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On the distribution of the periods of convex representations I

Abdelhamid Amroun

TL;DR

This work proves a central limit theorem for a class of Hölder cocycles with positive periods and finite exponential growth, and applies it to strictly convex irreducible representations of hyperbolic groups. By realizing the representation-theoretic periods $\lambda_1(\rho(\gamma))$ as cocycle periods and reparametrizing the geodesic flow to an Anosov flow, the authors obtain Gaussian fluctuations for counting conjugacy classes and for spectral-type invariants such as Cartan/Jordan projections. The results connect representation theory, hyperbolic dynamics, and probabilistic limit laws, providing CLTs for both period and norm data, and extending to dual cone functionals $\varphi$ in $\mathring{\mathcal{L}_{\rho}^{*}}$. Overall, the paper deepens the understanding of growth and fluctuation phenomena in convex representations of hyperbolic groups and Hitchin-type settings through dynamical systems techniques.

Abstract

We prove a central limit theorem for a class of Hölder continuous cocycles with an application to stricly convex and irreducible rational representations of hyperbolic groups, introduced by Sambarino [Quantitative properties of convexe representations. Comment. Math. Helv 89 (2014), 443-488].

On the distribution of the periods of convex representations I

TL;DR

This work proves a central limit theorem for a class of Hölder cocycles with positive periods and finite exponential growth, and applies it to strictly convex irreducible representations of hyperbolic groups. By realizing the representation-theoretic periods as cocycle periods and reparametrizing the geodesic flow to an Anosov flow, the authors obtain Gaussian fluctuations for counting conjugacy classes and for spectral-type invariants such as Cartan/Jordan projections. The results connect representation theory, hyperbolic dynamics, and probabilistic limit laws, providing CLTs for both period and norm data, and extending to dual cone functionals in . Overall, the paper deepens the understanding of growth and fluctuation phenomena in convex representations of hyperbolic groups and Hitchin-type settings through dynamical systems techniques.

Abstract

We prove a central limit theorem for a class of Hölder continuous cocycles with an application to stricly convex and irreducible rational representations of hyperbolic groups, introduced by Sambarino [Quantitative properties of convexe representations. Comment. Math. Helv 89 (2014), 443-488].
Paper Structure (12 sections, 17 theorems, 55 equations)

This paper contains 12 sections, 17 theorems, 55 equations.

Table of Contents

  1. Introduction and main results
  2. Strictly convex representations
  3. Definitions and auxilliary results
  4. Hölder continuous cocycles
  5. A central limit theorem for Hölder continuous cocycles
  6. Reparametrization of a Hölder continuous cocycle
  7. A central limit theorem for hyperbolic flows
  8. Proof of the main results
  9. Proof of Theorem \ref{['T1']}
  10. Proof of Theorem \ref{['T2']} and Theorem \ref{['T']}
  11. Proof of Theorem \ref{['Cor1']}
  12. Proof of Theorem \ref{['Cor2']}

Key Result

Theorem A

Let $\rho : \Gamma\longrightarrow PGL(d,\mathbb{R})$ be a strictly convex irreducible representation. There exist $h>0$, $L$ and $\sigma >0$ such that, for all $a,b \in \mathbb{R}$, as $t\rightarrow \infty$, and where we have set $\mathcal{N}(a,b) :=\frac{1}{\sqrt{2\pi}} \int_{a}^{b} e^{-\frac{x^2}{2}}dx$.

Theorems & Definitions (27)

  • Theorem A
  • Theorem 1.1: Sambarino sam1
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem 1.2: Sambarino sam1
  • Definition 2.1: Sambarino sam1 sam2
  • Definition 2.2: Sambarino sam2
  • Proposition 2.1: Sambarino sam1
  • Definition 2.3: Benoist Be
  • ...and 17 more