Seminar

In the hyperbolic plane H^2, horocycles are given by Euclidean circles tangent to the boundary. In 1936, Hedlund showed that in a compact hyperbolic surface Gamma\H^2, every horocycle is dense; this is one of the first theorems on orbit closures in homogeneous spaces.

Hedlund’s approach was significantly generalized by Margulis  in his proof of Oppenheim conjecture on values of quadratic forms  in 1989.

In 1991, Ratner gave a complete description of all possible orbit closures on a finite volume homogeneous space under the action of a subgroup generated by unipotent one-parameter subgroups, based on her measure classification theorem.

 We will discuss certain topological rigidity theorems in infinite volume hyperbolic manifolds of dimension 2 or 3 which can be approached by generalizing the ideas of Margulis.