Densely related groups

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

YVESCORNULIER ANDADRIENLEBOUDEC

Densely related groups

Tome XXVIII, n^o4 (2019), p. 619-653.

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Densely related groups

Yves Cornulier⁽¹⁾ and Adrien Le Boudec⁽²⁾

ABSTRACT. — We study the class of densely related groups. These are finitely generated (or more generally, compactly generated locally compact) groups satisfying a strong negation of being finitely presented, in the sense that new relations appear at all scales. Here, new relations means relations that do not follow from relations of smaller size. Being densely related is a quasi-isometry invariant among finitely generated groups.

We check that a densely related group has none of its asymptotic cones simply connected. In particular a lacunary hyperbolic group cannot be densely related.

We prove that the Grigorchuk group is densely related. We also show that a finitely generated group that is (infinite locally finite)-by-cyclic and which satisfies a law must be densely related. Given a classCof finitely generated groups, we consider the following dichotomy: every group in C is either finitely presented or densely related. We show that this holds within the class of nilpotent-by-cyclic groups and the class of metabelian groups. In contrast, this dichotomy is no longer true for the class of 3-step solvable groups.

RÉSUMÉ. — On s’intéresse à la classe des groupes densément présentés. Il s’agit des groupes de type fini (ou plus généralement des groupes localement compacts com- pactement engendrés) dans lesquels de nouvelles relations apparaissent à intervalles réguliers. Une relation est dite nouvelle si elle n’est pas conséquence de relations de longueur plus petites. Pour un groupe de type fini, être densément présenté est un invariant de quasi-isométrie.

On vérifie qu’un groupe densément présenté ne peut pas avoir de cône asymp- totique simplement connexe. En particulier un groupe lacunaire hyperbolique n’est jamais densément présenté.

On montre que le groupe de Grigorchuk est densément présenté. On prouve éga- lement que tout groupe de type fini (non virtuellement cyclique) qui est (localement fini)-par-Zet qui satisfait une loi, est densément présenté. Étant donnée une classe Cde groupes de type fini, on considère l’alternative suivante: tout groupe dans C

(*)Reçu le 10 novembre 2016, accepté le 24 mai 2017.

2010Mathematics Subject Classification:20F05, 20F65, 20F69, 20F16, 22D05.

(1) CNRS and Univ Lyon, Univ Claude Bernard Lyon 1, Institut Camille Jordan, 43 blvd du 11 novembre 1918, 69622 Villeurbanne, France — [email protected]

(2) CNRS, Unité de Mathématiques Pures et Appliquées, ENS Lyon, France — [email protected]

The authors were supported by ANR-14-CE25-0004 GAMME. This work was carried out when ALB was a F.R.S.-FNRS Postdoctoral Researcher.

Article proposé par Vincent Guirardel.

est soit finiment présenté, soit densément présenté. On montre que cette alterna- tive est satisfaite par la classe des groupes nilpotents-par-cyclique et la classe des groupes métabéliens. A contrario, cette dichotomie n’est plus vraie pour les groupes résolubles de classe 3.

1. Introduction

IfGis a group andS a generating subset, arelationinGis by definition an element of the kernel of the natural map FS G. A relation w ∈ FS

is said to be generated by a subset R ⊂ FS if w belongs to the normal subgroup ofFS generated byR. Equivalently,wcan be written as a product of conjugates of elements ofR^±1. To every pair (G, S), whereGis a group and S a generating subset, we study the relation range RS(G), which is the set of lengths of relations in G that are not generated by relations of smaller length. Here the length of an element ofF_S refers to the word length associated toS. The setRS(G) is finite precisely when the kernel ofF_S G is normally generated by elements of bounded length. WhenS is finite, this exactly means that the groupGis finitely presented. The relation range was introduced by Bowditch in [9], but has not been explicitly considered since then, except in the small cancelation case [30].

The study of the relation range takes part in a program to find “measures”

of the failure of finite presentability for finitely generated groups. Several approaches, of independent interest, have been carried out so far:

• In [5], several strengthenings of infinite presentability have been in- vestigated, in terms of the study of the poset of normal subgroups contained in the kernel of epimorphisms from finitely presented groups to a given finitely generated group. However, these prop- erties are not known to be quasi-isometry invariant, or are known not to be (see Remark 4.5).

• In [13], to one finitely generated group G, one associates the set of nonprincipal ultrafilters ω such that the asymptotic cone Cone^ω(G,(1/n)) is simply connected. This is obviously a quasi- isometry invariant.

The study of the relation range is in spirit closer to the latter one, but it is not based on asymptotic cones (although we make a connection at some point) and therefore more of combinatorial flavor. The relation range has a distinct behavior than the aforementioned invariant from [13], and may detect properties not seen by this invariant (see Remark 3.6).

We stress out that the relation range of a finitely generated group is not defined in terms of a presentation of the group. Moreover in general there is no connection between the relation range and the set of lengths of relators in a given presentation of the group, even in the case of a minimal presentation (see Remark 3.7).

We should mention that there exist other notions turning out to be strengthenings of infinite presentability, such as the negation of being of ho- mological type FP₂ over a given commutative ring. This is a quasi-isometry invariant; Alonso’s proof from [1] in the case of the ringZcarrying over the general case.

Definition 1.1. — A finitely generated group G is densely related if there existsc >0such that for alln>1, we haveRS(G)∩[n, cn]6=∅. (This does not depend on the finite generating subsetS, see Corollary 1.6.)

These are groups in which new relations (that is, not generated by rela- tions of smaller length) appear at all scales, in a sense precisely defined in the sequel. A typical example of a densely related group is the wreath product Γ =ZoZ, which admits the presentationht, x|[tⁿxt⁻ⁿ, x] = 1 for all n>1i.

By a result of Baumslag [3], this presentation is minimal. The relation [tⁿxt⁻ⁿ, x] is not generated by relations of smaller length, and therefore R_{t,x}(Γ) contains the integer 4n+ 4 for everyn>1.

