Abstract
We study the distribution of non-discrete orbits of geometrically finite groups in {text {SO}}(n,1) acting on {mathbb {R}}^{n+1}, and more generally on the quotient of {text {SO}}(n,1) by a horospherical subgroup. Using equidistribution of horospherical flows, we obtain both asymptotics for the distribution of orbits for the action of general geometrically finite groups, and we obtain quantitative statements with additional assumptions.
Highlights
We often seek to understand a group through the distribution of its orbits on a given space
/2, where the implied constant depends on v and
The dependence of T on x in Theorem 1.9 arises from the constant in Lemma 3.2, which is explicitly defined in that proof, and the precise Diophantine nature of x, through Theorem 2.15
Summary
We often seek to understand a group through the distribution of its orbits on a given space. Gorodnik and Nevo comprehensively studied the action of a lattice in a connected algebraic Lie group acting on infinite volume homogeneous varieties in [6], including obtaining quantitative results under appropriate assumptions. The case when has infinite covolume was recently studied by Maucourant and Schapira in [12], where they obtained an asymptotic version of Ledrappier’s result for convex cocompact subgroups of SL2(R), with a scaling factor permitted. They prove that an ergodic theorem like Ledrappier’s in the lattice case cannot be obtained in the infinite volume setting, because there is not even a ratio ergodic theorem. It measures the (1, n + 1) component of g
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