Description:

Title:
Distribution of dense lattice orbits in homogeneous spaces

Abstract:
Let H < G both be noncompact connected semisimple real algebraic groups where the former is maximal proper and Γ < G be a lattice. Gorodnik-Weiss showed that the distribution of dense Γ-orbits in the homogeneous space G/H is asymptotically the same as that of G-orbits which can then be described in terms of limiting ratios of certain volumes. One key ingredient in their proof is Shah's theorem which is derived from the famous Ratner's theorem. In a joint work with Zuo Lin, we prove an effective version of this result assuming that the effective version of Shah's theorem holds in general. Thanks to the pioneering work of Lindenstrauss-Mohammadi-Wang, such an effective version of Shah's theorem indeed holds for (G, H) = (SL_2(C), SL_2(R)) and Γ arithmetic. We thus obtain an unconditional theorem in this case.