A landmark result of Ratner states that if G is a Lie group, Γ a
lattice in G and if ut is a one-parameter Ad-unipotent subgroup of
G, then for any x∈G/Γ the orbit ut.x is equidistributed in a
periodic orbit of some subgroup L
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 Γ are as
above, and if H
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 SL2(C) and SL2(R)×SL(2,R).
