Translation Numbers of Groups Acting on Quasiconvex Spaces

We define a group to be translation discrete if it carries a metric in which the translation numbers of the non-torsion elements are bounded away from zero. We define the notion of quasiconvex space which generalizes the notion of both CAT(0) and Gromov–hyperbolic spaces. We show that a cocompact group of isometries acting properly discontinuously cocompactly on a proper quasiconvex metric space is translation discrete if and only if it does not contain an essential Baumslag-Solitar quotient. It follows that if such a group is either biautomatic or residually finite then it is translation discrete.