Generalized Unipotent Subgroups in Word Hyperbolic Groups
Lewis Bowen (Univ.Hawaii)

Abstract: Unipotent subgroups play a large role in the ergodic theory of semisimple Lie group actions. Word hyperbolic groups do not contain parabolic elements and therefore do not have unipotent subgroups. In spite of this, they do have "generalized subgroups" that have unipotent-like properties. These objects are measured equivalence relations on the space of horospheres. The main result is that if G is a word hyperbolic group acting ergodically on a probability space (X,mu) then the induced "action" of one of these "subgroups" is almost ergodic. I'll show that this can be applied to obtain a pointwise ergodic theorem for word hyperbolic groups with ball and spherical averages if the action space is finite.