Contracting elements and random walks

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>CAT</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Out</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {Out(Fₙ)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {n≥ 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.