in B . We call it a progression of rank r . Wesay it is a nilprogression in C -regular form for some C > • For all 1 ≤ i ≤ j ≤ r , and all choices of signs, we have[ u ± i , u ± j ] ∈ P (cid:18) u j +1 , . . . , u r ; CN j +1 N i N j , . . . , CN r N i N j (cid:19) . • The expressions u n . . . u n r r represent distinct elements as n , . . . , n r range over the integers with | n | ≤ N /C, . . . , | n r | ≤ N r /C .30or a nilprogression P in C -regular form, and ε ∈ (0 , P ( u , . . . , u r ; εN , . . . , εN r ) is also a nilprogression in C -regular form.We denote it by εP . We define the thickness of P as the minimum of N , . . . , N r and we denote it by thick ( P ). The set { u n . . . u n r r || n i | ≤ N i /C } is called the grid part of P , and is denoted by G ( P ).Let P m be a sequence of sets. If for α -large enough m , P m is a nilprogres-sion of rank r in C -regular form for some r ∈ N , C >
0, independent of m , wesay that the algebraic ultraproduct P = Q m → α P m is an ultra nilprogressionof rank r in C -regular form . We denote Q m → α εP m as εP . If ( thick ( P m )) m is unbounded, we say that P is a nondegenerate ultra nilprogression . Thealgebraic ultraproduct G ( P ) := Q m → α G ( P m ) is called the grid part of P .The goal of this section is to obtain the following theorem. Theorem 82.
Let A = Q m → α A m be a local NSS ultra approximate group.Assume there is a good Lie model σ : A → L . Then A contains a nonde-generate ultra nilprogression P of rank r := dim ( L ) in C -regular form, withthe property that for all standard ε ∈ (0 , U ε ⊂ L with σ − ( U ε ) ⊂ G ( εP ). Proof.
In the proof of ([7], Theorem 9.3), they construct via the short basistrick of [8], [16] and an induction on dim ( L ), a set P and they prove it isa nondegenerate ultra nilprogression of rank dim ( L ) in C -regular form. Wewill repeat that construction and show that for every standard ε ∈ (0 , U ε ⊂ L with σ − ( U ε ) ⊂ G ( εP ).Let ˆ B be a small convex set in l , the Lie algebra of L . Let A ′ , A ′′ , A ′′′ besub ultra approximate groups of A such that σ − ( exp ( ˆ B )) ⊂ A ′ ⊂ σ − ( exp ((1 . B )) ,σ − ( exp ( δ ˆ B )) ⊂ A ′′ ⊂ σ − ( exp ((1 . δ ˆ B )) ,σ − ( exp ( δ ˆ B/ ⊂ A ′′′ ⊂ σ − ( exp ((1 . δ ˆ B/ , where δ > u ∈ A ′ \{ id } be such that minimizes k u k A ′ (in this setting, k·k A ′ is a nonstandard real number). Let Z = { u n || n | ≤ / k u k A ′ } . The image σ ( Z ) is of the form φ ([ − , φ : [ − , → L , φ ( t ) = exp ( tv ), v ∈ l (see [7], Theorem 9.3).If ˆ B is small enough, then for all x ∈ ( A ′′ ) , k x k A ′′ ≤ δ , and then, byTheorem 75, if δ was chosen small enough, k [ u, x ] k A ′ = O ( k u k A ′ k x k A ′ ) < k u k A ′ for all x ∈ ( A ′′ ) .31ince k u k A ′ was minimal, u commutes with every element in ( A ′′ ) . Thatimplies that Z commutes with everyone in ( A ′′ ) and we can form the quo-tients π : A ′′ → A ′′ /Z and π : A ′′′ → A ′′′ /Z .Also, σ ( Z ) commutes with everyone in exp ( δB ), so by the