September 18
Speaker: Russell Ricks (University of Michigan)
Title: Flat strips in rank one CAT(0) spaces
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. This additional structure allows us to prove some results about the original CAT(0) space X. Here are three such results: First, with respect to the Patterson-Sullivan measure, almost every point in the boundary of X is isolated in the Tit s metric. Second, under the Bowen-Margulis measure, almost no geodesic bounds a flat strip of any positive width. Third, we characterize when the length spectrum 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 a few of the wrinkles involved for CAT(0) spaces.