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By work of Belyĭ [2] , the absolute Galois group G Q = Gal ( Q ‾ / Q ) of the field Q of rational numbers can be embedded into A = Aut ( F ˆ 2 ) , the automorphism group of the free profinite group F ˆ 2 on two generators. The image of G Q lies inside G T ˆ , the Grothendieck-Teichmuller group. While it is known that every abelian representation of G Q can be extended to G T ˆ , Lochak and Schneps [13] put forward the challenge of constructing irreducible non-abelian representations of G T ˆ . We do this virtually, namely by showing that a rich class of arithmetically defined representations of G Q can be extended to finite index subgroups of G T ˆ . This is achieved, in fact, by extending these representations all the way to finite index subgroups of A = Aut ( F ˆ 2 ) . We do this by developing a profinite version of the work of Grunewald and Lubotzky [7] , which provided a rich collection of representations for the discrete group Aut ( F d ) .

A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled., Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest Vinberg

The congruence subgroup problem for a finitely generated group Γ and for G ≤ Aut ⁢ ( Γ ) G\leq\mathrm{Aut}(\Gamma) asks whether the map G ^ → Aut ⁢ ( Γ ^ ) \hat{G}\to\mathrm{Aut}(\hat{\Gamma}) is injective, or more generally, what its kernel C ⁢ ( G , Γ ) C(G,\Gamma) is. Here X ^ \hat{X} denotes the profinite completion of 𝑋. In the case G = Aut ⁢ ( Γ ) G=\mathrm{Aut}(\Gamma) , we write C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) C(\Gamma)=C(\mathrm{Aut}(\Gamma),\Gamma) . Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] \bar{\Gamma}=\Gamma/[\Gamma,\Gamma] , and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) ≅ Z ( d ) \Gamma^{*}=\bar{\Gamma}/\mathrm{tor}(\bar{\Gamma})\cong\mathbb{Z}^{(d)} . Define Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) . \mathrm{Aut}^{*}(\Gamma)=\operatorname{Im}(\mathrm{Aut}(\Gamma)\to\mathrm{Aut}(\Gamma^{*}))\leq\mathrm{GL}_{d}(\mathbb{Z}). In this paper we show that, when Γ is nilpotent, there is a canonical isomorphism C ⁢ ( Γ ) ≃ C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) . C(\Gamma)\simeq C(\mathrm{Aut}^{*}(\Gamma),\Gamma^{*}). In other words, C ⁢ ( Γ ) C(\Gamma) is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut * ⁢ ( Γ ) \mathrm{Aut}^{*}(\Gamma) . In particular, in the case where Γ = Ψ n , c \Gamma=\Psi_{n,c} is a finitely generated free nilpotent group of class 𝑐 on 𝑛 elements, we get that C ⁢ ( Ψ n , c ) = C ⁢ ( Z ( n ) ) = { e } C(\Psi_{n,c})=C(\mathbb{Z}^{(n)})=\{e\} whenever n ≥ 3 n\geq 3 , and C ⁢ ( Ψ 2 , c ) = C ⁢ ( Z ( 2 ) ) = F ^ ω C(\Psi_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_{\omega} is the free profinite group on countable number of generators.

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

We initiate the study of property testing problems concerning relations between permutations. In such problems, the input is a tuple $(\sigma_1,\dotsc,\sigma_d)$ of permutations on $\{1,\dotsc,n\}$, and one wishes to determine whether this tuple satisfies a certain system of relations $E$, or is far from every tuple that satisfies $E$. If this computational problem can be solved by querying only a small number of entries of the given permutations, we say that $E$ is testable. For example, when $d=2$ and $E$ consists of the single relation $\mathsf{XY=YX}$, this corresponds to testing whether $\sigma_1\sigma_2=\sigma_2\sigma_1$, where $\sigma_1\sigma_2$ and $\sigma_2\sigma_1$ denote composition of permutations. We define a collection of graphs, naturally associated with the system $E$, that encodes all the information relevant to the testability of $E$. We then prove two theorems that provide criteria for testability and non-testability in terms of expansion properties of these graphs. By virtue of a deep connection with group theory, both theorems are applicable to wide classes of systems of relations. In addition, we formulate the well-studied group-theoretic notion of stability in permutations as a special case of the testability notion above, interpret all previous works on stability as testability results, survey previous results on stability from a computational perspective, and describe many directions for future research on stability and testability., Comment: 42 pages; this version was accepted to FOCS 2021

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In [BLu] we disproved this conjecture. In this paper we prove that for most groups $H$ the conjecture is actually true if we restrict to counting only non-uniform lattices., Comment: 23 pages, revised following referee's comments. Dedicated to Aner Shalev on his 60th birthday. This paper is related to our previous work arXiv:0905.1841 with which it shares some preliminaries