connectednessof L , σ ( Z ) lies in the center of L . For some open exp (9 δB ) ⊂ U ⊂ exp (10 δB ) , we can form the local quotient ˇ π : U → U/σ ( Z ). Thenˇ πσ : ( A ′′′ /Z ) → U/σ ( Z )is a good Lie model and A ′′ /Z has the NSS property (see [7], Theorem9.3), so we can apply the induction hypothesis and conclude that there isa nondegenerate P ( u , . . . , u r − ; N , . . . , N r − ) in C -regular form such that P ⊂ ( A ′′′ /Z ) ⊂ A ′′ /Z and for every ε >
0, there is an open W ε ⊂ U/σ ( Z )with (ˆ πσ ) − ( W ε ) ⊂ G ( εP ). To properly “lift” P to A ′′ , the following lemmais required. Lemma 83. (Lifting Lemma) . For every w ∈ A ′′ /Z , there is w ′ ∈ A ′′ with π ( w ′ ) = w , and k w ′ k A ′′ = O ( k w k A ′′ /Z ).The proof of the lemma can be found in ([7], Theorem 9.5).Construct P ( u , . . . , u r ; N , . . . , N r ), where u i ∈ A ′′ is a lift of u i thatminimizes k u i k A ′′ for i = 1 , . . . , r − N i := δ N i for i = 1 , . . . , r − u r := u , N r := δ / k u k A ′′ .For some small enough standard δ > P is a nondegenerate ultranilpro-gression in regular form (see [7], Theorem 9.3). We need to check that for all ε >
0, there is an open U ε ⊂ L such that σ − ( U ε ) ⊂ G ( εP ).By contradiction, assume that for some ε >
0, the element x of A ′′ \ G ( εP )with minimal norm k x k A ′′ satisfies σ ( x ) = id L . If that is the case, ˇ π ( σ ( x )) = id L/σ ( Z ) , and by our induction hypothesis, for all standard η > πx ∈ G ( ηP ). Therefore x = u n . . . u n r r , with | n i | ≤ ηN i /C for i = 1 , . . . , r − | n r | ≤ / k u r k A ′ .32lso, using Theorem 75, and the fact that N i = O (1 / k u i k A ′′ /Z ), we get k u n . . . u n r − r − k A ′′ = O r − X i =1 k u n i i k A ′′ ! = O r − X i =1 | n i |k u i k A ′′ ! = O η r − X i =1 N i k u i k A ′′ /Z ! = O ( η ) . Since η was arbitrary, we obtain that k u n . . . u n r − r − k A ′′ is infinitesimal.Using once more Theorem 75, k u n r r k A ′′ = O ( k x k A ′′ + k u n . . . u n r − r − k A ′′ ).This implies that k u n r r k A ′′ is infinitesimal and | n r | = o ( N r ) ≤ εN r /C .Also, since η was arbitrary, | n i | ≤ εN i /C for i = 1 , . . . , r −
1. Therefore x ∈ G ( εP ), which is a contradiction. Remark 84.
Note that from the proof of Theorem 82, the group L is nilpo-tent and the basis { l , . . . , l r } of l given byexp( l i ) = σ (cid:16) u ⌊ N i /C ⌋ i (cid:17) for i = 1 , . . . , r, is a Malcev basis (see [11] for the definition of a Malcev basis).Let r := dim ( X ). Applying Theorem 82 to ˜ t : Y → X , we obtain thefollowing. Proposition 85.
There is
C > , and for each n ∈ N , elements u , . . . , u r ∈ Z n , N ( n ) , . . . , N r ( n ) ∈ N withlim n → α N i ( n ) = ∞ for each i = 1 , . . . , r, satisfying that for every ε ∈ (0 , δ > α -large enough n , P n := P ( u , . . . , u r ; N ( n ) , . . . , N r ( n ))is a nilprogression in C -regular form, and { g ∈ ˆ π (Θ n ) | d ( t ( g ) , p ) < δ } ⊂ G ( εP n ) ⊂ { g ∈ ˆ π (Θ n ) | d ( t ( g ) , p ) < ε } . Malcev Theory.
