COORDINATES FOR THE TEICHMÜLLER SPACE OF PLANAR SURFACE N.E.C. GROUPS

Abstract

It is proved that every planar surface NEC group Γ admits a fundamental region, called a normal region, which is a hyperbolic right-angled polygon with several pairs of identified sides according to a pattern. Conversely, each such a polygon can be taken as a fundamental region of a planar surface NEC group. By means of these regions the Teichmüller space of Γ is studied, obtaining a parametrization by means of congruence classes of marked polygons. The parameters are certain lengths in these polygons, and from them explicit matrices of the generators of the associated group are obtained.