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In this final lecture, we establish a Ratner-type theorem for actions of the special linear group SL_2({\Bbb R}) on homogeneous spaces. More precisely, we show:

Theorem 1. Let G be a Lie group, let \Gamma < G be a discrete subgroup, and let H \leq G be a subgroup isomorphic to SL_2({\Bbb R}). Let \mu be an H-invariant probability measure on G/\Gamma which is ergodic with respect to H (i.e. all H-invariant sets either have full measure or zero measure). Then \mu is homogeneous in the sense that there exists a closed connected subgroup H \leq L \leq G and a closed orbit Lx \subset G/\Gamma such that \mu is L-invariant and supported on Lx.

This result is a special case of a more general theorem of Ratner, which addresses the case when H is generated by elements which act unipotently on the Lie algebra {\mathfrak g} by conjugation, and when G/\Gamma has finite volume. To prove this theorem we shall follow an argument of Einsiedler, which uses many of the same ingredients used in Ratner’s arguments but in a simplified setting (in particular, taking advantage of the fact that H is semisimple with no non-trivial compact factors). These arguments have since been extended and made quantitative by Einsiedler, Margulis, and Venkatesh.
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