Let P ( v , . . . , v r ; N , . . . , N r ) be a nilprogression in C -regular form. As-sume N i ≥ C for each i , and set Γ P to be the abstract group generatedby γ , . . . , γ r with relations [ γ i , γ j ] = γ β j +1 i,j j +1 . . . γ β ri,j r whenever i < j , where[ v i , v j ] = v β j +1 i,j j +1 . . . v β ri,j r and | β li,j | ≤ CN l N i N j . We say that P is good if Γ P isisomorphic to a lattice in a simply connected nilpotent Lie group, and eachelement of Γ P has a unique expression of the form γ n . . . γ n r r , with n , . . . , n r ∈ Z . Theorem 86. (Malcev Embedding)
Let r ∈ N , C >
0, and P ( v , . . . , v r ; N , . . . , N r ) a nilprogression in C -regular form in a group Γ. If thick ( P ) islarge enough depending on r, C , then P is good and the map v i → γ i extendsto an embedding ♯ : G ( P ) → Γ P . For A ⊂ G ( P ), we will denote its imageunder this embedding by A ♯ . Furthermore, there is a quasilinear polynomialgroup structure (see Definition 40) Q : R r × R r → R r of degree ≤ d ( r ) such that the multiplication in Γ P is given by γ n . . . γ n r r γ m . . . γ m r r = γ ( Q ( n,m )) . . . γ ( Q ( n,m )) r r for n, m ∈ Z r .Q is called the Malcev polynomial of P , and ( R r , Q ) the Malcev Lie group of P . Γ P is isomorphic, via γ i → e i , to the lattice ( Z r , Q | Z r × Z r ).The proof can be found in ([10], Section 4.2).By Theorem 86, for α -large enough n , the nilprogressions P n are goodwith Malcev polynomials ˆ Q n . Defne the maps t ♭ : G ( P n ) ♯ → X and t ♭N : G ( P n ) ♯ → N as t ♭ ( x ♯ ) = t ( x ) and t ♭N ( x ♯ ) = t N ( x ) . Lemma 87.
Let r ∈ N , C >
1, then there exist M , δ > P of rank r in C -regular form with thick ( P ) > M , we have G ( δ P ) ⊂ G ( P ) . Definition 88.
Let δ > r, C from Proposition85. For n , the Lie algebra of N , let { v , . . . , v r } be the basis such thatexp( v i ) := lim n → α t (cid:18) u j δ Ni ( n ) C k i (cid:19) for each i = 1 , . . . , r. By Remark 84, { v , . . . , v r } is a Malcev basis, and the map ψ : R r → N givenby ψ ( x , . . . , x r ) := exp( x v ) . . . exp( x r v r )is a diffeomorphism. Lemma 89.
The map R r × R r → R r given by( x, y ) → ψ − ( ψ ( x ) ψ ( y ))is a quasilinear polynomial of degree ≤ d ( r ). Proof.
By the Baker–Campbell–Hausdorff formula, after identifying n with R r via the basis { v , . . . , v r } , the map R r × R r → R r given by( x, y ) → exp − ( ψ ( x ) ψ ( y ))is polynomial of degree ≤ r . Also, the map R r → R r given by x → ψ − (exp( x ))is polynomial of degree bounded by a number depending only on r (see [11],Section 1.2). Therefore the composition is also polynomial of degree ≤ d ( r ).Quasilinearity is immediate from the definition.Let N ∈ N be given by Lemma 39 with d ( r ) given by Lemmas 86 and89, and δ > ξ : N → N as ξ ( n ) := N (cid:22) δ nCN (cid:23) . For n ∈ N , consider κ n : R r → R r given by κ n ( x , . . . , x r ) := ( x ξ ( N ( n )) , . . . , x r ξ ( N r ( n ))) . G n be the group ( R r , Q n ), where Q n : R r × R r → R r is the group structuregiven by Q n ( x, y ) := κ − n ( ˆ Q n ( κ n ( x ) , κ n ( y ))) . Note that after identifying Γ P n with Z n , the map κ − n : Γ P n → G n is an injec-tive morphism. For A ⊂ Γ P n , we will denote its image under this embeddingby A ♮ . Also defineΩ := (cid:26) − , . . . , − N , , N , . . . , (cid:27) × r ⊂ R r . Consider the maps ω n : Ω → Γ P n defined as ω n ( x , . . . , x r ) = γ x ξ ( N ( n ))1 . . . γ x r ξ ( N r ( n )) r . Finally define ω α : Ω → N as ω α = ψ | Ω . Consider the following diagram.Ω × Ω ( G ( δ P n ) ♯ ) × G ( P n ) ♯ R r Ω × Ω N × N N R rω n id ∗ t ♭N t ♭N κ − n idω α ∗ ψ − The first row of the diagram is the polynomial Q n , while the second rowis the polynomial Q . Commutativity of the diagram does not hold in general,but it holds in the limit, as the following proposition shows. Proposition 90.
