Action rigidity for free products of hyperbolic manifold groups
Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, that residually finite groups...
| Asıl Yazarlar: | , |
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| Materyal Türü: | Journal article |
| Dil: | English |
| Baskı/Yayın Bilgisi: |
Association des Annales de l'Institut Fourier
2023
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| _version_ | 1826313794204205056 |
|---|---|
| author | Stark, ER Woodhouse, DJ |
| author_facet | Stark, ER Woodhouse, DJ |
| author_sort | Stark, ER |
| collection | OXFORD |
| description | Two groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, that residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. An important component of the proof is a generalization of Leighton’s graph covering theorem. The main theorem depends on residual finiteness, and we show that finite extensions of uniform lattices in rank-1 symmetric spaces that are not residually finite would give counterexamples. |
| first_indexed | 2024-03-07T07:59:05Z |
| format | Journal article |
| id | oxford-uuid:77555bdd-85c9-4b2b-8f56-c3695d5cc4f7 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:22:12Z |
| publishDate | 2023 |
| publisher | Association des Annales de l'Institut Fourier |
| record_format | dspace |
| spelling | oxford-uuid:77555bdd-85c9-4b2b-8f56-c3695d5cc4f72024-08-12T11:06:37ZAction rigidity for free products of hyperbolic manifold groupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:77555bdd-85c9-4b2b-8f56-c3695d5cc4f7EnglishSymplectic ElementsAssociation des Annales de l'Institut Fourier2023Stark, ERWoodhouse, DJTwo groups have a common model geometry if they act properly and cocompactly by isometries on the same proper geodesic metric space. We consider free products of uniform lattices in isometry groups of rank-1 symmetric spaces and prove, within each quasi-isometry class, that residually finite groups that have a common model geometry are abstractly commensurable. Our result gives the first examples of hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. An important component of the proof is a generalization of Leighton’s graph covering theorem. The main theorem depends on residual finiteness, and we show that finite extensions of uniform lattices in rank-1 symmetric spaces that are not residually finite would give counterexamples. |
| spellingShingle | Stark, ER Woodhouse, DJ Action rigidity for free products of hyperbolic manifold groups |
| title | Action rigidity for free products of hyperbolic manifold groups |
| title_full | Action rigidity for free products of hyperbolic manifold groups |
| title_fullStr | Action rigidity for free products of hyperbolic manifold groups |
| title_full_unstemmed | Action rigidity for free products of hyperbolic manifold groups |
| title_short | Action rigidity for free products of hyperbolic manifold groups |
| title_sort | action rigidity for free products of hyperbolic manifold groups |
| work_keys_str_mv | AT starker actionrigidityforfreeproductsofhyperbolicmanifoldgroups AT woodhousedj actionrigidityforfreeproductsofhyperbolicmanifoldgroups |