Contracting elements and random walks

2011/12/12 by Sisto, Alessandro · 3 citations
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1112.2666

Abstract

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 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 Out(F_n), n≥ 3. 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. Also, we show that each (weakly) contracting element is contained in a hyperbolically embedded elementary subgroup.

Cited by

• Growth gaps and exponential genericity in acylindrically hyperbolic groups
• Boundary actions of CAT(0) spaces: topological freeness and applications to C^*-algebras
• On Similarity Structure Groups and their W^* and C^*-Algebras

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