For every x, y ∈ Ω,lim n → α κ − n ( ω n ( x ) ω n ( y )) = ψ − ( ω α ( x ) ω α ( y )) . (3) Proof.
We will first show that for any sequence x ♯n ∈ G ( P n ) ♯ , we havelim n → α κ − n ( x ♯n ) = lim n → α ψ − ( t ♭N ( x ♯n )) . (4)We can decompose the sequence x n = u p n, . . . u p n,r r ∈ G ( P n ) as x n = x n, . . . x n,r , with x n,j = u p n,j j .And by Lemma 71,lim n → α t N ( x n ) = lim n → α t N ( x n, ) . . . lim n → α t N ( x n,r )= exp (cid:18) lim n → α Cp n, δ N ( n ) v (cid:19) . . . exp (cid:18) lim n → α Cp n,r δ N r ( n ) v r (cid:19) .
36n the other hand, by definition,lim n → α κ − n ( x n ) = Cδ lim n → α (cid:18) p n, N ( n ) , . . . , p n,r N r ( n ) (cid:19) . Hence ψ (cid:16) lim n → α κ − n ( x n ) (cid:17) = lim n → α t N ( x n ) , establishing Equation 4. Finally, for x, y ∈ Ω,lim n → α t N ( ω n ( x ) ω n ( y )) = lim n → α t N ( ω n ( x )) lim n → α t N ( ω n ( y ))= ω α ( x ) ω α ( y ) . Combining this with Equation 4, we obtain Equation 3.
Remark 91.
By Proposition 90 and Lemma 39, we see that Q n convergeswell to Q , and by Lemma 38, the corresponding Lie algebras converge wellto n . In this section we finish the proof of Theorem 56 by establishing the followingproposition.
Proposition 92.
For α -large enough n , there are groups Λ n ≤ π ( X n ) andsurjective morphisms Λ n → Λ.Recall that Z n = Γ n /H n . Let η > α -large enough n , D n := { g ∈ Z n | d ( g ( p n H n ) , p n H n ) < η } ⊂ G ( δ P n ) . Let ˜ Z n be the abstract group generated by D n , with relations s = s s ∈ ˜ Z n whenever s, s , s ∈ D n and s = s s in Z n . Remark 93.
By Theorem 86, for α -large enough n , ˜ Z n = Γ P n , and byTheorem 44, there is a regular η/ X ′ n → X n /H n whoseGalois group is the kernel of the canonical map Φ n : Γ P n → Z n .37 emark 94. By Theorem 86,
Ker (Φ n ) ∩ D ♯n = { e Γ Pn } , and for every g ∈ Γ P n with d (Φ n ( g )( p n H n ) , p n H n ) < ε < η, there is w ∈ G ( P n ) ♯ such that d (Φ n ( w )( p n H n ) , p n H n ) < ε, and gw ∈ Ker (Φ n ).Let { λ , . . . , λ ℓ } ֒ → Λ ≤ N be a basis of Λ as a free abelian group. Since N is a simply connected nilpotent Lie group, there is M ∈ N such that the M -th roots of the λ i lie in B N ( p, η/ i ∈ { , . . . , ℓ } pick a sequence λ i ( n ) ∈ G ( P n ) ♯ ⊂ Γ ♮P n ≤ G n = ( R r , Q n )such that lim n → α t ♭N ( λ i ( n )) = λ i M .
