On the properness of some algebraic equations appearing in Fuchsian Groups

Abstract

In this paper we give necessary and sufficient conditions on three orientation-preserving hyperbolic isometries T1,T2,T3 so that for any point x ∈ H2 the orthogonal bisectors of the three segments with endpoints x, Ti(x) intersect. This is equivalent to proving the properness of certain algebraic sets. As a corollary, we give a new proof for the existence and density of generic fundamental polygons for cocompact Fuchsian groups.