Journal of Modern Dynamics (JMD)

Effective equidistribution of translates of maximal horospherical measures in the space of lattices

Pages: 229 - 254, Volume 10, 2016      doi:10.3934/jmd.2016.10.229

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Kathryn Dabbs - Department of Mathematics, University of Texas, 1 University Station, Austin, TX 78712, United States (email)
Michael Kelly - Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, United States (email)
Han Li - Department of Mathematics and Computer Science, Wesleyan University, 265 Church Street, Middletown, CT 06459, United States (email)

Abstract: 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 the maximal horospherical measures. In this paper we prove the corresponding effective equidistribution results in the space of unimodular lattices. We also prove the corresponding results for probability measures with absolutely continuous densities in rank two and three. Then we address the problem of determining the error terms in two counting problems also considered by Mohammadi and Salehi-Golsefidy. In the first problem, we determine an error term for counting the number of lifts of a closed horosphere from an irreducible, finite-volume quotient of the space of positive definite $n\times n$ matrices of determinant one that intersect a ball with large radius. In the second problem, we determine a logarithmic error term for the Manin conjecture of a flag variety over $\mathbb{Q}$.

Keywords:  Effective equidistribution, horospherical measures, horospherical subgroups, convergence cone.
Mathematics Subject Classification:  Primary: 37C85, 37P30, 22E40.

Received: June 2015;      Revised: February 2016;      Available Online: July 2016.