Contracting elements and random walks
By Alessandro Sisto. Journal für die Reine und Angewandte Mathematik, 2011
125 citations recorded by the source corpus.
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper {\mathrm{CAT}(0)} spaces, elements acting hyperbolically on the Bass–Serre tree in graph manifold groups. We also define a related notion of weakly contracting element, and show that those coincide with hyperbolic elements in groups acting acylindrically on hyperbolic spaces and with iwips in {\mathrm{Out}(F_{n})} , {n\geq 3} . We show that each weakly contracting element is contained in a hyperbolically embedded elementary subgroup, which allows us to answer a problem in [16]. We prove that any simple random walk in a non-elementary finitely generated subgroup containing a (weakly) contracting element ends up in a non-(weakly-)contracting element with exponentially decaying probability.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f7853c36606b88177e11135f0a3818b2837396c1
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Citations in the corpus
- GROUPS ACTING ACYLINDRICALLY ON HYPERBOLIC SPACES (2017, unknown): “uch that Out(F n) acts on it and the action of fsatises the weak proper discontinuity condition. Thus Out(F n) is acylindrically hyperbolic by Theorem 2.14. (c) Groups acting on CAT(0) spaces. Sisto [59] showed that if a group Gacts properly on a proper CAT(0) space and contains a rank one element, ”
- Statistically convex-cocompact actions of groups with contracting elements (2016, background): “lic isometry in hyperbolic spaces, signifying a kind of negative curvature in directions. The reader should distinguish our denition of contracting element with an other one in literatures (eg. [7], [82]), where the projection is an abstract one but we use the shortest point projection. A big dieren”
- Hyperbolicities in CAT(0) cube complexes (2017, unknown): “[Sis11] Let G be a non virtually cyclic group acting properly discontinuously on a CAT(0) space.”