Connections between hyperlinearity, stability and character rigidity for higher rank lattices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915359562399744 |
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| author | Dogon, Alon Vigdorovich, Itamar |
| author_facet | Dogon, Alon Vigdorovich, Itamar |
| contents | Let $Γ$ be an irreducible lattice in a semisimple Lie group of real rank at least $2$. Suppose that $Γ$ has property (T;FD), that is, its finite dimensional representations have a uniform spectral gap. We show that if $Γ$ is (flexibly) Hilbert--Schmidt stable then: $(a)$ infinite central extensions $\widetildeΓ$ of $Γ$ are not hyperlinear, and $(b)$ every character of $Γ$ is either finite-dimensional or induced from the center (character rigidity). As a consequence, a positive answer to the following question would yield an explicit example of a non-hyperlinear group: If two representations of the modular group $\mathrm{SL}_2(\mathbb{Z})$ almost agree on a specific congruence subgroup $H$ under a commensuration, must they be close to representations that genuinely agree on $H$? |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20843 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Connections between hyperlinearity, stability and character rigidity for higher rank lattices Dogon, Alon Vigdorovich, Itamar Group Theory Operator Algebras 22E40, 20H05, 22D25, 20E26, 43A35 (Primary) 22D55, 46L53, 20C25 (Secondary) Let $Γ$ be an irreducible lattice in a semisimple Lie group of real rank at least $2$. Suppose that $Γ$ has property (T;FD), that is, its finite dimensional representations have a uniform spectral gap. We show that if $Γ$ is (flexibly) Hilbert--Schmidt stable then: $(a)$ infinite central extensions $\widetildeΓ$ of $Γ$ are not hyperlinear, and $(b)$ every character of $Γ$ is either finite-dimensional or induced from the center (character rigidity). As a consequence, a positive answer to the following question would yield an explicit example of a non-hyperlinear group: If two representations of the modular group $\mathrm{SL}_2(\mathbb{Z})$ almost agree on a specific congruence subgroup $H$ under a commensuration, must they be close to representations that genuinely agree on $H$? |
| title | Connections between hyperlinearity, stability and character rigidity for higher rank lattices |
| topic | Group Theory Operator Algebras 22E40, 20H05, 22D25, 20E26, 43A35 (Primary) 22D55, 46L53, 20C25 (Secondary) |
| url | https://arxiv.org/abs/2506.20843 |