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Register a new userHyperbolic generalized triangle groups, property (T) and finite simple quotients
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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.
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Let $G$ be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this paper we describe several approaches for constructing continuous families of periodic quotients of $G$ with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of $G$. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer $n$ such that $G/G^n$ is an infinite group of exponent $n$. The fourth approach starts with a large group $G$ and produces a continuum of pairwise non-isomorphic periodic residually finite quotients. Speaking of a particular application, we use each of these methods to give a positive answer to a question of Wiegold from Kourovka Notebook.
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.
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 prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, dots, h_k$, there exists another element $gamma eq 1$ such that for all $i$, $langle h_i, gamma rangle = langle h_i rangle* langle gamma rangle$. We also obtain that if a collection of subgroups $H_1, dots, H_k$ is a hyperbolically embedded collection, then there is $gamma eq 1$ such that for all $i$, $langle H_i, gamma rangle = H_i * langle gamma rangle$.
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