Recall that Gromov proved that if a finitely generated groupG has all its asymptotic cones simply connected, thenGis finitely presented and has a polynomially bounded Dehn function [21, §5.F]. In turns out that simple connectedness ofoneasymptotic cone ofGalready has consequences, namely that G cannot be densely related (see Proposition 3.16 for a more precise result):

Theorem 1.2 (Corollary 3.17). — Densely related finitely generated groups have no simply connected asymptotic cone.

Any (non-trivial) wreath product is an example of a densely related group (Proposition 4.11). The following construction provides other kind of exam- ples: Let G be a non-Hopfian group, and let ϕ : G → G be a surjective endomorphism ofGwith non-trivial kernel. We denote byK_ϕthe increasing union of the normal subgroups ker(ϕⁿ), n > 1, and by G_ϕ = G/K_ϕ the associated quotient.

Proposition 1.3 (Corollary 4.20). — Let G be a compactly generated locally compact group. Assume that ϕ is a continuous surjective and non- injective endomorphism ofGsuch that Kϕ is a closed subgroup of G. Then the groupGϕ=G/Kϕ is densely related.

Examples of groups covered by Proposition 1.3 are the semidirect prod- uctsZ[1/pq]oZ, where the action is by multiplication byp/q, wherep, q>2 are coprime integers. See Section 4.3 for more examples.

Proposition 1.3 is a particular case of a more general construction inves- tigated in Section 4.3. LetGbe a group with a finite index normal subgroup H, which come with a finite set Φ = {ϕi} of surjective homomorphisms ϕ_i : H → G. Under appropriate assumptions on G, H and Φ (see Sec- tion 4.3), this naturally defines an increasing sequence of normal subgroups K_nCG, so that the groupsG/K_n form a directed system. We show that the associated direct limitG/K_∞is always densely related (Theorem 4.15). An example of group which may be obtained via this construction is the Grig- orchuk groupGintroduced in [17], so that we obtain the following result.

Theorem 1.4. — The Grigorchuk groupGis densely related. In partic- ular no asymptotic cone ofGis simply connected.

Finitely generated groups that are not densely related are calledlacunary presented. Of course this includes finitely presented groups, but here we are especially interested by infinitely presented ones.

The idea of considering sequences of relations of sparse lengths was used by Bowditch in order to provide a continuum of pairwise non-quasi-isometric infinitely presented small cancelation groups [9]. It was then shown by Thomas and Velickovic that this yields the first example of a finitely gener- ated group having non-homeomorphic asymptotic cones (more precisely, one asymptotic cone that is a real tree, and one asymptotic cone that is not sim- ply connected) [31]. This motivated the introduction oflacunary hyperbolic groups (groups with at least one asymptotic cone a real tree) by Olshan- skii, Osin and Sapir [26]. They showed that this class of groups is actually very large. These are examples of groups that are lacunary presented, not all of which being finitely presented. However, the class of lacunary presented groups is much larger (see Remark 3.19).

Given A, B ⊂R+, we write A 4 B if there exists c > 0 such that for every a ∈ A, there exists an element of B in the interval [c⁻¹a, ca]. We write A∼B when A 4B and B 4A, and we say that A, B are at finite multiplicative Hausdorff distance. This defines an equivalence relation for subsets ofR+, which identifies all non-empty finite subsets.

The following result is a particular case of Theorem 2.3, which holds in the more general context of connected graphs. We refer to Section 2.3 for the relevant terminology.

Theorem 1.5. — Let G, SandG⁰, S⁰ be two pairs of groups and gener- ating subsets. If the metric space(G, d_S)is a large-scale Lipschitz retract of (G⁰, d_S⁰), then R_S(G)4R_S⁰(G⁰).

Corollary 1.6(Bowditch). — If the metric spaces(G, d_S)and(G⁰, d_S⁰) are quasi-isometric, thenR_S(G)∼ R_S⁰(G⁰).

Corollary 1.6 implies in particular that ifGis endowed with a (possibly discrete) locally compact topology, and ifS is a compact generating subset, then the setR_S(G) does not depend, up to finite multiplicative Hausdorff distance, on the choice ofS. We denote byR(G) the associated equivalence class, and we call R(G) the relation range of the group G. According to Corollary 1.6, the relation rangeR(G) in an invariant of the quasi-isometry class ofG.

In this setting, we have the following two extreme situations:

• R(G) is finite. WhenGis a finitely generated group, the fact that R(G) is finite exactly means thatGis finitely presented. More gen- erally in the locally compact setting, the groups having a finite re- lation range are precisely the compactly presented groups. For an introduction to compactly presented groups, we refer the reader to the book [12].

• R(G)∼N. By definition the groupsGsuch thatR(G)∼Nare the densely related groups.

It is natural to ask which subsets ofN are, up to multiplicative Haus- dorff distance, the relation range of a finitely generated group. The easy answer is: all of them. Indeed we can start from the above presentation of ZoZand choose an arbitrary subset of relators. Namely, given any subset I⊂Nr{0}, defining ΓI =ht, x|[tⁿxt⁻ⁿ, x] = 1 for alln∈Ii, the relation range of (Γ_I,{t, x}) is equal to{0} ∪(4I+ 4), and every non-empty subset of Nis at finite Hausdorff distance to a subset of this form. As a consequence, the relation range distinguishes continuum many quasi-isometry classes of groups, a fact that was established by Bowditch in [9] (using small cancela- tion rather than the previous presentation). Bowditch’s strategy, combined with work of Olshanskii–Osin–Sapir [26], actually shows that every subset I⊂Nsuch thatINis the relation range of a finitely generated lacunary hyperbolic group (see Section 4.1).

Still, it remains natural to ask the same question within more restricted classes of groups. First recall that ifP,Qare properties, a group is said to be P-by-Qwhen there is a normal subgroup withP whose associated quotient hasQ.

