Random unitary representations of surface groups, II : The large n limit

Magee, Michael

doi:10.2140/gt.2025.29.1237

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Random unitary representations of surface groups, II: The large $n$ limit

Michael Magee

Geometry & Topology 29 (2025) 1237–1281

DOI: 10.2140/gt.2025.29.1237

Abstract

Let $Σ_{g}$ be a closed surface of genus $g \geq 2$ 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 $S U (n)$ , 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 nonidentity element of $Γ_{g}$ is bounded as $n \to \infty$ . The proof involves an interplay between Dehn’s work on the word problem in $Γ_{g}$ and classical invariant theory.

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Keywords

random unitary representation, surface group

Mathematical Subject Classification

Primary: 14H60, 22D10, 46L54

Secondary: 20C30, 20C35, 32G15, 70S15

References

Publication

Received: 2 December 2021

Revised: 3 January 2023

Accepted: 14 April 2023

Published: 31 May 2025

Proposed: David Fisher

Seconded: John Lott, Leonid Polterovich

Authors

Michael Magee
Department of Mathematical Sciences Durham University Durham United Kingdom
https://www.mmagee.net/

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Volume 29, issue 3 (2025)