Abstract
We generalize Gruber–Sisto’s construction of the coned-off graph of a small cancellation group to build a partially ordered set T C of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned-off graph. In almost all cases T C is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions[G ↷ X][G ↷ Y] in this poset, there is an embeddeding ι : P(ω) → T C such that ι(∅) = [G ↷ X] and ι(ℕ) = [G ↷ Y]. We use this poset to prove that there are uncountably many quasi-isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both.
| Original language | English |
|---|---|
| Pages (from-to) | 325–363 |
| Number of pages | 39 |
| Journal | Geometriae Dedicata |
| Volume | 212 |
| Issue number | 1 |
| Early online date | 26 Aug 2020 |
| DOIs | |
| Publication status | Published - Jun 2021 |
Bibliographical note
Acknowledgements:The authors are grateful to Rémi Coulon and Dominik Gruber for interesting conversations, and for sharing with us the results of their paper. The authors thank the anonymous referee for useful comments. The first author was partially supported by the NSF RTG award DMS-1502553 and by the NSF grant DMS-1803368. The second author was supported by the NSF grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester, and by a Titchmarsh Research Fellowship from the University of Oxford.
Keywords
- Hyperbolic spaces
- Acylindrical actions
- Small cancellation groups
- Largest actions