Robert Suzzi Valli

Combinatorial Growth of Reciprocal Geodesics in the Modular Group

Consider the modular surface, that is, the (2,3,∞) triangle orbifold. A reciprocal geodesic on the modular surface is a closed geodesic that begins and ends at the order-two cone point, traversing its image twice. In this talk we consider an exhaustion of the modular surface by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so called low lying reciprocal geodesics) converges to the growth rate of the full set of reciprocal geodesics on the modular surface. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics. This is joint work with Ara Basmajian.