Proposition 1.7. — Every subset of N is (equivalent to) the relation range of a (3-nilpotent locally finite)-by-abelian finitely generated group.

Note in particular that infinitely presented solvable groups can be la- cunary presented. In contrast, non-elementary lacunary hyperbolic groups are never solvable. Indeed, more generally they cannot satisfy a law [16, Cor. 6.13]. Recall that a groupGsatisfies a law if there exists a non-trivial group wordw(x1, . . . , xk) in a free group such thatw(g1, . . . , gk) = 1 for all g1, . . . , gk∈G.

Now given a certain class of groups, we want to study the behavior of the relation range within this class. Proposition 1.7 says that the class of amenable (and even solvable) groups is not yet sufficiently restrictive, whereby the necessity to focus on smaller classes of groups. The following result provides classes of finitely generated groups for which the behavior of the relation range is completely understood.

Theorem 1.8. — Let Gbe a finitely generated group. Assume that one of the following holds:

(1) Gis metabelian, or more generally center-by-metabelian;

(2) Gis nilpotent-by-cyclic.

ThenGis either finitely presented or densely related.

In combination with Theorem 1.2, this implies the following:

Corollary 1.9. — LetGbe as in Theorem 1.8. IfGadmits one simply connected asymptotic cone, thenGis finitely presented.

Recall that it follows from Bieri–Strebel theorem that a (non virtually cyclic) finitely generated groupGthat is (locally finite)-by-cyclic is infinitely presented [7]. By a construction due to Olshanskii, Osin and Sapir [26, §3.5], (locally finite)-by-cyclic groups are not always densely related. Moreover, although we will not go into this direction here, the flexibility of their con- struction suggests that the relation range of (locally finite)-by-cyclic groups may be arbitrary. The following result shows that, under the additional as- sumption that the group satisfies a law, the relation range is forced to be as large as possible.

Theorem 1.10. — Let G=NoZbe a finitely generated group, where N is an infinite locally finite subgroup. IfGsatisfies a law, thenGis densely related. In particularGhas no simply connected asymptotic cone.

The existence of a law in a finitely generated group or in some large enough subgroup was already known to have consequences at the level of

asymptotic cones. For instance ifGsatisfies a law then no asymptotic cone ofGcan be tree-graded in the sense of Drutu and Sapir [16, Cor. 6.13]. Also if Ghas a subgroupH of relative exponential growth that satisfies a law, then no asymptotic cone ofGcan be real tree [26, Th. 3.18 (c)], [23, Th. 1.4]. The idea is that the existence of a law prevents ultraproducts from containing free subgroups, and this has consequences on their possible transitive isometric group actions. The approach of Theorem 1.10 is different in the sense that the existence of a law is interpreted directly at the level of the group (rather than ultraproducts) in order to produce appropriate relations.

A common feature of the proofs of Theorem 1.8 and Theorem 1.10 is the study of the relation range of finitely generated groups with a given homomorphism onto the groupZ. Recall that givenπ:GZ, we say that the action ofZcontracts into a finitely generated subgroup ofGif there is a decomposition ofGas an ascending HNN-extension over a finitely generated group whose associated homomorphism ontoZis equal toπ.

Theorem 1.11. — Let G=M oZbe a finitely generated group satis- fying a law. Then either the action of Zcontracts into a finitely generated subgroup ofM, or the groupGis densely related.

Finally we end this introduction with the following problem. While the class of finitely generated linear groups (that is, isomorphic to a subgroup of GL_n(K) for some n and some field K) contains both finitely presented groups and densely related groups (e.g. the wreath productZoZ), we do not know any other behavior of the relation range within this class of groups.

Question 1.12. — Is it true that every finitely generated linear group is either finitely presented or densely related?

We point out that a positive answer to this question would imply the non-existence of infinitely presented linear lacunary hyperbolic groups, and the existence of such groups was asked by Olshanskii, Osin and Sapir in [26].

We refer to Section 3.4 for details.

Outline

This article is organized as follows. Section 2 introduces the analogue of the relation range for graphs, and contains the proof of Theorem 1.5. In Section 3 we establish preliminary results on the relation range and make the connection with asymptotic cones (Theorem 1.2). Finally the study of the relation range for specific classes of groups is carried out in Section 4, which contains the proofs of all other results stated in the introduction.

Acknowledgments

We are grateful to Goulnara Arzhantseva and Pierre de la Harpe for useful comments, to Romain Tessera for interesting discussions related this work, and to Henry Wilton for pointing out [14]. We also thank Mikhail Ershov for pointing out some inaccuracy in a preliminary version of this article, and the referee for useful suggestions.

2. Relation range of graphs

Following [9], this section introduces the analogue of the relation range in the setting of connected graphs.

2.1. Equivalence relation on subsets of R+

Given two subsetsA, B ofR+, we writeA4Bif there existsc >0 such that there is an element ofB in the interval [c⁻¹a, ca] for everya∈A. The relationA∼B defined byA4B and B4Ais an equivalence relation on the set of subsets ofR₊. Equivalently, we have A ∼B if and only ifA, B are at finite multiplicative Hausdorff distance, i.e. log(A) and log(B) are at finite Hausdorff distance. The class of a subsetA will be denoted [A]. We leave as an exercise the verification that the operation [A]∪[B] = [A∪B] is well defined, and that the relation defined by [A]⊂[B] if there existA⁰∼A andB⁰ ∼B such thatA⁰ ⊂B⁰, is a partial order on the set of equivalence classes.

2.2. Definition for graphs

LetXbe a graph. A pathαof lengthn>0 inX is a sequence of vertices x₀, . . . , x_n such thatx_i andx_i+1are adjacent for everyi= 0, . . . , n−1. We say thatx₀ and x_n are respectively the inital point and the endpoint ofα.

We say thatαis a loop ifx₀=x_n.

