Counting geodesics of given commutator length
Viveka Erlandsson (Univ. Bristol, Reino Unido)

It is a classical result by Huber that the number of closed geodesics of length bounded by L on a closed hyperbolic surface is asymptotic to eL=L as L grows. This result has been generalized in many directions, for example by counting certain subsets of closed geodesics. Mirzakhani obtained the asymptotic growth of simple geodesics of given type, for example those that bound an embedded genus g surface. In another direction, the growth of geodesics that are homologically trivial was obtained independenttly by Phillips-Sarnak and Katsura-Sunada A homologically trivial curve can be written as a product of commutators, and in this talk we will look at those that can be written as a product of g commutators and obtain their asymptotic growth. As a consequence we also get the growth of geodesics that bound an immersed genus g surface. As a special case, our methods give a geometric proof of Huber’s classical theorem. This is joint work with Juan Souto.