Maps between relatively hyperbolic spaces and between their boundaries
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866910939104673792 |
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| author | Mackay, John M. Sisto, Alessandro |
| author_facet | Mackay, John M. Sisto, Alessandro |
| contents | We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion.
We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_11902 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Maps between relatively hyperbolic spaces and between their boundaries Mackay, John M. Sisto, Alessandro Geometric Topology Group Theory Metric Geometry 20F65, 20F67, 20F69, 31L10, 51F99 We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective) quasi-isometric embeddings between relatively hyperbolic groups/spaces that coarsely respect peripherals, and quasisymmetric embeddings between their boundaries satisfying suitable conditions. Further, we establish a similar correspondence regarding maps with at most polynomial distortion. We use this to characterise groups which are hyperbolic relative to some collection of virtually nilpotent subgroups as exactly those groups which admit an embedding into a truncated real hyperbolic space with at most polynomial distortion, generalising a result of Bonk and Schramm for hyperbolic groups. |
| title | Maps between relatively hyperbolic spaces and between their boundaries |
| topic | Geometric Topology Group Theory Metric Geometry 20F65, 20F67, 20F69, 31L10, 51F99 |
| url | https://arxiv.org/abs/2012.11902 |