### To Infinity and Beyond in Moduli Space

By Maxime Fortier-Bourque

March 6, 2013

#### Abstract

We start with a compact surface S of genus g > 1 assembled from cylinders. Then we modify S by stretching each cylinder by a factor e^t. This gives a geodesic path S_t in moduli space which goes to infinity. For each t, S_t admits a unique complete hyperbolic metric compatible with its conformal structure. We prove that the geometry of S_t in that metric stabilizes as t goes to infinity in the sense that its Fenchel-Nielsen coordinates converge. There will be pictures.