A threshold for relative hyperbolicity in random right-angled Coxeter groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918146682650624 |
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| author | Behrstock, Jason Ciceksiz, Recep Altar Falgas-Ravry, Victor |
| author_facet | Behrstock, Jason Ciceksiz, Recep Altar Falgas-Ravry, Victor |
| contents | We consider the random right-angled Coxeter group $W_Γ$ whose presentation graph $Γ\sim \mathcal{G}_{n,p}$ is an Erd{\H o}s--Rényi random graph on $n$ vertices with edge probability $p=p(n)$. We establish that $p=1/\sqrt{n}$ is a threshold for relative hyperbolicity of the random group $W_Γ$. As a key step in the proof, we determine the minimal number of pairs of generators that must commute in a right-angled Coxeter group which is not relatively hyperbolic, a result which is of independent interest.
We also show that there is an interval of edge probabilities of width $Ω(1/\sqrt{n})$ in which the random right-angled Coxeter group has precisely cubic divergence. This interval is between the thresholds for relative hyperbolicity (whence exponential divergence) and quadratic divergence. Moreover, a simple random walk on any Cayley graph of the random right-angled Coxeter group for $p$ in this interval satisfies a central limit theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12959 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A threshold for relative hyperbolicity in random right-angled Coxeter groups Behrstock, Jason Ciceksiz, Recep Altar Falgas-Ravry, Victor Group Theory Combinatorics Geometric Topology Probability 05C80 20F55 20F67 20F65 We consider the random right-angled Coxeter group $W_Γ$ whose presentation graph $Γ\sim \mathcal{G}_{n,p}$ is an Erd{\H o}s--Rényi random graph on $n$ vertices with edge probability $p=p(n)$. We establish that $p=1/\sqrt{n}$ is a threshold for relative hyperbolicity of the random group $W_Γ$. As a key step in the proof, we determine the minimal number of pairs of generators that must commute in a right-angled Coxeter group which is not relatively hyperbolic, a result which is of independent interest. We also show that there is an interval of edge probabilities of width $Ω(1/\sqrt{n})$ in which the random right-angled Coxeter group has precisely cubic divergence. This interval is between the thresholds for relative hyperbolicity (whence exponential divergence) and quadratic divergence. Moreover, a simple random walk on any Cayley graph of the random right-angled Coxeter group for $p$ in this interval satisfies a central limit theorem. |
| title | A threshold for relative hyperbolicity in random right-angled Coxeter groups |
| topic | Group Theory Combinatorics Geometric Topology Probability 05C80 20F55 20F67 20F65 |
| url | https://arxiv.org/abs/2407.12959 |