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

Magee, Michael¹

¹Department of Mathematical Sciences, Durham University, Durham, United Kingdom

Geometry & topology, Tome 29 (2025) no. 3, pp. 1237-1281

Résumé

Let Σg be a closed surface of genus g ≥ 2 and Γg denote the fundamental group of Σg. We establish a generalization of Voiculescu’s theorem on the asymptotic ∗-freeness of Haar unitary matrices from free groups to Γg. We prove that, for a random representation of Γg 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 nonidentity element of Γg is bounded as n →∞. The proof involves an interplay between Dehn’s work on the word problem in Γg and classical invariant theory.

Détail

DOI : 10.2140/gt.2025.29.1237

Keywords: random unitary representation, surface group

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