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Random Unitary Representations of Surface Groups I: Asymptotic expansions

Magee, Michael

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Abstract

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.

Citation

Magee, M. (2022). Random Unitary Representations of Surface Groups I: Asymptotic expansions. Communications in Mathematical Physics, 391(1), 119-171. https://doi.org/10.1007/s00220-021-04295-5

Journal Article Type Article
Acceptance Date Nov 30, 2021
Online Publication Date Dec 31, 2021
Publication Date 2022-04
Deposit Date Jul 9, 2021
Publicly Available Date Feb 7, 2022
Journal Communications in Mathematical Physics
Print ISSN 0010-3616
Electronic ISSN 1432-0916
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 391
Issue 1
Pages 119-171
DOI https://doi.org/10.1007/s00220-021-04295-5

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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 http://creativecommons.org/licenses/by/4.0/.




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