Special affine representations for hyperbolic groups

TL;DR

This paper extends special representations to Gromov hyperbolic groups with complementary series, establishing their cohomological properties, dynamical behavior, and irreducibility of associated affine actions, revealing new insights into hyperbolic group representations.

Contribution

It introduces a new class of special representations for hyperbolic groups with complementary series and analyzes their cohomology, dynamics, and irreducibility, providing novel results even for classical lattices.

Findings

Existence of non-trivial reduced cohomology class [c] for these representations

An analogue of Kuhn-Vershik's formula is established

Proves cocycle equidistribution and irreducibility of affine actions

Abstract

In this paper we extend the construction of special representations to Gromov hyperbolic groups which admits complementary series. We prove that these representations have a natural non-trivial reduced cohomology class $[c]$. An analogue of Kuhn-Vershik's formula is established and as a by-product a characterisation of hyperbolic groups that admit complementary series. Investigating dynamical properties of the cohomology class $[c]$ we prove an cocycle equidistribution theorem \'a la Roblin-Margulis and deduce the irreducibility of the associated affine actions. The irreducibility of the affine actions associated to the canonical class $[c]$ is original even in the case of uniform lattices in $SO(n,1)$, $SU(n,1)$ or $SL_2(\mathbb{Q}_p)$ with $n\ge 1$ and $p$ prime.