By Equation 4, lim n → α λ i ( n ) = ψ − (cid:18) λ i M (cid:19) . Since Q n converges well to Q ,lim n → α ( λ i ( n )) M = ψ − ( λ i ) . Therefore, if R from Definition 68 was chosen large enough,lim n → α t (cid:0) Φ n (cid:0) λ i ( n ) M (cid:1)(cid:1) = lim n → α t (cid:16) (Φ n ( λ i ( n ))) M (cid:17) = (cid:16) lim n → α t (Φ n ( λ i ( n ))) (cid:17) M = (cid:16) lim n → α t ♭ ( λ i ( n )) (cid:17) M = (cid:18) λ i M (cid:19) M = e X . By Remark 94, for α -large enough n , there are w n,i ∈ G ( P n ) ♯ such that λ i ( n ) M w n,i ∈ Ker (Φ n ) , n → α t (Φ n ( w n,i )) = lim n → α t ♭ ( w n,i ) = e X . Since Q n converges well to Q ,lim n → α λ i ( n ) M w n,i = λ i for each i ∈ { , . . . , ℓ } , by Remark 91 and Lemma 41,lim n → α log n ( λ i ( n ) M w n,i ) = log( λ i ) for each i ∈ { , . . . , ℓ } , where log n , log , denote the logarithm maps with respect to Q n and Q , re-spectively. Therefore for α -large enough n , the set { log n ( λ ( n ) M w n, ) , . . . , log n ( λ ℓ ( n ) M w n,ℓ ) } is linearly independent. Also, for i, j ∈ { , . . . , ℓ } ,lim n → α [ λ i ( n ) M w n,i , λ j ( n ) M w n,j ] = [ lim n → α λ i ( n ) M w n,i , lim n → α λ j ( n ) M w n,j ]= λ − i λ − j λ i λ j = e N . By Remark 94, for α -large enough n ,[ λ i ( n ) M w n,i , λ j ( n ) M w n,j ] = e Γ Pn , and the group h λ ( n ) M w n, , . . . , λ ℓ ( n ) M w n,ℓ i ≤ Ker (Φ n )is a free abelian group of rank ℓ . This implies that there is a subgroupˆΛ n ≤ Ker (Φ n ) isomorphic to Λ. By Theorem 46, Remark 76, and Remark93, for α -large enough n , there is a regular covering ˜ X n → X n with Galoisgroup Ker (Φ n ), so there is a surjective map π ( X n , p n ) → Ker (Φ n ) . Letting Λ n ≤ π ( X n , p n ) be the preimage of ˆΛ n , we have surjective mapsΛ n → Λ . Nilpotent Locally Compact Groups of Isome-tries.
The goal of this section is to prove the following theorem, which combinedwith Theorem 56, yields Theorem 12.
Theorem 95.
Let ( X n , p n ) be a sequence of proper length spaces converg-ing in the pointed Gromov–Hausdorff sense to a proper length space ( X, p ).Assume there is a sequence of groups Γ n acting discretely and almost transi-tively on X n . If X is semilocally simply connected, then it is homeomorphicto a topological manifold.We start by proving a Lemma analogous to Lemma 58 in the infinitedimensional setting. Lemma 96.
Let (
Y, q ) be a proper pointed length space, and Γ a groupacting transitively by isometries. Then for every r >
0, there are elements γ , . . . , γ K ∈ Γ, with K depending on r , such that B ( q, r ) ⊂ K [ j =1 B ( γ j q, r ) . Proof.
The collection of balls { B ( γq, r ) } γ ∈ Γ is such that the union of theirinteriors covers B ( q, r ). By compactness of B ( q, r ), finitely many suffice.Fix 0 < δ < A n ( δ ) := { g ∈ Γ n | d ( gp n , p n ) < (0 . δ } . If we have good enough GH approximations between B X n ( p n ,
10) and B X ( p, B X n ( p n , (2 . δ ) can be covered by K ( δ ) translates of B X n ( p n , (0 . δ ),where K ( δ ) comes from applying Lemmma 58 to X . This implies that( ˇ A n ( δ )) can be covered by K translates of ˇ A n ( δ ).For any fixed M ∈ N , by Lemma 42, if n is large enough, there aresymmetric generating sets ˇ S n ⊂ Γ n with ˇ S Mn ⊂ ˇ A n ( δ ). Applying Theorem 60with K = K to ˇ A n ( δ ), we get that for large enough n , there are subgroups G n ( δ ) ≤ Γ n such that • [Γ n : G n ( δ )] ≤ I for some I ( δ ) ∈ N independent of n .40 G n ( δ ) ( s ) ⊂ { g ∈ G n ( δ ) | d ( gp n , p n ) < δ } for some s ( δ ) ∈ N indepen-dent of n .Since [Γ n : G n ( δ )] ≤ I , the squence of groups G n ( δ ) also acts almosttransitively. Since the conclussion of Theorem 95 does not involve the groupsΓ n , we can replace them by the groups G n ( δ ) and assume thatΓ ( s ) n ⊂ { g ∈ Γ n | d ( gp n , p n ) < δ } for all large enough n and some s depending on δ . Remark 97.