Assume now thatXis a connected graph, and choose a base pointx₀∈X. For everyn>0, we consider the subgroupπ₁⁽ⁿ⁾(X, x₀) ofπ₁(X, x₀) generated by loops of the formp⁻¹·α·p, where pis a path with initial pointx0, and αis a loop (based at the endpoint of p) of length at most n. The subgroup π₁⁽ⁿ⁾(X, x0) is normal in π1(X, x0), and clearly π₁⁽ⁿ⁾(X, x0) is a subgroup

of π^(m)₁ (X, x₀) for m > n. Note that π₁(X, x₀) is the increasing union of the subgroupsπ₁⁽ⁿ⁾(X, x₀). We take the convention thatπ₁⁽⁻¹⁾(X, x₀) is the trivial subgroup.

Definition 2.1. — We denote by Φ(X) the set of integersn>0 such thatπ₁⁽ⁿ⁾(X, x0)properly containsπ⁽ⁿ⁻¹⁾₁ (X, x0). This does not depend onx0.

For simplicity we will use the same notation for Φ(X) and its∼-class.

Remark 2.2. — For every integer k>1, the set Φ(X) is at finite multi- plicative Hausdorff distance to the set of integersn such that π₁^(kn)(X, x0) properly containsπ⁽ⁿ⁻¹⁾₁ (X, x0).

2.3. Quasi-isometry invariance

In this paragraph we study the behavior of Φ(X) under large scale Lips- chitz retracts of connected graphs.

A map f : X → Y between two metric spaces is (C, C⁰)-LS-Lipschitz (where LS stands for Large-Scale) if

d(f(x), f(x⁰))6max(Cd(x, x⁰), C⁰) for allx, x⁰ ∈X;

andf is LS-Lipschitz if it is (C, C⁰)-LS-Lipschitz for some constantsC, C⁰. The spaceX is anLS-Lipschitz retract ofY if there are LS-Lipschitz maps X →Y →X whose composite map is at bounded distance to the identity map ofX.

Theorem 2.3. — Let X, Y be connected graphs. Assume that X is an LS-Lipschitz retract ofY. Then [Φ(X)]⊂[Φ(Y)]. In particular, ifX andY are quasi-isometric then[Φ(X)] = [Φ(Y)].

Let us proceed to prove the theorem. It will follow from the more quan- titative Lemma 2.5.

IfX is a metric space and c>0, let Rc(X) be the Rips complex: this is the simplicial complex whose set of vertices isX and there is an-simplex between anyn+ 1 points pairwise at distance 6c. (We only consider the Rips complex as a topological space, when endowed with the usual inductive limit topology.)

Iff :X →Y maps any two points at distance6c to points at distance 6c⁰, then it induces a map ¯f :R_c(X)→ R_c⁰(Y), defined to be equal tof on vertices and extended to be affine on simplices; then ¯f is continuous and f 7→f¯is functorial. In particular:

• iff is (C, C⁰)-LS-Lipschitz then for allc>C⁰/C,f induces a map f¯:Rc(X)→RcC(Y);

• iff has distance6kto the identity, i.e.d(f(x), x)6kfor allx∈X, thend(f(x), f(x⁰))6d(x, x⁰) + 2k, so this defines ¯f fromRc(X) to R_c+2k(X) for all c>0.

Lemma 2.4. — If f has distance 6kto the identity then f¯:R_c(X)→ R_c+2k(X)is homotopic to¯i, whereiis the identity ofX.

Proof. — We write coordinates in an (n+ 1) simplex of vertices (x₀, . . . , x_n) as P∗

iλ_ix_i, whereλ_i∈[0,1] and Pλ_i = 1. Note that when two of the x_iare equal this is meaningful and lies in a smaller simplex. Fort∈[0,1], we defineγ_t:P∗

iλ_ix_i7→P∗

i(1−t)λ_ix_i+P∗

itλ_if(x_i). We need to check this is meaningful. First, if (x_i) forms a simplex, thend(x_i, f(x_j))6d(x_i, x_j) +k andd(f(xi), f(xj))6d(xi, xj) + 2k, hence the xi andf(xj) together form a simplex inRc+2k (possibly there are some equalitiesxi=f(xj)). We also need to check that the definition ofγtmatches between different simplices, and this is the case.

To check continuity, we first observe that by definition of the inductive limit topology, it is enough to check continuity on finite subcomplexes (i.e. on F ×[0,1] where F ranges over finite subcomplexes of Rc(X)). Since such a finite subcomplex maps into a finite subcomplex, this is equivalent to checking the continuity when the simplices are endowed with some standard metric, for instance the Euclidean metric with edges of length 1.

The formula being Lipschitz in all variables (λ_i) andt, the map (t, x)7→

γt(x) is continuous and defines a homotopy between ¯f and ¯i(note that the xi lying on a discrete set, there is no continuity issue at this level).

For two path-connected metric spacesZ1, Z2, a continuous mapf :Z1→ Z2 is said to be π1-injective (respectivelyπ1-surjective) if the induced map f_∗ at the level of fundamental groups is injective (respectively surjective).

Now we fix a base-point onX and we need the following condition onX: there existsc0such that for allc>c0,

the inclusion mapR_c0(X)→R_c(X) isπ₁-surjective. (∗) This is ensured by some coarse geodesic assumptions such as: for everyc>c₀, for any two points at distance6c+ 1 there is a third point at distance 6c from both. This in particular holds withc₀= 1 in a combinatorial connected graph.

We fix another metric spaceY that also satisfies (∗) (with the samec0).

We also fix inverse quasi-isometriesf, gbetweenX and Y. We can assume

(increasingc₀ if necessary) that f : X → Y and g : Y → X are (C, Cc₀)- LS-Lipschitz and (C⁰, c₀)-LS-Lipschitz for some C, C⁰ > 1. We assume all these maps are basepoint-preserving and thatg◦f has distance 6c₀/2 to the identity ofX. (We use nothing on f◦g, hence we just suppose thatX is a QI-retract ofY.)

