**Title:** Rigidity for actions on infinite volume homogeneous spaces

**Speaker:** Professor Alex Furman

**Speaker Info:** University of Illinois at Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

We shall discuss measurable rigidity phenomena for group actions on infinite homogeneous spaces, such as the following result: THM: Suppose that two abstractly isomorphic lattices L_1 and L_2 in SL(2,R) admit a measurable map T of the plane R^2 which intertwines their linear actions. Then L_1 and L_2 are necessarily conjugate and T is a linear map realizing this conjugation.This particular theorem was found by Y.Shalom and T.Steger, who have deduced this and similar rigidity results from the study of unitary representations of lattices. We shall discuss a different purely dynamical approach which gives a broader picture of measurable rigidity on certain infinite measure spaces.

Curiously, the above theorem can be considered as a dual to the horocycle rigidity results of M. Ratner (1982). In this case this duality is fruitful in posing questions but does not seem to help in proofs.

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