**Joint seminar with Geometric Group Theory**

Abstract: Let X be a proper, geodesically complete CAT(0) space under a geometric (that is, properly discontinuous, cocompact, and isometric) group action on X; further assume X admits a rank one axis. Using the Patterson-Sullivan measure on the boundary, we construct a generalized Bowen-Margulis measure on the space of geodesics in X. However, in order to construct this measure, we must prove a couple structural results about the original CAT(0) space X. First, with respect to the Patterson-Sullivan measure, almost every point in the boundary of X is isolated in the Tits metric. Second, under the Bowen-Margulis measure, almost no geodesic bounds a flat strip of any positive width. Then, with the generalized Bowen-Margulis measure, we can characterize when the length spectrum of X is arithmetic (that is, the set of translation lengths is contained in a discrete subgroup of the reals). In this talk, we will discuss the constructions and some of the issues involved.