Geometry and Topology Seminar

Let X be a proper, geodesically complete CAT(0) space under a properly discontinuous, cocompact, and isometric group action; 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.  First, with respect to the Patterson-Sullivan measure, almost every point in the boundary 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.