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

Magee, Michael

Abstract

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 non-identity 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.

Citation

Magee, M. (in press). Random Unitary Representations of Surface Groups II: The large n limit. Geometry & Topology,

• Journal Article Type: Article
• Acceptance Date: Apr 15, 2023
• Deposit Date: Jul 19, 2023
• Publicly Available Date: Jul 19, 2023
• Journal: Geometry & Topology
• Print ISSN: 1465-3060
• Electronic ISSN: 1364-0380
• Publisher: Mathematical Sciences Publishers (MSP)
• Peer Reviewed: Peer Reviewed
• Public URL: https://durham-repository.worktribe.com/output/1168582
• Publisher URL: https://msp.org/gt/2024/28-2/index.xhtml