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Special affine representations for hyperbolic groups

Kevin Boucher

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 á 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.

Special affine representations for hyperbolic groups

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 . 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 we prove an cocycle equidistribution theorem á la Roblin-Margulis and deduce the irreducibility of the associated affine actions. The irreducibility of the affine actions associated to the canonical class is original even in the case of uniform lattices in , or with and prime.

Paper Structure

This paper contains 14 sections, 33 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Statement of the results.
  3. Organisation
  4. Acknowledgments
  5. Notations and conventions
  6. preliminaries
  7. Hyperbolic spaces and groups
  8. The Patterson-Sullivan theory and equidistribution
  9. Equidistribution and counting
  10. Construction of the special representations
  11. Special representations and cohomology
  12. Special representations and reduced cohomology
  13. Irreducibility of the special affine actions
  14. Positive operators on reflexive Banach spaces.

Key Result

Theorem 1

Assume the operator $I'$ is positive definite on $L^2_0({\partial} \Gamma, \nu)$. Then the associated unitary representation, $(\pi_1,H_1')$, is mixing and $c$ has non-trivial reduced cohomology class.

Theorems & Definitions (70)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1
  • Example 1.3
  • Theorem : Boucher:2020aa Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.4
  • Theorem 4
  • Definition 2.1: MR3551185
  • ...and 60 more