Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups
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| Format: | Preprint |
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2025
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| author | Behrstock, Jason Ciceksiz, R. Altar Falgas-Ravry, Victor |
| author_facet | Behrstock, Jason Ciceksiz, R. Altar Falgas-Ravry, Victor |
| contents | We consider random right-angled Coxeter groups, $W_Γ$, whose presentation graph $Γ$ is taken to be an Erdős--Rényi random graph, i.e., $Γ\sim \mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of these random groups.
We resolve a conjecture of Susse and determine the connectivity threshold for square percolation on the random graph $Γ\sim \mathcal{G}_{n,p}$. We use this result to determine a large range of $p$ for which the random right-angled Coxeter group $W_Γ$ has a unique cubical coarse median structure. Until recent work of Fioravanti, Levcovitz and Sageev, there were no non-hyperbolic examples of groups with cubical coarse rigidity; our present results show the property is in fact typically satisfied by a random RACG for a wide range of the parameter $p$, including $p=1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_18165 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups Behrstock, Jason Ciceksiz, R. Altar Falgas-Ravry, Victor Probability Combinatorics Group Theory Geometric Topology 05C80 20F55 20F67 20F65 We consider random right-angled Coxeter groups, $W_Γ$, whose presentation graph $Γ$ is taken to be an Erdős--Rényi random graph, i.e., $Γ\sim \mathcal{G}_{n,p}$. We use techniques from probabilistic combinatorics to establish several new results about the geometry of these random groups. We resolve a conjecture of Susse and determine the connectivity threshold for square percolation on the random graph $Γ\sim \mathcal{G}_{n,p}$. We use this result to determine a large range of $p$ for which the random right-angled Coxeter group $W_Γ$ has a unique cubical coarse median structure. Until recent work of Fioravanti, Levcovitz and Sageev, there were no non-hyperbolic examples of groups with cubical coarse rigidity; our present results show the property is in fact typically satisfied by a random RACG for a wide range of the parameter $p$, including $p=1/2$. |
| title | Connectivity for square percolation and coarse cubical rigidity in random right-angled Coxeter groups |
| topic | Probability Combinatorics Group Theory Geometric Topology 05C80 20F55 20F67 20F65 |
| url | https://arxiv.org/abs/2502.18165 |