For a metric space Z, we denote by Ψ(Z) is the set of r such that Rr(Z) → R2r(Z) is not π1-injective. Note that when Z is a combinato- rial connected graph, we have Ψ(Z) ∼Φ(Z), so that the following lemma implies Theorem 2.3.

Lemma 2.5. — There exists r0 > 0 and s>1 such that if r >r0 and R_r(X) → R_2r(X) is not π₁-injective, then R_Cs−1r(Y) → R_2Cr(Y) is not π₁-injective.

In particular, up to finite Hausdorff multiplicative distance, we have the inclusion ofΨ(X)intoΨ(Y).

Proof. — Fix a positive numberqsatisfyingq6min((C⁰C)⁻¹,1/2). Fix r > q⁻¹c0 such thatRr(X)→R2r(X) is notπ1-injective.

Assume by contradiction thatRCqr(Y)→R2Cr(Y) isπ1-injective. Then we have the following diagram of continuous maps, where the horizontal ar- rows are standard inclusions and the top square commutes (for each vertical arrow, a simple inequality shows that it is well-defined).

R_qr(X) −−−−→ R_2r(X)



 y

f¯



 y

f¯

RCqr(Y) −−−−→ R2Cr(Y)



 y^¯g RCC⁰qr(X)

The upper square induces a commutative square of maps between the fundamental groups. Pick a non-trivial element γ ∈ π1(Rr(X)) having a trivial image inπ1(R2r(X)). The assumption (∗) implies thatγcan be chosen to lie inπ1(Rqr(X)).

Composing (right and then down), we see that γ has a trivial image in π1(RCqr(Y)). The injectivity of RCqr(Y)→ R2Cr(Y) then implies that f¯(γ) ∈ π1(RCqr(Y)) is trivial. Again composing by g, it follows that ¯g◦ f¯maps γ to a trivial element in π1(RC⁰Cqr(X)). On the other hand, by Lemma 2.4, ¯g ◦ f¯, as a map from Rqr(X) to R_max(C⁰_Cqr,qr+c0)(X), is homotopic to the identity. Hence the identity maps γ to a trivial element in π1(R_max(C⁰_Cqr,qr+c0)(X)). The upper bound on q implies that

max(C⁰Cqr, qr+c₀)6r. Hence we obtain that the image ofγinπ₁(R_r(X))

is trivial, which is a contradiction.

3. Relation range of a group 3.1. Definition

Let G be a group, and S a generating subset of G, so that we have a short exact sequence

1−→N −→FS−→G−→1,

whereF_S is the free group on S. For everyn>0, we let N_n be the normal subgroup ofF_S generated as such by elements ofN or word length at most n. By definition (N_n) is an increasing sequence of normal subgroups of F_S ascending to the subgroupN.

Definition 3.1. — We denote byRS(G)the set of integersn>0such that Nn properly contains Nn−1. We call RS(G) the relation range of the groupGwith respect to the generating subsetS.

Definition 3.2. — A group Gis said to be:

(1) boundedly presentedoverS if RS(G)is finite;

(2) densely related over S if RS(G) ∼ N. Otherwise G is said to be lacunary presentedoverS.

Recall that the Cayley graph Cay(G, S) of G with respect to S is the graph withGas set of vertices, and (x, y) is an edge if there iss∈S^±1such thaty=xs. Since loops in Cay(G, S) correspond to relations inG, we have the following.

Lemma 3.3. — RS(G) = Φ(Cay(G, S)).

Theorem 2.3 has the following consequence.

Corollary 3.4. — Let G be a compactly generated locally compact group, andS a compact generating subset. Then the∼-class ofRS(G)does not depend onS, and is actually a quasi-isometry invariant of the group G.

More generally [R(H)] ⊂[R(G)] whenever H is a large-scale Lipschitz retract ofG(e.g. a group retract).

Recall that a group retract of G is a subgroup H 6 Gsuch that there exists a homomorphismG→H whose restriction toH is the identity.

Definition 3.5. — IfGis a compactly generated locally compact group, we callR(G) = [RS(G)] the relation rangeof G, which does not depend on the choice of compact generating subsetS. We say thatGisdensely related ifR(G)∼N, and lacunary presentedotherwise.

Remark 3.6. — Following [13], ifGis compactly generated locally com- pact group, letν(G) be the set of non-principal ultrafilters onNsuch that the asymptotic cone Cone^ω(G,(1/n)) is simply connected. This is a quasi- isometry invariant, and its complement ν(G)^c is related in spirit to the re- lation range. However, it behaves differently: there exist finitely presented groups for which ν(G)^c can be non-empty, or even be the set of all non- principal ultrafilters. For instance, any lattice Γ in the 3-dimensional Lie group SOL has ν(Γ) empty (that is, ν(Γ)^c is the set of all ultrafilters), in spite of being finitely presented. Hence if we consider a family of groups (Λi) achieving all possible relation ranges (up to equivalence), then so does the family (Γ×Λi), but all these groups have ν empty and thus ν does not distinguish these groups.

Remark 3.7 (On minimal presentations and relation range). — LetGbe a finitely generated group with a minimal presentationG=hS|Ri, meaning that no relator belongs to the normal subgroup generated by other relators.

Then in general there is no relation between the relation rangeRS(G) and the setLR={|r|S :r∈R}, in the sense that both inclusions [RS(G)]⊂[LR] and [LR]⊂[RS(G)] may fail:

(1) Let H be a finitely generated densely related group having a min- imal presentation H =hS|(r_n)i, and form the group G=H ∗Z.

Since H is a retract of G, the group G remains densely related by Corollary 3.4. On the other hand G admits the presentation G = hS, a|(a^unr_na^−un)i for arbitrary (u_n), and this presentation remains minimal. In particular if (u_n) grows very fast, we have a densely related group with a minimal presentation such thatL_R is not∼-equivalent toN.

