Geometry of pseudocharacters

Abstract

If G is a group, a pseudocharacter f: G-->R is a function which is "almost" a homomorphism. If G admits a nontrivial pseudocharacter f, we define the space of ends of G relative to f and show that if the space of ends is complicated enough, then G contains a nonabelian free group. We also construct a quasi-action by G on a tree whose space of ends contains the space of ends of G relative to f. This construction gives rise to examples of "exotic" quasi-actions on trees.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper26.abs.htm

Acylindrical hyperbolicity of groups acting on trees

“…Note that β Groups from [74] and [58] are not finitely presented. On the other hand, there is an evidence that for finitely presented groups, existence of non-trivial quasi-cocycles may be used to construct non-elementary actions on hyperbolic spaces [62]. This motivates the following question:…”

Actions of certain arithmetic groups on Gromov hyperbolic spaces

“…Each such "projective pseudocharacter" gives rise to a quasi-action on R; no two such are equivalent. Moreover, these often give rise to quasi-actions on more complicated trees [17]. The groups SL.2; O/ where O has infinitely many units are more rigid.…”

Large scale geometry of commutator subgroups

“…A group whose Cayley graph is quasi-isometric to an unbounded tree has more than one end (see eg Manning [13], especially Sections 2.1 and 2.2). Hence Theorem A and Theorem B together imply the following: Corollary 5.3 Let G be a nonelementary torsion-free word-hyperbolic group.…”