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| Main Author: | |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2101.03224 |
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Table of Contents:
- 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 $\mathsf{SU}(n)$, 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 $Γ_{g}$ 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.