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Zeta functions in higher Teichmuller theory

Mark Pollicott, Richard Sharp

Abstract

In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in PSL(d, R), for d >= 3. These are natural generalizations of the wellknown classical Selberg zeta function and L-function for Fuchsian groups, corresponding to the case d=2.We show that these complex functions have meromorphic extensions to the entire complex plane C.

Zeta functions in higher Teichmuller theory

Abstract

In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in PSL(d, R), for d >= 3. These are natural generalizations of the wellknown classical Selberg zeta function and L-function for Fuchsian groups, corresponding to the case d=2.We show that these complex functions have meromorphic extensions to the entire complex plane C.
Paper Structure (18 sections, 24 theorems, 57 equations, 2 figures)

This paper contains 18 sections, 24 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Representations and zeta functions
  3. Selberg and Ruelle zeta functions for $\mathrm{PSL}(2, \mathbb R)$
  4. Higher rank Teichmüller theory
  5. Anosov representations
  6. Zeta functions for projective Anosov representations
  7. Meromorphic extensions of the zeta functions
  8. Surface groups and symbolic dynamics
  9. Action on projective space
  10. Surface groups and coding
  11. Symbolic dynamics
  12. Semi-flows and entropy
  13. Extensions of Zeta Functions
  14. Partitions
  15. Transfer operators
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be the fundamental group of a compact oriented surface of genus at least $2$ and let $\rho: \Gamma \to \mathrm{SL}(d, \mathbb R)$ ($d \geq 3$) be a projective Anosov representation. Then $Z(s, \rho)$ converges for $Re(s)$ sufficiently large, and extends to a meromorphic function in the

Figures (2)

  • Figure 1: Here $\rho$ is a representation into $\mathrm{SL}(3,\mathbb R)$ and $\Lambda \subset \mathbb R P^2$ is the associated limit set. For $g \in \Gamma \setminus \{1\}$, the point $\xi_g$ is an attracting fixed point for $\widehat{\rho}(g): \mathbb R P^2 \to : \mathbb R P^2$.
  • Figure 2: The contractions $\psi_i : P_j \to P_i$ satisfy $\psi_i(U_j) \subset U_i$ for the neighbourhoods of the complexifications (suggested in the figure).

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 2.1: Selberg
  • Theorem 2.2
  • Example 2.3: Fuchsian representations
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 42 more