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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.02138 |
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| _version_ | 1866914133816901632 |
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| author | Dubashinskiy, Mikhail |
| author_facet | Dubashinskiy, Mikhail |
| contents | Let $Γ$ be a discrete finitely presented group. Pick any system $S$ of generators in $Γ$. In Cayley graph $\mathrm{Cay}(Γ)=\mathrm{Cay}(Γ, S)$ with edge set $E$, glue with oriented polygons all the group relations translated to all the points of $Γ$; denote the obtained simply connected complex by $\mathrm{Cay}^{(2)}(Γ)$. We study non-negative Hodge--Laplace operator $Δ_1$ on edge functions which is defined via complex $\mathrm{Cay}^{(2)}(Γ)$; $Δ_1$ acts on $$ \ell^2_{0,c}(E):= \mathrm{clos}_{\ell^2(E)} \left\{\mbox{finitely supported closed $1$-(co)chains in }\mathrm{Cay}^{}(Γ)\right\}. $$ We prove the following implication in the spirit of Kesten Theorem: if $Δ_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $Γ$ either has exponential growth or is virtually $\mathbb Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coexact 1-Laplacian spectral gap and exponential growth of a group Dubashinskiy, Mikhail Spectral Theory Group Theory Metric Geometry Primary: 58J50, Secondary: 20F65, 53C23 Let $Γ$ be a discrete finitely presented group. Pick any system $S$ of generators in $Γ$. In Cayley graph $\mathrm{Cay}(Γ)=\mathrm{Cay}(Γ, S)$ with edge set $E$, glue with oriented polygons all the group relations translated to all the points of $Γ$; denote the obtained simply connected complex by $\mathrm{Cay}^{(2)}(Γ)$. We study non-negative Hodge--Laplace operator $Δ_1$ on edge functions which is defined via complex $\mathrm{Cay}^{(2)}(Γ)$; $Δ_1$ acts on $$ \ell^2_{0,c}(E):= \mathrm{clos}_{\ell^2(E)} \left\{\mbox{finitely supported closed $1$-(co)chains in }\mathrm{Cay}^{}(Γ)\right\}. $$ We prove the following implication in the spirit of Kesten Theorem: if $Δ_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $Γ$ either has exponential growth or is virtually $\mathbb Z$. |
| title | Coexact 1-Laplacian spectral gap and exponential growth of a group |
| topic | Spectral Theory Group Theory Metric Geometry Primary: 58J50, Secondary: 20F65, 53C23 |
| url | https://arxiv.org/abs/2511.02138 |