(2) Consider the partial presentation of the lamplighter group H=D

t, x

[t^(2n)!xt^−(2n)!, x] = 1 for alln>1E .

Again form the groupG=H∗Z, whose relation range is equivalent to{4(2n)!+4} ∼ {(2n)!}(Lemma 4.8). Denotingrn= [t^(2n)!xt^−(2n)!, x], consider the presentation G = ht, x, a|a^unrna^−un = 1i, where un is chosen so that the length ofa^unrna^−un is equal to (2n+ 3)!

for alln>1. This presentation remains minimal, but the∼-class of LR is not included inside the relation range ofG.

3.2. Relation range of quotients

IfN is a normal subgroup of a groupGandS generatesG, we denote by XS(N, G) the set of integersnsuch that there is an element ofN of length noutside the normal subgroup of G normally generated by elements ofN of length at mostn−1.

Proposition 3.8. — Let 1 → N → G → Q → 1 be a short exact sequence of groups. Assume thatS is a generating subset ofG such thatG is boundedly presented overS. ThenRS(Q) =XS(N, G).

Before giving the proof, let us derive the following consequence:

Corollary 3.9. — Let1→N→G→Q→1be a short exact sequence of groups. IfGis finitely presented, thenR(Q)∼ XS(N, G).

Proposition 3.8 will follow from Lemmas 3.10 and 3.11.

Lemma 3.10. — Let 1→N →G→Q→1 be a short exact sequence of groups, and letS be a generating subset ofG. ThenXS(N, G)⊂ RS(Q).

Proof. — We denote by NQ the kernel of FS → Q. Let x ∈ N be an element of lengthn>1 that is not in the normal closure inGof the set of elements ofNof length at mostn−1. Letw∈FS of lengthnthat represents xinG. Thenw lies inNQ, and we claim wdoes not belong to the normal closure inF_S of the set of elements of N_Q of length at mostn−1.

Argue by contradiction and assume that there is a decompositionw = Qα_ir_iα⁻¹_i , where r_i ∈ N_Q has length at most n−1. If x_i (resp. a_i) is the element of G represented byri (resp. αi), then by pushing the above decomposition inGwe obtainx=Q

aixia⁻¹_i . Contradiction.

The following shows that the converse inclusion in the previous lemma also holds when the extension is boundedly presented.

Lemma 3.11. — Let 1→N →G→Q→1 be a short exact sequence of groups. Assume thatS is a generating subset ofGsuch thatG is boundedly presented overS. ThenRS(Q)⊂ XS(N, G).

Proof. — Again we let N_Q (resp.N_G) be the kernel of F_S → Q (resp.

F_S →G). Letr>1 such thatN_G is normally generated by its elements of length at mostr. Let n>r+ 1 such that there is a word w∈N_Q ⊂F_S of lengthn, and such thatwdoes not belong to the normal closure of elements ofNQ of length at mostn−1. We shall prove thatnbelongs toXS(N, G).

We letxbe the element represented by the word w in G. Then x∈N, and we claim that xdoes not belong to the subgroup normally generated

by elements of N of length at mostn−1. Indeed, assume that in G there is a decompositionx=Qa_ix_ia⁻¹_i , wherea_i ∈Gand x_i ∈N has length at mostn−1. For every i, we letw_i∈F_S representing x_i inGand such that

|w_i|_S6n−1, and we choose α_i∈F_S such thatα_irepresentsa_i inG. Then w(Q

αiwiα⁻¹_i )⁻¹ ∈NG. SinceNG is normally generated by its elements of length at mostr and n−1 >r by assumption, we deduce thatw belongs to the normal closure of the set of elementst∈N_Q such that|t|S 6n−1.

Contradiction.

3.3. Lacunary approximations

Given a locally compact groupG, we call approximation ofGa sequence of continuous surjective homomorphisms with discrete kernelG0 → G1 → . . . and G_i → G such that the obvious diagrams commute, and such that the kernelN ofG₀→Gis equal to the union of kernelsN_iofG₀→G_i, and such that allG_i are compactly presented.

Given such an approximation, fix a compact generating subset ofG₀. For i > 1, let ρ_i ∈ [1,+∞] be the smallest length of an element in N rN_i. Lets_i be the smallest numberssuch that N_i is normally generated by the intersection ofN_iwith thes-ball inG₀; thens_i<∞becauseG_iis compactly presented.

We say that the approximation islacunary ifρ_i/s_i→+∞.

Proposition 3.12. — A compactly generated locally compact group is lacunary presented if and only if it admits a lacunary approximation. More- over, in the discrete case,G0 can be chosen to be free.

Proof. — This is easy. Start from any compactly presented groupG₀such that Gcan be written as a quotient of G₀ with discrete kernelN (we can chooseG0 free in the discrete case); fix a compact generating subset ofG0. First defineHnas the quotient ofG0with the normal closureNnofN∩B(n), whereB(n) is then-ball in G0. ThenN =SNn (ascending union).

Assume now thatGis lacunary presented. Then for every integersc, n₀>

1, we can findn>n0 such that [n, cn] contains no element of the relation range. Therefore, we can extract from (Hn) a subsequence satisfying the required conditions.

Conversely, the existence of a lacunary approximation implies that there is no new relation of size betweensi and ρi and since the ratioρi/si tends to infinity, this implies that the relation range is not at finite multiplicative

distance toN.

Let us provide an application of lacunary approximations to the Hopfian property, extending the recent result by Coulon and Guirardel [14] that lacunary hyperbolic groups are Hopfian. Recall that a group is Hopfian if all its surjective endomorphisms are injective.

Proposition 3.13. — Let G be a finitely generated group with a lacu- nary approximation by Hopfian groups. ThenGis Hopfian.

