On the Hausdorff dimension of subsets of geodesics in hyperbolic surfaces

Given a closed hyperbolic surface S, we can describe the set of geodesics of S as pairs of points in the boundary of hyperbolic space together with the action of the fundamental group of S. In 1985, Birman and Series showed that the set of geodesics with a finite number of self-intersections has zero Hausdorff dimension.

In this talk I will present two generalizations of this result: first, I will show that geodesics whose self-intersection angles are bounded from below form a set of Hausdorff dimension zero. Secondly, the set of geodesics that do not bound a triangle also has Hausdorff dimension zero.