Seminar 9.25.14 Ricks

Joint seminar with Geometric Group Theory
Speaker: Russell Ricks (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.  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.

Seminar 9.18.14 Yang

Speaker: Lei Yang (Yale)

Title: Equidistribution of evolution of curves in homogeneous space under diagonal flow

Abstract:  In this talk, we consider a compact analytic curve φ : I → H = SO(n, 1) and embed it into some homogeneous space G/Λ where H ⊂ G and HΛ is dense in G. Fix a maximal R-split Cartan subgroup A = {at : t ∈ R}, we wonder under which condition the expanded curves {atφ(I) : t > 0} tend to be equidistributed. It turns out that it is true if the image of φ(I)is not contained in any proper totally geodesic submanifold of H. It answers a question of Nimish Shah and extends his previous result. And then we will talk the main idea to prove the same result if we replace the analytic curve by only smooth curve, this project is ongoing and joint with Nimish Shah.