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Register a new userDistinguishing geometries using finite quotients
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We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to $hat{mathbb{Z}}^2$. This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for which the profinite completion of the fundamental group has a non-trivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-$p$ subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-$p$.
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Let $C(Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $Gamma$. We investigate the extent to which $C(Gamma)$ determines $Gamma$ when $Gamma$ is a group of geometric interest. If $Gamma_1$ is a lattice in ${rm{PSL}}(2,R)$ and $Gamma_2$ is a lattice in any connected Lie group, then $C(Gamma_1) = C(Gamma_2)$ implies that $Gamma_1$ is isomorphic to $Gamma_2$. If $F$ is a free group and $Gamma$ is a right-angled Artin group or a residually free group (with one extra condition), then $C(F)=C(Gamma)$ implies that $FcongGamma$. If $Gamma_1<{rm{PSL}}(2,Bbb C)$ and $Gamma_2< G$ are non-uniform arithmetic lattices, where $G$ is a semi-simple Lie group with trivial centre and no compact factors, then $C(Gamma_1)= C(Gamma_2)$ implies that $G cong {rm{PSL}}(2,Bbb C)$ and that $Gamma_2$ belongs to one of finitely many commensurability classes. These results are proved using the theory of profinite groups; we do not exhibit explicit finite quotients that distinguish among the groups in question. But in the special case of two non-isomorphic triangle groups, we give an explicit description of finite quotients that distinguish between them.
We show that low-density random quotients of cubulated hyperbolic groups are again cubulated (and hyperbolic). Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the statement that hyperplane stabilizers grow exponentially more slowly than the ambient cubical group.
These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis proof of Knesers simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.
We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdans property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k-fold generalized triangle groups, i.e. groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitive on the set of triangles, and such that edge-stabilizers are cyclic of order k.
Let $IA_n$ denote the group of $IA$-automorphisms of a free group of rank $n$, and let $mathcal I_n^b$ denote the Torelli subgroup of the mapping class group of an orientable surface of genus $n$ with $b$ boundary components, $b=0,1$. In 1935 Magnus proved that $IA_n$ is finitely generated for all $n$, and in 1983 Johnson proved that $mathcal I_n^b$ is finitely generated for $ngeq 3$. It was recently shown that for each $kinmathbb N$, the $k^{rm th}$ terms of the lower central series $gamma_k IA_n$ and $gamma_kmathcal I_n^b$ are finitely generated when $n>>k$; however, no information about finite generating sets was known for $k>1$. The main goal of this paper is to construct an explicit finite generating set for $gamma_2 IA_n = [IA_n,IA_n]$ and almost explicit finite generating sets for $gamma_2mathcal I_n^b$ and the Johnson kernel, which contains $gamma_2mathcal I_n^b$ as a finite index subgroup.
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