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Random Unitary Representations of Surface Groups I: Asymptotic expansions

Magee, Michael (2022) 'Random Unitary Representations of Surface Groups I: Asymptotic expansions.', Communications in Mathematical Physics, 391 (1). pp. 119-171.


In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let Σg denote a topological surface of genus g≥2. We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of π1(Σg) under a random representation of π1(Σg) into SU(n). Each such expected value involves a contribution from all irreducible representations of SU(n). The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

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Publisher statement:Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
Date accepted:30 November 2021
Date deposited:07 February 2022
Date of first online publication:31 December 2021
Date first made open access:07 February 2022

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