Growth tight actions
By Goulnara Arzhantseva, Christopher H. Cashen, Jing Tao. 2014
53 citations recorded by the source corpus.
We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grig- orchuk and de la Harpe. Given a nitely generated, non-elementary group G acting on a G{space X , we prove that if G contains a strongly contracting el- ement and if G is not too badly distorted in X , then the action of G on X is a growth tight action. It follows that ifX is a cocompact, relatively hyperbolic G{space, then the action of G onX is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some innite, innite index normal subgroups is relatively hyperbolic, and, conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that certain CAT(0) graphs of groups and snowake groups admit cocompact, growth tight actions. These provide rst examples of non-relatively hyperbolic groups ad- mitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmuller space with the Teichmuller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/48953722b3cf733184c489cafef5fd822f4c569c
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Statistically convex-cocompact actions of groups with contracting elements (2016, methodology): “stants M1;M2 > 0 such that !(OM1;M2 ) < !(G), then the action of Gon Y is called statistically convex-cocompact (SCC). Remark. (1)This underlying concept was introduced by Arzhantseva et al. in [2] as a generalization of a parabolic gap condition due to Dal’bo et al. [28]. We propose the parti”