Iterating this process with distinct δ , for proving Theorem 95,we can assume that for every k ∈ N , there is s ( k ) ∈ N such that for all largeenough n , we have Γ ( s ) n ⊂ { g ∈ Γ n | d ( gp n , p n ) ≤ / k +1 } . Which, by Lemma 63, implies for large enough n ,Γ ( s ) n ⊂ { g ∈ Γ n | d ( gq, q ) ≤ / k for all q ∈ X n } . Let Γ α be the equivariant ultralimit of the sequence Γ n . We identifyit with a closed subgroup of Iso ( X ) acting transitively. We will now useAndrew Gleason and Hidehiko Yamabe solution of Hilbert’s fifth problem tofind nilpotent Lie groups close to X . Theorem 98. [13], [31]. Let Γ be a locally compact Hausdorff group. Thenthere is an open subgroup G such that for any neighborhood U of the identity,there is a compact normal subgroup H ⊳G with H ⊂ U , and G/H a connectedLie group.Let G ≤ Γ α be the open subgroup given by Theorem 98. By the followingtheorem of Berestovskii, G still acts transitively on X . Theorem 99. [3]. Let Y be a proper length space, and Γ a closed subgroupof Iso ( Y ) acting transitively. If G ≤ Γ is an open subgroup, then G actstransitively as well.Since open subgroups are also closed, G is still a locally compact group.For every k ∈ N , set U k := { g ∈ G | d ( gp, p ) ≤ / k } . H ⊳ G to be a compact subgroup with H ⊂ U , and G/H a Liegroup. We would like to define inductively a sequence of nested compactsubgroups . . . ⊳ H k +1 ⊳ H k ⊳ . . . H ⊳ G such that H k is normal in G , H k ⊂ U k , and G/H k is a Lie group for every k . We can do it thanks to our choice of G and the following theorem byViktor Glushkov. Theorem 100. [14]. Let G be a locally compact group, and H , H be twoclosed normal subgroups such that G/H and G/H are both Lie groups.Then G/ ( H ∩ H ) is also a Lie group.Let ρ k : G → Iso ( X/H k ) be the induced action. Note that Ker ( ρ k +1 ) ≤ Ker ( ρ k ) and Ker ( ρ k ) ⊂ U k for each k , so we can replace H k by Ker ( ρ k ) andassume that G/H k acts faithfully on X/H k for each k .Let G k := G/H k and π k : G → G k be the canonical projection. Since G k is a Lie group, it has a neighborhood W k of the identity that contains nonontrivial subgroups. If ˜ W k := ( π k ) − ( W k ), then every subgroup containedin ˜ W k is automatically contained in H k . If ℓ ∈ N is large enough, { g ∈ G | d ( gq, q ) ≤ / ℓ for all q ∈ X } ⊂ ˜ W k , so by Remark 97, there is s ∈ N , depending on k , such that G ( s ) ⊂ H k ,and G k is nilpotent. By Lemma 67, G k acts freely on X/H k . Lemma 101. G acts freely on X . Proof. If gq = q for some g ∈ G , q ∈ X , then for each k ∈ N , gH k ∈ G k fixesthe class of q in X/H k . Since G k acts freely, g ∈ H k . This holds for every k ,so g ∈ ∞ \ k =1 H k = { e } . Remark 102.
