Conformal dynamics at infinity for groups with contracting elements
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917889225785344 |
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| author | Yang, Wenyuan |
| author_facet | Yang, Wenyuan |
| contents | This paper develops a theory of conformal density at infinity for groups with contracting elements. We start by introducing a class of convergence boundary encompassing many known hyperbolic-like boundaries, on which a detailed study of conical points and Myrberg points is carried out. The basic theory of conformal density is then established on the convergence boundary, including the Sullivan shadow lemma and a Hopf--Tsuji--Sullivan dichotomy. This gives a unification of the theory of conformal density on the Gromov and Floyd boundary for (relatively) hyperbolic groups, the visual boundary for rank-1 CAT(0) groups, and Thurston boundary for mapping class groups. Besides that, the conformal density on the horofunction boundary provides a new important example of our general theory. Applications include the identification of Poisson boundary of random walks, the co-growth problem of divergent groups, measure theoretical results for CAT(0) groups and mapping class groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04861 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Conformal dynamics at infinity for groups with contracting elements Yang, Wenyuan Group Theory Dynamical Systems Geometric Topology Metric Geometry This paper develops a theory of conformal density at infinity for groups with contracting elements. We start by introducing a class of convergence boundary encompassing many known hyperbolic-like boundaries, on which a detailed study of conical points and Myrberg points is carried out. The basic theory of conformal density is then established on the convergence boundary, including the Sullivan shadow lemma and a Hopf--Tsuji--Sullivan dichotomy. This gives a unification of the theory of conformal density on the Gromov and Floyd boundary for (relatively) hyperbolic groups, the visual boundary for rank-1 CAT(0) groups, and Thurston boundary for mapping class groups. Besides that, the conformal density on the horofunction boundary provides a new important example of our general theory. Applications include the identification of Poisson boundary of random walks, the co-growth problem of divergent groups, measure theoretical results for CAT(0) groups and mapping class groups. |
| title | Conformal dynamics at infinity for groups with contracting elements |
| topic | Group Theory Dynamical Systems Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2208.04861 |