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$ , the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$ . Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $p$ -adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism $\rho_{n_k}\colon \Gamma\rightarrow\operatorname{Sym}(n_k)$ (i.e., for every $g,h\in\Gamma$, $\lim_{k\rightarrow\infty}d_{n_k}( \rho_{n_k}(gh),\rho_{n_k}(g)\rho_{n_k}(h))=0$) is close to an actual homomorphism $\varphi_{n_k} \colon\Gamma\rightarrow\operatorname{Sym}(n_k)$. Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and P\u{a}unescu showed the same for abelian groups and raised many questions, especially about P-stability of amenable groups. We develop P-stability in general, and in particular for amenable groups. Our main tool is the theory of invariant random subgroups (IRS), which enables us to give a characterization of P-stability among amenable groups, and to deduce stability and instability of various families of amenable groups., Comment: 24 pages; v2 includes minor updates and new references

Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. After explaining their connection to the Ramanujan conjecture we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to "golden gates" which are of importance in quantum computation., To appear in the proceedings of the Ramanujan Centenary Celebration at the Royal Society

If $$\Gamma $$ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example $${{\,\mathrm{SL}\,}}_n({\mathbb {Z}})$$ , $$n \ge 3$$ ) and $$\Lambda $$ is a finitely generated group that is elementarily equivalent to $$\Gamma $$ , then $$\Lambda $$ is isomorphic to $$\Gamma $$ .

Let l be a prime number, K a finite extension of ℚ l , and D a finite-dimensional central division algebra over K. We prove that the profinite group G = D × / K × ${G=D^\times /K^\times }$ is finitely sliceable, i.e. G has finitely many closed subgroups H 1,...,Hn of infinite index such that G = ⋃ i = 1 n H i G ${G=\bigcup _{i=1}^nH_i^G}$ . Here, H i G = { h g ∣ h ∈ H i , g ∈ G } ${H_i^G=\lbrace h^g\mid h\in H_i, \, g\in G\rbrace }$ . On the other hand, we prove for l ≠ 2 that no open subgroup of GL2(ℤ l ) is finitely sliceable and we give an arithmetic interpretation to this result, based on the possibility of realizing GL2(ℤ l ) as a Galois group over ℚ. Nevertheless, we prove that G = GL2(ℤ l ) has an infinite slicing, that is G = ⋃ i = 1 ∞ H i G ${G=\bigcup _{i=1}^\infty H_i^G}$ , where each Hi is a closed subgroup of G of infinite index and Hi ∩ Hj has infinite index in both Hi and Hj if i ≠ j.

We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface., 34 pages, 8 figures

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group $\Gamma$ with respect to a sequence of groups $\left\{G_{n}\right\}_{n=1}^{\infty}$, equipped with bi-invariant metrics $\left\{d_{n}\right\}_{n=1}^{\infty}$. We consider the case $G_{n}=\operatorname{U}\left(n\right)$ (resp. $G_{n}=\operatorname{Sym}\left(n\right)$), equipped with the normalized Hilbert-Schmidt metric $d_{n}^{\operatorname{HS}}$ (resp. the normalized Hamming metric $d_{n}^{\operatorname{Hamming}}$). Our main result is that if $\Gamma$ is infinite, hyperlinear (resp. sofic) and has Property $\operatorname{(T)}$, then it is not stable with respect to $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ (resp. $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$). This answers a question of Hadwin and Shulman regarding the stability of $\operatorname{SL}_{3}\left(\mathbb{Z}\right)$. We also deduce that the mapping class group $\operatorname{MCG}\left(g\right)$, $g\geq 3$, and $\operatorname{Aut}\left(\mathbb{F}_n\right)$, $n\geq 3$, are not stable with respect to $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$. Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on $\operatorname{U}\left(n\right)$ and the (unnormalized) $p$-Schatten metrics, since many groups with Property $\operatorname{(T)}$ are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim. We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to $\left(\operatorname{U}\left(n\right),d_{n}^{\operatorname{HS}}\right)$ and $\left(\operatorname{Sym}\left(n\right),d_{n}^{\operatorname{Hamming}}\right)$., Comment: 20 pages; v2 includes new results in Section 3 and an appendix

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak., Comment: 12 pages, 2 figures. V2 incorporates changes suggested by the referee. This is the final version accepted for publication

We show that for a fixed k, Gromov random groups with any positive density have no non-trivial degree-k representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when the density is less than 1/6 such groups have a faithful linear representation over the rationals, a.a.s., 10 pages

