Groups Acting on Quasiconvex Spaces and Translation Numbers

We prove that groups acting geometrically on δ-quasiconvex spaces contain no essential Baumslag-Solitar quotients as subgroups. This implies that they are translation discrete, meaning that the translation numbers of their nontorsion elements are bounded away from zero. The notion of translation numbers is used by many authors, such as J. Alonso and M. Bridson [1], G. Conner [2], S.M. Gersten and H. Short [3] and M. Gromov [5]. If a group is equipped with a word metric, then the translation number of an element in the group may be viewed as the “length” of that element in the group. In “Hyperbolic Groups”, Gromov claims that the translation numbers in negatively curved groups are rational with bounded denominator (see also [6]). Conner [2] proves a similar result for CAT(0) groups, namely that they are translation discrete, meaning that the translation numbers of their nontorsion elements are bounded away from zero. He also gives necessary and sufficient conditions for translation discreteness of quasiconvex groups i.e. groups acting geometrically on quasiconvex spaces. In this paper we prove that these groups, which include all negatively curved and all CAT(0) groups, contain no essential Baumslag-Solitar quotients as subgroups, which in turn implies that they are translation discrete. Definition. A geodesic triangle in a metric space X is δ-midpoint convex, δ > 0, if the distance between the midpoints of any two sides of the triangle is no more than half the length of the remaining side plus δ. Definition. A geodesic metric space X is δ-quasiconvex, if all geodesic triangles in X are δ-midpoint convex. Using the tripod definition of negatively curved space, one can see that every δ-thin triangle is 2δ-midpoint convex. Therefore, every negatively curved space is quasiconvex. On the other hand, every geodesic triangle in a CAT(0) space is 0-midpoint convex and thus every CAT(0) space is δ-quasiconvex, for every δ. Received by the editors January 5, 1999. 2000 Mathematics Subject Classification. Primary 20F65. This paper forms a part of the author’s Ph.D. dissertation written under the direction of P. Bowers at Florida State University. c ©2000 American Mathematical Society