Effective equidistribution results for horospherical actions on the space of lattices

In the first part of this thesis, we are concerned with effective equidistribution of translates of horospherical measures in the space of lattices. Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient conditions under which certain translates of homogeneous measures converge, and they determined the limiting measures in the cases of convergence. The class of measures they considered includes maximal horospherical measures. In this thesis, we prove the corresponding effective equidistribution results in the space of unimodular lattices: SLn(R)/SLn(Z). We also prove the corresponding results for probability measures with absolutely continuous densities in rank two and three. In the second part of this thesis, we consider more general homogeneous spaces: (M semi-direct product W)/Γ, where M is semisimple and W is unipotent. We prove a mixing result for expanding horopsherical subgroups of M semi-direct product W, and we use this result to classify measures invariant under the action of horospherical subgroups of SLn(R).Mathematic