Proof. — First, we can find an epimorphism from a free groupG₋₁toG₀ and then shift indices to assume thatG₀ is free. Next, letf be a surjective endomorphism ofG. SinceG0 is free, we can lift it to an endomorphism ˆf of G0; then ˆf maps the 1-ball into thek-ball for some k. We can describe Gi as the quotient ofG0 by the normal closure of a certain subsetRiof the s_i-ball. Then ˆf(R_i) belongs to the ks_i-ball and also belongs to the kernel of G0 → G. If i is large enough, say i > i0, so that ρi > ksi, it follows that ˆf(Ri) belongs to the kernel ofG0 →Gi. Thus for i>i0, ˆf factors to an endomorphismfi of Gi, still liftingf. Then sincef is surjective, we can write generators as elements of the image inG, and lift this toGi for, say i>i1>i0. Thusfi is a surjective endomorphism for alli>i1. Since Gi is Hopfian, it follows thatf_i is an automorphism for all i>i₁, and therefore

f is an automorphism as well.

Corollary 3.14 (Coulon–Guirardel [14]). — Finitely generated lacu- nary hyperbolic groups are Hopfian.

Proof. — Sela proved that torsion-free hyperbolic groups are Hopfian [29], and this was more recently extended to arbitrary hyperbolic groups by Re- infeldt and Weidmann in the preprint [28]. Given that finitely generated lacunary hyperbolic groups admit lacunary approximations by hyperbolic groups [26, Th. 3.3], we conclude by Proposition 3.13.

Proposition 3.13 can also be applied beyond the lacunary hyperbolic case.

For instance, those partial finite presentations of ZoZ are easily seen to be residually finite (and hence Hopfian), so the lacunary presented groups obtained in this way are Hopfian groups.

3.4. Asymptotic cones of densely related groups

The following terminology and notation are essentially borrowed from [27]. We let I2 = [0,1]×[0,1] be the unit Euclidean square of dimension two, and denote by∂I2 its boundary. A collection of squaresD1, . . . , Dk is defined to be a partition ofI2 ifDi∩Dj =∂Di∩∂Dj wheneveri6=j, and ifI2 is the union of the squaresDi.

IfX is a geodesic metric space, a loop inX is by definition a continuous map α : ∂I₂ → X, and we freely identify a loop with its image in X. A partitionπofαis a continuous map extendingαto∂D₁∪. . .∪∂D_k, where D₁, . . . , D_k is a partition of I₂. We define the mesh of π as the maximal length of the pathsπ(∂D_i).

Lemma 3.15. — Let X be a geodesic metric space, and denote by C = Cone^ω(X,(xn),(sn))an asymptotic cone ofX. Then

(a) any loop inC is theω-limit of a sequence of loops inX;

(b) ifαis a loop inCwhich is theω-limit of a sequence of loops(αn)in X, then any partition of αis the ω-limit of a sequence of partitions ofα_n.

Proof. — Statement (a) is proved in [22, Prop. 2.2] for paths rather than loops, but the proof can be easily adapted to realize any loop in C as the ω-limit of a sequence of loops inX.

Statement (b) is obtained similarly, working in each square of the parti-

tion.

Recall that by a theorem of Gromov [21, §5.F], if a finitely generated group G has all its asymptotic cones simply connected, then G is finitely presented (and has a polynomially bounded Dehn function). One cannot hope to obtain the same conclusion if we weaken the hypothesis by requir- ing thatone asymptotic cone of Gis simply connected, as for example any non-hyperbolic lacunary hyperbolic group is infinitely presented [26, Appen- dix]. Recall that a group is lacunary hyperbolic if it admits (at least) one asymptotic cone that is a real tree [26].

We next show that ifGhas one asymptotic cone that is simply connected, then the groupGis lacunary presented. The proof follows the same strategy as the proof of the direct implication of Theorem 4.4 in [15], which says that simple connectedness of all asymptotic cones implies a certain division property for loops (see [15] for the relevant definition).

Proposition 3.16. — Let Gbe a compactly generated locally compact group, and(s_n)a sequence of positive numbers. Assume that the asymptotic cone Cone^ω(G,(sn)) is simply connected for some ultrafilter ω such that limωsn= +∞. Then{sn:n∈N} 6⊂ R(G).

Proof. — LetS be a compact generating subset ofG. We argue by con- tradiction, and assume that{sn :n∈N} ⊂ R(G). This implies that there exist a constantc >0 and a sequence of relationsrn∈FS so that

c⁻¹sn 6|rn|S 6csn,

and r_n is not generated by relations of smaller length. By construction, the sequence of loops α_n : ∂I₂ → Cay(G, S) parametrized proportionally to the length associated to r_n yields a loop α : ∂I₂ → Cone^ω(G,(s_n)).

Since Cone^ω(G,(s_n)) is supposed to be simply connected, the map α :

∂I₂ → Cone^ω(G,(s_n)) can be extended to a continuous function σ : I₂ → Cone^ω(G,(sn)). Now sinceI₂is compact, the mapσis uniformly continuous, and there existsη >0 so thatdω(σ(t), σ(u)) is at mostc⁻¹/5 as soon as the distance between t and u is at mostη. Let us consider the partition of I2

given by the net

{(aη, bη) :a, b∈Z,06a, b61/η}.

Since the mesh of this partition is equal to 4η, the restriction of σ to this partition yields a partition of the loopαin Cone^ω(G,(s_n)) of mesh at most 4c⁻¹/5.

From a geometric point of view, the fact thatrn does not belong to the normal subgroup ofFS generated by relations of length at most |rn|S −1 implies that the mesh of any partition of the loopαn is at least|rn|S, so in particular at least c⁻¹sn. Now we claim that this implies that the mesh of any partition of the loopαis at leastc⁻¹. Indeed according to Lemma 3.15, any partitionπof the loopαis theω-limit of a sequence of partitionsπ_n of the loopα_nin Cay(G, S). Being the limit over the ultrafilterωof the mesh of π_n rescaled bys_n, the mesh ofπis at leastc⁻¹by the previous observation.

This readily gives a contradiction with the previous paragraph.

