Geometry of pseudocharacters

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 quasiaction 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.