One-ended spanning subforests and treeability of groups
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917352973533184 |
|---|---|
| author | Conley, Clinton T. Gaboriau, Damien Marks, Andrew S. Tucker-Drob, Robin D. |
| author_facet | Conley, Clinton T. Gaboriau, Damien Marks, Andrew S. Tucker-Drob, Robin D. |
| contents | We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. This provides the first examples of one-ended nonamenable groups which are measure strongly treeable. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_07431 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | One-ended spanning subforests and treeability of groups Conley, Clinton T. Gaboriau, Damien Marks, Andrew S. Tucker-Drob, Robin D. Group Theory Dynamical Systems Logic Operator Algebras Probability 37A20 (03E15, 22F10, 05C10) We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. This provides the first examples of one-ended nonamenable groups which are measure strongly treeable. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests. |
| title | One-ended spanning subforests and treeability of groups |
| topic | Group Theory Dynamical Systems Logic Operator Algebras Probability 37A20 (03E15, 22F10, 05C10) |
| url | https://arxiv.org/abs/2104.07431 |