Projective Deformations of Triangle Tilings

Abstract

A hyperbolic triangle group is the group generated by reflections in the sides of a triangle in hyperbolic space. For a given hyperbolic triangle group, we find a one-parameter group of representations into GL(3,$\R$) and associated invariant cones. We show that the representations are faithful and that the cones are sharp. We then apply the results of Guichard to approximate the H\"older continuity of the boundaries of the cones. We conjecture that this may be directly calculated by considering only the Coxeter elements of the triangle group.