# Construction of geometric cone-manifold structures with

#### by Dan Mathews

Institution: The Univsersity of Melbourne
Date: Wed 13th July 2005
Time: 2:00 PM
Location: Theatre 1, Old Geology Building

Abstract: We know that a geometric structure on a manifold $M$ induces a holonomy
representation $\rho$ of $\pi_1(M)$ into a group of isometries. We can
consider, conversely, this space of representations and ask which amongst
them are holonomy representations. The question can be asked for various
geometries, and under various geometric conditions. We consider
the question for 2-dimensional hyperbolic geometry, and for hyperbolic
structures allowing certain cone-type singularities. We prove results
regarding the representations and hyperbolic cone-manifold structures of
the punctured torus and the genus 2 closed surface. We also shed some new
light on the theorem of Goldman classifying which representations arise as
the holonomy of complete hyperbolic structures.