Edge subdivisions and the $L^2$-homology of right-angled Coxeter groups
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912131541106688 |
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| author | Avramidi, Grigori Okun, Boris Schreve, Kevin |
| author_facet | Avramidi, Grigori Okun, Boris Schreve, Kevin |
| contents | If $L$ is a flag triangulation of $S^{n-1}$, then the Davis complex $Σ_L$ for the associated right-angled Coxeter group $W_L$ is a contractible $n$-manifold. A special case of a conjecture of Singer predicts that the $L^2$-homology of such $Σ_L$ vanishes outside the middle dimension. We give conditions which guarantee this vanishing is preserved under edge subdivision of $L$. In particular, we verify Singer's conjecture when $L$ is the barycentric subdivision of the boundary of an $n$-simplex, and for general barycentric subdivisions of triangulations of $S^{2n-1}$. Using this, we construct explicit counterexamples to a torsion growth analogue of Singer's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08009 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Edge subdivisions and the $L^2$-homology of right-angled Coxeter groups Avramidi, Grigori Okun, Boris Schreve, Kevin Geometric Topology Group Theory If $L$ is a flag triangulation of $S^{n-1}$, then the Davis complex $Σ_L$ for the associated right-angled Coxeter group $W_L$ is a contractible $n$-manifold. A special case of a conjecture of Singer predicts that the $L^2$-homology of such $Σ_L$ vanishes outside the middle dimension. We give conditions which guarantee this vanishing is preserved under edge subdivision of $L$. In particular, we verify Singer's conjecture when $L$ is the barycentric subdivision of the boundary of an $n$-simplex, and for general barycentric subdivisions of triangulations of $S^{2n-1}$. Using this, we construct explicit counterexamples to a torsion growth analogue of Singer's conjecture. |
| title | Edge subdivisions and the $L^2$-homology of right-angled Coxeter groups |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/2411.08009 |