Hyperbolic generalized triangle groups, property (T) and finite simple quotients

TL;DR

This paper constructs explicit hyperbolic groups with Kazhdan's property (T) that are shorter than previous examples and also have finite simple quotients of arbitrarily large rank, combining these properties for the first time.

Contribution

It introduces new, shorter hyperbolic groups with property (T) and demonstrates their ability to have large finite simple quotients, advancing understanding of hyperbolic group properties.

Findings

Constructed shorter hyperbolic groups with property (T)

Demonstrated existence of hyperbolic groups with large finite simple quotients

First examples combining property (T) with arbitrarily large simple quotients

Abstract

We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan's 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.