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Spectral gaps, critical exponents and representations of negatively curved groups

Kevin Boucher

Abstract

In this paper we introduce a notion of Poincaré exponent for isometric representations of discrete groups on Hilbert spaces. Similarly as growth exponents control the geometry this exponent is shown to control the size of spectral gaps. Following similar ideas as Patterson and Sullivan it is used in the case of negatively curved groups to construct weakly contained boundary representations reflecting the spectral properties of the original representation analogously as complementary series representations in the case of semi-simple Lie groups. This is exploited to deduced sharp estimates on spectral invariants. A quantitive property (T) á la Cowling is also established proving uniform bound on the mixing rate of representations of hyperbolic groups with property (T). Along the way some properties of boundary representations are discussed. A original characterisation of the positivity of the so-called Knapp-Stein operators and certain fusion rules on the boundary complementary series representations are established.

Spectral gaps, critical exponents and representations of negatively curved groups

Abstract

In this paper we introduce a notion of Poincaré exponent for isometric representations of discrete groups on Hilbert spaces. Similarly as growth exponents control the geometry this exponent is shown to control the size of spectral gaps. Following similar ideas as Patterson and Sullivan it is used in the case of negatively curved groups to construct weakly contained boundary representations reflecting the spectral properties of the original representation analogously as complementary series representations in the case of semi-simple Lie groups. This is exploited to deduced sharp estimates on spectral invariants. A quantitive property (T) á la Cowling is also established proving uniform bound on the mixing rate of representations of hyperbolic groups with property (T). Along the way some properties of boundary representations are discussed. A original characterisation of the positivity of the so-called Knapp-Stein operators and certain fusion rules on the boundary complementary series representations are established.
Paper Structure (34 sections, 63 theorems, 188 equations)

This paper contains 34 sections, 63 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Boundary representations
  3. A example: homogeneous dynamics of $SL_2({\mathbb{R}})$
  4. Statement of results
  5. Critical exponent
  6. Critical exponent and negatively curved groups
  7. Critical exponent and boundary representations
  8. Organisation of the paper
  9. Notations and symbols
  10. Acknowledgments
  11. Preliminaries
  12. Strong hyperbolicity and consequences
  13. Orbital distribution
  14. $s$-density representations and summation estimates
  15. Nets and positive families of vectors
  16. ...and 19 more sections

Key Result

Theorem 1

Let $[\pi,v]$ be a positive cyclic representation. Then for all $f\in {\mathbb{C}}[\Gamma]$ one has: where $r(f)$ stands for the smallest $r$ such that $\text{supp}(f)\subset B_r$ the $d_\Gamma$-ball of radius $r$ around $e\in \Gamma$.

Theorems & Definitions (140)

  • Remark 1.1
  • Example 1.2
  • Example 1.3: Positive cyclic representations
  • Remark 1.4
  • Theorem 1
  • Definition : MR3666050
  • Remark 1.5
  • Corollary 1.6
  • Corollary
  • Theorem 2
  • ...and 130 more