The scale function for locally compact groups acting on non-positively curved spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912158371020800 |
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| author | Reid, Colin D. |
| author_facet | Reid, Colin D. |
| contents | Let $G$ be a totally disconnected, locally compact (t.d.l.c.) group. The scale $s_G(g)$ of $g \in G$ in the sense of Willis is given by the minimum value of the index $|gUg^{-1}:U \cap gUg^{-1}|$ as $U$ ranges over the compact open subgroups; the theory associated to the scale has been very successful in describing general dynamical features of automorphisms of t.d.l.c. groups.
We focus on the case where $G$ acts properly and continuously by isometries on a geodesic space $X$, where $X$ is complete CAT(0) or proper and Gromov-hyperbolic, and $g \in G$ is hyperbolic. In this context, we find geometric descriptions of the parabolic and contraction groups, tidy subgroups, and structures in the $G$-action that encode the scale, including criteria for $g$ to have scale $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The scale function for locally compact groups acting on non-positively curved spaces Reid, Colin D. Group Theory 22D05, 20F67 Let $G$ be a totally disconnected, locally compact (t.d.l.c.) group. The scale $s_G(g)$ of $g \in G$ in the sense of Willis is given by the minimum value of the index $|gUg^{-1}:U \cap gUg^{-1}|$ as $U$ ranges over the compact open subgroups; the theory associated to the scale has been very successful in describing general dynamical features of automorphisms of t.d.l.c. groups. We focus on the case where $G$ acts properly and continuously by isometries on a geodesic space $X$, where $X$ is complete CAT(0) or proper and Gromov-hyperbolic, and $g \in G$ is hyperbolic. In this context, we find geometric descriptions of the parabolic and contraction groups, tidy subgroups, and structures in the $G$-action that encode the scale, including criteria for $g$ to have scale $1$. |
| title | The scale function for locally compact groups acting on non-positively curved spaces |
| topic | Group Theory 22D05, 20F67 |
| url | https://arxiv.org/abs/2412.11352 |