It was shown in a previous work of the first named author with De Chiffre, Glebsky and Thom that there exists a finitely presented group which cannot be approximated by almost-homomorphisms to the unitary groups $U(n)$ equipped with the Frobenius norms (a.k.a as $L^2$ norm, or the Schatten-2-norm). In his ICM18 lecture, Andreas Thom asks if this result can be extended to general Schatten-p-norms. We show that this is indeed the case for $1, Comment: 12 pages

This paper describes in basic terms what a "Thin Group" is, as well as its uses in various subjects., Comment: 6 pages, 2 pictures

To every $Q$-irreducible representation $r$ of a finite group $H$, there corresponds a simple factor $A$ of $Q[H]$ with an involution $\tau$. To this pair $(A,\tau)$, we associate an arithmetic group $\Omega$ consisting of all $(2g-2)\times (2g-2)$ matrices over a natural order of $A^{op}$ which preserve a natural skew-Hermitian sesquilinear form on $A^{2g-2}$. We show that if $H$ is generated by less than $g$ elements, then $\Omega$ is a virtual quotient of the mapping class group $Mod(\Sigma_g)$, i.e. a finite index subgroup of $\Omega$ is a quotient of a finite index subgroup of $\Mod(\Sigma_g)$. This shows that the mapping class group has a rich family of arithmetic quotients (and "Torelli subgroups") for which the classical quotient $Sp(2g, Z)$ is just a first case in a list, the case corresponding to the trivial group $H$ and the trivial representation. Other pairs of $H$ and $r$ give rise to many new arithmetic quotients of $Mod(\Sigma_g)$ which are defined over various (subfields of) cyclotomic fields and are of type $Sp(2m), SO(2m,2m),$ and $SU(m,m)$ for arbitrarily large $m$., Comment: 46 pages, 1 figure, minor edits, added some references and changed the discussion of some other references

This paper answers a question raised by Grothendieck in 1970 on the "Grothendieck closure" of an integral linear group and proves a conjecture of the first author made in 1980. This is done by a detailed study of the congruence topology of arithmetic groups, obtaining along the way, an arithmetic analogue of a classical result of Chevalley for complex algebraic groups. As an application we also deduce a group theoretic characterization of thin subgroups of arithmetic groups., 24 pages

A long standing problem in the area of error correcting codes asks whether there exist good cyclic codes. Most of the known results point in the direction of a negative answer. The uncertainty principle is a classical result of harmonic analysis asserting that given a non-zero function $f$ on some abelian group, either $f$ or its Fourier transform $\hat{f}$ has large support. In this note, we observe a connection between these two subjects. We point out that even a weak version of the uncertainty principle for fields of positive characteristic would imply that good cyclic codes do exist. We also provide some heuristic arguments supporting that this is indeed the case., Comment: 20 pages

Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways (cf. [Lub94], [HLW06], [Lub12] and the references therein). In the last decade, a theory of "high dimensional expanders" has begun to emerge. The goal of the current paper is to describe some paths of this new area of study., Comment: Paper to be presented as a Plenary talk at ICM 2018

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $G$ over a local field $F$. We show that if $T$ is any $k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple combinatorial property (collision-free), the associated random walk on the $n$-vertex Ramanujan complex has cutoff at time $\log_k n$. The high dimensional case, unlike that of graphs, requires tools from non-commutative harmonic analysis and the infinite-dimensional representation theory of $G$. Via these, we show that operators $T$ as above on Ramanujan complexes give rise to Ramanujan digraphs with a special property ($r$-normal), implying cutoff. Applications include geodesic flow operators, geometric implications, and a confirmation of the Riemann Hypothesis for the associated zeta functions over every group $G$, previously known for groups of type $\widetilde A_n$ and $\widetilde C_2$., Comment: 27 pages

Let $2 \leq a \leq b \leq c \in \mathbb{N}$ with $��=1/a+1/b+1/c$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of $T$, as well as positive results showing that many finite simple groups are quotients of $T$.

The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether $\widehat{Aut\left(\Gamma\right)}\to Aut(\hat{\Gamma})$ is injective, or more generally, what is its kernel $C\left(\Gamma\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $\Gamma=F_{2}$ and $\Phi_{2}$) and a new result for $\Gamma=\Phi_{3}$: 1. $C\left(F_{2}\right)=\left\{ e\right\}$ when $F_{2}$ is the free group on two generators. 2. $C\left(\Phi_{2}\right)=\hat{F}_{\omega}$ when $\Phi_{n}$ is the free metabelian group on $n$ generators, and $\hat{F}_{\omega}$ is the free profinite group on $\aleph_{0}$ generators. 3. $C\left(\Phi_{3}\right)$ contains $\hat{F}_{\omega}$. Results 2. and 3. should be contrasted with an upcoming result of the first author showing that $C\left(\Phi_{n}\right)$ is abelian for $n\geq4$., Comment: 20 pages

We determine the structure of finitely generated residually finite groups in which the number of subgroups of each finite index n is bounded by a fixed power of n. © 1993 Hebrew University.

Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(\Lambda)$ be the number of congruence subgroups of $\Lambda$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log n} = \frac{3-2\sqrt{2}}{4}.$ The proof is based on the Bombieri-Vinogradov `Riemann hypothesis on the average' and on the solution of a new type of extremal problem in combinatorial number theory. Similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple Lie groups. If $G$ is such a Lie group and $\Gamma$ is an irreducible lattice of $G$ it turns out that the subgroup growth of $\Gamma$ is independent of the lattice and depends only on the Lie type of the direct factors of $G$. It can be calculated easily from the root system. The most general case of this result relies on the Generalized Riemann Hypothesis but many special cases are unconditional. The proofs use techniques from number theory, algebraic groups, finite group theory and combinatorics., Comment: 11 pages

The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this paper is to begin to develop such a framework for $k$-regular simplicial complexes of general dimension $d$. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of $k$-regular simplicial complexes as quotients of one universal object: the $k$-regular $d$-dimensional arboreal complex, which is itself a simplicial complex originating in one specific group depending only on $d$ and $k$. Along the way we answer a question from [PR12] on the spectral gap of higher dimensional Laplacians and prove a high dimensional analogue of Leighton's graph covering theorem. This approach also suggests a random model for $k$-regular $d$-dimensional multicomplexes., 40 pages, 7 figures

Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel [6] combined with Mostow rigidity imply that there exists a finite group G = G ( M ) such that any finite subgroup of Homeo + ( M ) is isomorphic to a subgroup of G. Borel [6] asked if there exist M's with G ( M ) trivial and if the number of conjugacy classes of finite subgroups of Homeo + ( M ) is finite. We answer both questions: (1) For every finite group G there exist M's with G ( M ) = G , and (2) the number of maximal subgroups of Homeo + ( M ) can be either one, countably many or continuum and we determine (at least for dim ⁡ M ≠ 4 ) when each case occurs. Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity (for dim M ≠ 4 ) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.

A general sieve method for groups is formulated. It enables one to "measure" subsets of a finitely generated group. As an application we show that if $\Gamma$ is a finitely generated non virtually-solvable linear group of characteristic zero then the set of proper powers in $\Gamma$ is exponentially small. This is a far reaching strengthening of the main result of \cite{HKLS}., Comment: 33 pages, 4 figures

We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for SL2 which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different. If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the classification of the finite simple groups. Our result does not hold if the profinite groups are not finitely generated. We give concrete examples of non-isomorphic profinite groups which are elementarily equivalent.

There is a constantC0C_0such that all nonabelian finite simple groups of ranknnoverFq\mathbb {F}_q, with the possible exception of the Ree groups2G2(32e+1)^2G_2(3^{2e+1}), have presentations with at mostC0C_0generators and relations and total length at mostC0(log⁡n+log⁡q)C_0(\log n +\log q). As a corollary, we deduce a conjecture of Holt: there is a constantCCsuch thatdim⁡H2(G,M)≤Cdim⁡M\dim H^2(G,M) \leq C\dim Mfor every finite simple groupGG, every primeppand every irreducibleFpG{\mathbb F}_p G-moduleMM.

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property τ . As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.

Let X be a d-dimensional simplicial complex with N faces of dimension d − 1. Suppose that every (d − 1)-face of X is contained in at least k d + 2 faces of X, of dimension d. Extending the classical Moore bound for graphs, it is shown that X must contain a ball B of radius at mostlog k−d Nwhose d-dimensional homology Hd(B) is non-zero. The Ramanujan complexes constructed by Lubotzky, Samuels and Vishne are used to show that this upper bound on the radius of B cannot be improved by more than a multiplicative constant factor. This implies the classical Moore bound. Theorem A. g(G) < 2 logk−1 n +2 . Let dG(u, v) be the distance between the vertices u and v in the graph metric, and let Br(v )= {u ∈ V : dG(u, v) r} denote the ball of radius r around v. Define the acyclicity radius rv(G )o fG at the vertex v to be the maximal r such that the induced graph G(Br(v)) is acyclic. Let r(G) = minv∈V rv(G); then r(G )= � g(G)/2 �− 1. The asymptotic version of Moore's bound is equivalent to the following statement. Theorem A1. If δ(G )= k 3, then for every v ∈ V rv(G) � log k−1 n� . (1.2)

We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt., Comment: 44 pages, to appear in Groups, Geometry and Dynamics

We go back and forth between, on the one hand, presentations of arithmetic and Kac-Moody groups and, on the other hand, presentations of profinite groups, deducing along the way new results on both.