The dynamics on homogeneous spaces has many interesting connections to
number theory. One of the main problems here is to understand the
distribution of closed orbits for subgroups H of the ambient Lie group
G. We will explain this problem in detail in the case of the horocycle
where it is still possible to draw pictures.|
In joint work with G.Margulis and A.Venkatesh we prove an error rate in the equidistribution for semisimple subgroups H acting on congruence quotients of G. This makes use of spectral gap in the form of property (tau). However, the proof of our theorem can also be used to prove all cases of property (tau) except for groups of type A_1.
We will discuss the relationship between spectral gap, effective decay of matrix coefficients, and effective equidistribution, as well as the main ideas of our argument.
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