Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, January 23, 2017 - 12:00 to 12:45

Location:

Gilkey 115

Institution:

Graduate Student

Abstract:

Every compact connected 2-manifold N can be expressed as the orbit space determined by a group of isometries G acting freely and properly discontinuously on a Riemannian 2-manifold M which is either the 2-sphere, the Euclidean plane, or the PoincarĂ© disk. In each case, M is simply connected, and so the fundamental group of N is isomorphic to G. In this talk, we express the isometries of the PoincarĂ© disk as complex-valued functions, and determine the groups of isometries which yield as orbit spaces the orientable surfaces of genus at least 2 and the nonorientable surfaces of genus at least 3. We then use these functions to find relations which must hold in the fundamental groups of these surfaces.