### Directions in Orbits of Discrete Hyperbolic Subgroups

Ergodic Theory and Dynamical Systems Seminar

18th October 2018, 2:00 pm – 3:00 pm

Howard House, 4th Floor Seminar Room

I will discuss a powerful theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume exceptions. This result holds for general geometrically finite subgroups (i.e subgroups generating infinite volume hyperbolic manifolds).

Unlike in the lattice case, orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather, they may equidistribute on a fractal subset. Understanding the behavior of these orbits near the boundary has proved crucial in the study of Patterson-Sullivan theory, affine sieve techniques and hyperbolic geometry.

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