### Or Landesberg (Yale University)

Tuesday, November 29, 2022, 14:30 – 15:30, Math -101

Abstract:

Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood.

We consider $\mathbb{Z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of horocycle orbit closures is known. Based on an ongoing joint work with James Farre and Yair Minsky.