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| Format: | Preprint |
| Veröffentlicht: |
2021
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2101.03224 |
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| _version_ | 1866915333961416704 |
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| author | Magee, Michael |
| author_facet | Magee, Michael |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_03224 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Random Unitary Representations of Surface Groups II: The large $n$ limit Magee, Michael Representation Theory Mathematical Physics Geometric Topology Operator Algebras 14H60, 20C30, 20C35, 22D10, 32G15, 46L54, 57M20, 70S15 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. |
| title | Random Unitary Representations of Surface Groups II: The large $n$ limit |
| topic | Representation Theory Mathematical Physics Geometric Topology Operator Algebras 14H60, 20C30, 20C35, 22D10, 32G15, 46L54, 57M20, 70S15 |
| url | https://arxiv.org/abs/2101.03224 |