Quasi-Isometries for certain Right-Angled Coxeter Groups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866916134287048704 |
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| author | Edletzberger, Alexandra |
| author_facet | Edletzberger, Alexandra |
| contents | We construct the JSJ tree of cylinders $T_c$ for finitely presented, one-ended, two-dimensional right-angled Coxeter groups (RACGs) splitting over two-ended subgroups in terms of the defining graph of the group, generalizing the visual construction by Dani and Thomas given for hyperbolic RACGs. Additionally, we prove that $T_c$ has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided $K_4$. By use of the structure invariant of $T_c$ introduced by Cashen and Martin, we obtain a quasi-isometry-invariant of these RACGs, essentially determined by the defining graph. Furthermore, we refine the structure invariant to make it a complete quasi-isometry-invariant in case the JSJ decomposition of the RACG does not have any rigid vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_10463 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quasi-Isometries for certain Right-Angled Coxeter Groups Edletzberger, Alexandra Group Theory We construct the JSJ tree of cylinders $T_c$ for finitely presented, one-ended, two-dimensional right-angled Coxeter groups (RACGs) splitting over two-ended subgroups in terms of the defining graph of the group, generalizing the visual construction by Dani and Thomas given for hyperbolic RACGs. Additionally, we prove that $T_c$ has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided $K_4$. By use of the structure invariant of $T_c$ introduced by Cashen and Martin, we obtain a quasi-isometry-invariant of these RACGs, essentially determined by the defining graph. Furthermore, we refine the structure invariant to make it a complete quasi-isometry-invariant in case the JSJ decomposition of the RACG does not have any rigid vertices. |
| title | Quasi-Isometries for certain Right-Angled Coxeter Groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2112.10463 |