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On the distribution of the periods of convex representations II

Abdelhamid Amroun

TL;DR

This work proves a central limit theorem for the periods of $P$-convex Zariski dense representations $\rho: \Gamma \to G$ of a hyperbolic group $\Gamma$ into a semisimple Lie group $G$. By developing the Busemann-Iwasawa cocycle, Gromov products on flag varieties, and Ledrappier’s correspondence for Hölder cocycles, the authors reduce period statistics to a reparametrized geodesic flow and apply the prime orbit theorem and Cantrell-Scharp CLTs to obtain Gaussian fluctuations with explicit constants. The results unify period counting, entropy-maximizing reparametrizations of flows, and Cartan/λ-projections, and apply to broad classes including Hitchin-like representations. This connects Lie-theoretic invariants (periods, Cartan projections) with probabilistic limit laws and has potential implications for dynamics of convex representations and their equidistribution properties.

Abstract

Let $ρ: Γ\longrightarrow G$ be a Zariski dense irreducible convex representation of the hyperbolic group $Γ$, where G is a connected real semisimple algebraic Lie group. We establish a central limit type theorem for the periods of the representation $ρ$.

On the distribution of the periods of convex representations II

TL;DR

This work proves a central limit theorem for the periods of -convex Zariski dense representations of a hyperbolic group into a semisimple Lie group . By developing the Busemann-Iwasawa cocycle, Gromov products on flag varieties, and Ledrappier’s correspondence for Hölder cocycles, the authors reduce period statistics to a reparametrized geodesic flow and apply the prime orbit theorem and Cantrell-Scharp CLTs to obtain Gaussian fluctuations with explicit constants. The results unify period counting, entropy-maximizing reparametrizations of flows, and Cartan/λ-projections, and apply to broad classes including Hitchin-like representations. This connects Lie-theoretic invariants (periods, Cartan projections) with probabilistic limit laws and has potential implications for dynamics of convex representations and their equidistribution properties.

Abstract

Let be a Zariski dense irreducible convex representation of the hyperbolic group , where G is a connected real semisimple algebraic Lie group. We establish a central limit type theorem for the periods of the representation .
Paper Structure (18 sections, 18 theorems, 60 equations)

This paper contains 18 sections, 18 theorems, 60 equations.

Table of Contents

  1. Introduction and main results
  2. Convex representations
  3. Cocycles, Periods and Gromov products
  4. The Busemann-Iwasawa cocycle
  5. Periods
  6. Gromov product on $\mathcal{F}_{P^{-}}\times \mathcal{F}_P$
  7. Gromov product on $\partial \Gamma \times \partial \Gamma$
  8. Proximality
  9. Proof of Lemma \ref{['UD']}
  10. Hölder cocycles and Ledrappier's correspondance
  11. Hölder continuous cocycles
  12. A central limit theorem for Hölder cocycles
  13. Reparametrization of a Hölder continuous cocycle
  14. A central limit theorem for hyperbolic flows
  15. Proof of the main results
  16. ...and 3 more sections

Key Result

Theorem A

Let $\rho : \Gamma \rightarrow G$ be Zariski dense and $P$-convex irreducible representation, and fix $\varphi$ in the interior of $\mathcal{L}_P^{*}$, where $P$ is a parabolic subgroup of $G$. There exist $L_P^{\varphi}$ and $\sigma_P^{\varphi} >0$ such that, as $t\rightarrow \infty$.

Theorems & Definitions (27)

  • Theorem A: Theorem \ref{['T4']}
  • Theorem B: Theorem \ref{['TT4']}
  • Definition 2.1: Sambarino samb1
  • Proposition 3.1: Q samb3
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 4.1
  • Proposition 4.1: Sambarino samb2
  • ...and 17 more