By Lemma 101, we can identify G with X .Let s ∈ N be such that G ( s )1 = { e } . For every k ∈ N , there is a continu-ous surjective morphism G k → G with compact kernel H /H k . This impliesthat G ( s ) k ≤ H /H k , and since compact subgroups of connected nilpotent42ie groups are central, G ( s +1) k = { e } . Therefore, G ( s +1) ≤ H k for all k ∈ N ,and G ( s +1) = { e } , so G is nilpotent and Theorem 11 is thereby proved.To finish the proof of Theorem 95, we need to consider two separate cases.The first case is when for only finitely many k , the connected component of H k /H k +1 is nontrivial. In this case, let k be such that H j /H j +1 is discretefor all j ≥ k . Let δ > B G k ( e, δ ) contains nonontrivial subgroups and B G k ( e, δ ) → G k has no content. By Lemma 49and induction, the balls B G k ( e, δ ) contain no nontrivial subgroups if k ≥ k .For ℓ ∈ N large enough, H ℓ /H ℓ +1 ⊂ { g ∈ G ℓ +1 | d ( g, e ) ≤ / ℓ +1 } ⊂ B G ℓ +1 ( e, δ ) , hence the groups H ℓ are trivial. This implies that G = G ℓ for all suffi-ciently large ℓ , and G is a Lie group.We are left to the case when for infinitely many k , the connected compo-nents of the groups H k /H k +1 are nontrivial. We will assume this, which willultimately contradict the semilocal simple connectedness of G .Since G is semilocally simply connected, there is δ > B G ( e, δ ) → G has no content. By our assumption, there is k ∈ N with 1 / k − ≤ δ , and the connected component of H k − /H k is nontrivial.By Lemma 48, there is a nontrivial loop in G k based at e whose image iscontained in B G k ( e, / k − ). Our contradiction will arise from lifting thisloop to a nontrivial small loop in G .By Lemma 52, there is a noncontractible loop γ : [0 , → G k based at e with length ( γ ) ≤ / k − . Since G k is semilocally simply connected, there is ε > G k whose uniform distance to eachother is less than ε are automatically homotopic to each other.Choose ℓ ∈ N large enough so that 1 / ℓ − ≤ ε and let f : G ℓ → G k be the natural map, which by Lemma 54 is a metric submersion. UsingLemma 55, we get a curve ˜ γ : [0 , → G ℓ with ˜ γ (0) = e , f ◦ ˜ γ = γ , and length (˜ γ ) ≤ / k − . Since f (˜ γ (1)) = e , we have ˜ γ (1) ∈ H k /H ℓ . For each m ∈ N , define the curve β m : [0 , → G ℓ as β m ( t ) := ˜ γ (1) m − ˜ γ ( t ). Observethat β m (1) = β m +1 (0), so we can define the curves ˜ γ m as β m ∗ . . . ∗ β .Let m be the smallest integer such that ˜ γ m (1) lies in the connectedcomponent of the identity of H k /H ℓ , and let β : [0 , → H k /H ℓ be a (notnecessarily rectifiable) curve from ˜ γ m (1) to e . Then the curve β ∗ ˜ γ m is acurve whose image lies in B G ℓ ( e, / k − ), and the composition f ◦ ( β ∗ ˜ γ m ) ishomotopic to γ ∗ . . . ∗ γ ( m times). Since π ( G k ) has no torsion, f ◦ ( β ∗ ˜ γ m )43s noncontractible in G k . By Lemma 51, there is a loop ˜ γ in G ℓ based at e with length (˜ γ ) ≤ / k − , such that f ◦ ˜ γ is noncontractible in G k .By Lemma 55, there is a curve c : [0 , → B G ( e, / k − ) with c (0) = e , c (1) ∈ H ℓ , and π k ◦ c noncontractible in G k . Take a minimizing path c from c (1) to e ∈ G . Then the curve c := c ∗ c is a loop in G satisfying that c ([0 , ⊂ B G ( e, δ/ d U ( π k ◦ c, π k ◦ ˜ c , [0 , ≤ / ℓ − for some suitable reparametrization ˜ c of c . This implies that π k ◦ c is non-contractible in G k , and the composition B G ( e, δ ) → G → G k has nontrivialcontent, which is a contradiction. References [1] Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin. Alexan-drov geometry.
Book in preparation , 2017.[2] Itai Benjamini, Hilary Finucane, and Romain Tessera. On the scal-ing limit of finite vertex transitive graphs with large diameter. arXivpreprint arXiv:1203.5624 , 2012.[3] VN Berestovskii. Homogeneous buseman g-spaces.
Siberian Mathemat-ical Journal , 23(2):141–150, 1982.[4] VN Berestovskii. Homogeneous manifolds with intrinsic metric. i.
Siberian Mathematical Journal , 29(6):887–897, 1988.[5] VN Berestovskii. Homogeneous manifolds with intrinsic metric. ii.
Siberian Mathematical Journal , 30(2):180–191, 1989.[6] VN Berestovskii. Locally compact homogeneous spaces with inner met-ric.
