Geometry of pseudocharacters

Abstract

If $G$ is a group, a pseudocharacter $f:G\to ℝ$ 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.