Conformal dimension bounds for certain Coxeter group Bowditch boundaries
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912332414713856 |
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| author | Field, Elizabeth Gupta, Radhika Lyman, Robert Alonzo Stark, Emily |
| author_facet | Field, Elizabeth Gupta, Radhika Lyman, Robert Alonzo Stark, Emily |
| contents | We give upper and lower bounds on the conformal dimension of the Bowditch boundary of a Coxeter group with defining graph a complete graph and edge labels at least three. The lower bounds are obtained by quasi-isometrically embedding Gromov's round trees in the Davis complex. The upper bounds are given by exhibiting a geometrically finite action on a CAT(-1) space and bounding the Hausdorff dimension of the visual boundary of this space. Our results imply that there are infinitely many quasi-isometry classes within each infinite family of such Coxeter groups with edge labels bounded from above. As an application, we prove there are infinitely many quasi-isometry classes among the family of hyperbolic groups with Pontryagin sphere boundary. Combining our results with work of Bourdon--Kleiner proves the conformal dimension of the boundaries of hyperbolic groups in this family achieves a dense set in $(1,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conformal dimension bounds for certain Coxeter group Bowditch boundaries Field, Elizabeth Gupta, Radhika Lyman, Robert Alonzo Stark, Emily Geometric Topology Group Theory 20F65 We give upper and lower bounds on the conformal dimension of the Bowditch boundary of a Coxeter group with defining graph a complete graph and edge labels at least three. The lower bounds are obtained by quasi-isometrically embedding Gromov's round trees in the Davis complex. The upper bounds are given by exhibiting a geometrically finite action on a CAT(-1) space and bounding the Hausdorff dimension of the visual boundary of this space. Our results imply that there are infinitely many quasi-isometry classes within each infinite family of such Coxeter groups with edge labels bounded from above. As an application, we prove there are infinitely many quasi-isometry classes among the family of hyperbolic groups with Pontryagin sphere boundary. Combining our results with work of Bourdon--Kleiner proves the conformal dimension of the boundaries of hyperbolic groups in this family achieves a dense set in $(1,\infty)$. |
| title | Conformal dimension bounds for certain Coxeter group Bowditch boundaries |
| topic | Geometric Topology Group Theory 20F65 |
| url | https://arxiv.org/abs/2504.12404 |