Journal of Generalized Lie Theory and Applications , 9(1), 2015.[7] Emmanuel Breuillard, Ben Green, and Terence Tao. The structure ofapproximate groups.
Publications math´ematiques de l’IH ´ES , 116(1):115–221, 2012.[8] P Buser and H Karcher. Gromov’s almost flat manifolds. soc. math. defrance.
Asth´erisque Fascicules , 1981.449] Claude Chevalley.
Theory of Lie groups . Courier Dover Publications,2018.[10] Anthony E Clement, Stephen Majewicz, and Marcos Zyman.
The The-ory of Nilpotent Groups , volume 43. Springer, 2017.[11] Laurence Corwin and Frederick P Greenleaf.
Representations of nilpo-tent Lie groups and their applications: Volume 1, Part 1, Basic theoryand examples , volume 18. Cambridge university press, 2004.[12] Tsachik Gelander. A metric version of the jordan–turing theorem. Tech-nical report, 2012.[13] Andrew M Gleason. On the structure of locally compact groups.
Pro-ceedings of the National Academy of Sciences of the United States ofAmerica , 35(7):384, 1949.[14] Viktor Mikhaiovich Glushkov. Structure of locally bicompact groupsand hilbert’s fifth problem.
Uspekhi Matematicheskikh Nauk , 12(2):3–41, 1957.[15] Mikhael Gromov. Groups of polynomial growth and expanding maps.
Publications Math´ematiques de l’Institut des Hautes ´Etudes Scien-tifiques , 53(1):53–78, 1981.[16] Mikhail Gromov. Almost flat manifolds.
Journal of Differential Geom-etry , 13(2):231–241, 1978.[17] Mikhail Gromov.
Metric structures for Riemannian and non-Riemannian spaces . Springer Science & Business Media, 2007.[18] Ehud Hrushovski. Stable group theory and approximate subgroups.
Journal of the American Mathematical Society , 25(1):189–243, 2012.[19] Vitali Kapovitch, Anton Petrunin, and Wilderich Tuschmann. Nilpo-tency, almost nonnegative curvature, and the gradient flow on alexan-drov spaces.
Annals of Mathematics , pages 343–373, 2010.[20] Vitali Kapovitch and Burkhard Wilking. Structure of fundamentalgroups of manifolds with ricci curvature bounded below. arXiv preprintarXiv:1105.5955 , 2011. 4521] Enrico Le Donne. A primer on carnot groups: homogenous groups,carnot-carath´eodory spaces, and regularity of their isometries.
Analysisand Geometry in Metric Spaces , 5(1):116–137, 2018.[22] Enrico Le Donne and Alessandro Ottazzi. Isometries of carnot groupsand sub-finsler homogeneous manifolds.
The Journal of Geometric Anal-ysis , 26(1):330–345, 2016.[23] John Lott and C´edric Villani. Ricci curvature for metric-measure spacesvia optimal transport.
Annals of Mathematics , pages 903–991, 2009.[24] Deane Montgomery and Leo Zippin.
Topological transformation groups .Courier Dover Publications, 2018.[25] Alexander Nagel, Elias M Stein, and Stephen Wainger. Balls and metricsdefined by vector fields i: Basic properties.
Acta Mathematica , 155:103–147, 1985.[26] Pierre Pansu. Croissance des boules et des g´eod´esiques ferm´ees dans lesnilvari´et´es.
Ergodic Theory and Dynamical Systems , 3(3):415–445, 1983.[27] Christina Sormani and Guofang Wei. Hausdorff convergence and uni-versal covers.
Transactions of the American Mathematical Society ,353(9):3585–3602, 2001.[28] Christina Sormani and Guofang Wei. The covering spectrum of a com-pact length space.
Journal of Differential Geometry , 67(1):35–77, 2004.[29] Christina Sormani and Guofang Wei. Universal covers for hausdorff lim-its of noncompact spaces.
Transactions of the American MathematicalSociety , 356(3):1233–1270, 2004.[30] Alan M Turing. Finite approximations to lie groups.
Annals of Mathe-matics , pages 105–111, 1938.[31] Hidehiko Yamabe. A generalization of a theorem of gleason.