Linearity and indiscreteness of amalgamated products of hyperbolic groups

TL;DR

This paper investigates the linearity and discreteness properties of amalgamated products of hyperbolic groups, proving linearity in certain cases and constructing examples that lack discrete faithful representations, thus advancing understanding of hyperbolic group structures.

Contribution

It establishes that doubles of Anosov groups along maximal cyclic subgroups are always linear and provides new examples of non-linear hyperbolic groups, extending previous research.

Findings

Double of Anosov group along maximal cyclic subgroup is linear

Constructed examples of hyperbolic groups without discrete faithful representations

Extended previous work on non-linear hyperbolic groups

Abstract

We discuss the linearity and discreteness of amalgamated products of linear word-hyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups which do not admit any discrete and faithful representation in rank 1. We also build new examples of non-linear word-hyperbolic groups, elaborating on a previous work of Canary--Stover--Tsouvalas.