Arithmetic Groups

A landmark result of Ratner states that if $G$ is a Lie group, $\Gamma$ a lattice in $G$ and if $u_t$ is a one-parameter $Ad$-unipotent subgroup of $G$, then for any $x \in G/\Gamma$ the orbit $u_t.x$ is equidistributed in a periodic orbit of some subgroup $L <G$, and moreover that the orbit of $x$ under $u_t$ is contained in this periodic $L$ orbit.

A key motivation behind Ratner's equidistribution theorem for one-parameter unipotent flows has been to establish Raghunathan's conjecture regarding the possible orbit closures of groups generated by one-parameter unipotent groups; using the equidistribution theorem Ratner proved that if $G$ and $\Gamma$ are as above, and if $H<G$ is generated by one parameter unipotent groups then for any $x \in G/\Gamma$ one has that $\overline{H.x}=L.x$ where $H < L < G$ and $L.x$ is periodic. Important special cases of Raghunathan's conjecture were proven earlier by Margulis and by Dani and Margulis by a different, more direct, approach.

These results have had many beautiful and unexpected applications in number theory, geometry and other areas. A key challenge has been to quantify and effectify these results. Beyond the case of actions of horospheric groups where there are several fully quantitative and effective results available, results in this direction have been few and far between. In particular, if $G$ is semisimple and $U$ is not horospheric no quantitative form of Ratner's equidistribution theorem was known with any error rate, though there has been some progress on understanding quantitatively density properties of such flows with iterative logarithm error rates.

In these two talks, we report on a fully quantitative and effective equidistribution result for orbits of one-parameter unipotent groups in quotients of $SL_2(C)$ and $SL_2(R) \times SL(2,R)$.