Random Unitary Representations of Surface Groups II: The large n limit

2021/01/08 by Michael Magee, Magee, Michael · 1 citation
Computer Science · Mathematics · #14H60 #20C30 #20C35 #22D10 #32G15 #46L54 #57M20 #70S15 #Advanced Algebra and Geometry #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Representation Theory (math.RT) #Topological and Geometric Data Analysis

paper · doi:10.48550/arxiv.2101.03224

openalex publication_date 2021/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Σ_g be a closed surface of genus g≥ 2 and Γ_g denote the fundamental group of Σ_g. We establish a generalization of Voiculescu's theorem on the asymptotic *-freeness of Haar unitary matrices from free groups to Γ_g. We prove that for a random representation of Γ_g into SU(n), with law given by the volume form arising from the Atiyah-Bott-Goldman symplectic form on moduli space, the expected value of the trace of a fixed non-identity element of Γ_g is bounded as n→∞. The proof involves an interplay between Dehn's work on the word problem in Γ_g and classical invariant theory.

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