Corollary 3.17. — Let G be a densely related compactly generated group. Then none of the asymptotic cones ofGare simply connected.

Since a real tree is a simply connected metric space, Corollary 3.17 also admits the following consequence, which can also be derived from Propo- sition 3.12 using the combinatorial characterization of lacunary hyperbolic groups from [26] in the discrete case, and from [23] in the locally compact case.

Corollary 3.18. — Any lacunary hyperbolic locally compact group is lacunary presented.

Remark 3.19. — The class of lacunary presented groups is much larger than the class of lacunary hyperbolic groups. For instance, in the discrete case, it contains all finitely presented groups and is stable under direct powers and direct products with finitely presented groups (Corollary 4.9), which are essentially never lacunary hyperbolic (recall that being direct limits of hyperbolic groups, lacunary hyperbolic groups contain no copy ofZ²).

Remark 3.20. — (See also Question 1.12) We do not know any example of a finitely generated linear group that is neither finitely presented nor densely

related. If it is true that every finitely generated infinitely presented linear group is densely related, then this implies according to Corollary 3.18 that every finitely generated lacunary hyperbolic group that is linear is actually a hyperbolic group (because a finitely presented lacunary hyperbolic group is hyperbolic [26, Appendix]). Whether this last assertion is true was asked in [26].

4. Specific classes of groups and examples 4.1. Classes of groups with various relation ranges 4.1.1. Lacunary hyperbolic groups

Recall from Corollary 3.18 that every finitely generated lacunary hy- perbolic group is lacunary presented. The following proposition, the proof of which is a combination of works of Bowditch [9] and Olshanskii–Osin–

Sapir [26], says that this is actually the only restriction on the relation range of a lacunary hyperbolic group.

Proposition 4.1. — For every subset I of N with I N, there exists a lacunary hyperbolic groupΓ withR(Γ)∼I.

Proof. — Upon changingIinto someI⁰∼I, we may easily find a presen- tation Γ =ha, b|(rn)niso that{|rn|:n∈N}=I⁰and satisfying theC⁰(1/7) small cancelation condition. This last property means that{rn :n∈N} is a set of reduced words stable under taking cyclic conjugates, and such that for every n ∈ N and every m 6= n, the largest common prefix of rn and rm has length at most (1/7)|rn|. TheC⁰(1/7) condition implies on the one hand that the relation range of the group Γ is given by the set of lengths of relators [9, Lem. 5], i.e.R(G)∼I⁰ ∼I, and on the other hand that Γ is lacunary hyperbolic sinceI⁰ N [26, Prop. 3.12].

4.1.2. Solvable groups

If Gis a group and ` : G→ R+ is a length function on G, we denote byX`(G) the set of integersn such that there isg ∈Gsuch that`(g) =n andgdoes not belong to the subgroup ofGgenerated by elements of length at mostn−1. Note that ifZ is a central subgroup of a group G andS is a generating subset ofG, then we haveX`S(Z) =XS(Z, G) (see Section 3.2 for the definition ofXS(Z, G)).

The proof of the following lemma is routine, and we omit it.

Lemma 4.2. — If `<sub>1</sub>, `₂ are bi-Lipschitz equivalent length functions on G, then X`1(G)∼ X`1(G).

In the sequel we denote byRthe ringF_p[t, t⁻¹,(t+1)⁻¹], and we consider Abels’ groupA₄(R) consisting of upper triangular matrices of GL₄(R) whose first and last diagonal entries are equal to 1 (see [13]).

Fori>0, we denote by z_i ∈ A₄(R) the element whose diagonal entries are equal to 1, and whose only non-zero off-diagonal entry has coordinate (1,4) and is equal totⁱ. We note that everyz_i lies in the center ofA₄(R). If I⊂N, we letZI =hzii_i∈I. The groupZI is abelian and isomorphic to the groupF^(I)p . Every z∈ZI can be uniquely represented asP

k>0akt^k, where all but finitely many ak ∈ Fp are equal to zero. We consider the length function onZI defined by`(z) = sup{k:ak6= 0}.

Lemma 4.3. — We have X_`(Z_I) =I.

Proof. — We have`(zi) =ifor everyi∈I, and clearlyzidoes not belong to the subgroup generated by thezk fork < i. This showsI ⊂ X`(ZI), and the converse is clear since`:Z_I →R+ take values inI.

Recall that Grigorchuk gave, using growth exponents, a continuum of pairwise non quasi-isometric finitely generated groups of intermediate growth [18]. In view of Corollary 3.4, the following proposition provides a continuum of quasi-isometry classes of finitely generated solvable groups, a fact that was established in [13] using asymptotic cones. This was also obtained in [10] using compression of embeddings intoL^p-spaces.

Proposition 4.4. — LetI⊂N, and letQI be the quotient of the group A4(R)by its central subgroupZI. ThenR(QI)∼I.

Proof. — LetSbe a finite generating subset of the groupA₄(R). Since the groupA₄(R) is finitely presented [13, Th. 5.1], we haveR_S(Q_I) =X_S(Z_I, G) according to Corollary 3.9. Now since ZI is central in A₄(R), if follows that XS(ZI, G) = X`<sub>S</sub>(ZI) = X`(ZI), where the last equality follows from Lemma 4.2 (since the metrics ` and `S are bi-Lipschitz equivalent on ZI, see for instance [13, §4.3]). ThereforeRS(QI) =X`(ZI), and the conclusion

then follows from Lemma 4.3.

Remark 4.5. — This completes a remark in the introduction. In [5], the class of extrinsic condensation group is studied: these are finitely generated groupsGsuch that for every finitely presented group H and surjective ho- momorphismp:H →G, the kernel ofpcontains uncountably many normal subgroups of H. This property is a strengthening of being infinitely pre- sented but is not a quasi-isometry invariant, and is actually not even closed under taking finite index overgroups. For instance, ifBis Abels’ group (de- notedA4/Z in [5, Example 5.13]); here it is rather, for some prime p, the