A modular group and Riemann surfaces of genus 2

Abstract

Let T(S) be the Teichmiiller space of Compact Riemann surfaces of genus 2 and let M(S) be the corresponding Teichmiiller modular group. In this paper, we determine a fundamental region R(S) for M(S) acting on T(S). R(S) is constructed as a subspace of T(S), as it is described in [8-10]. We make use of and extend the results in [12] to fully describe the action of M(S) on the parameters determining T(S). Our construction also depends strongly on a theorem of Bers [3]. In Section I, we recall some basic definitions, and determine our notation. In Section II, we describe the moduli space for a torus with a hole, the action of the corresponding modular group on this space, and determine a fundamental domain for this modular group. In Section III we describe the space T(S) and in Section IV we discuss the modular group and study its action on T(S). In Section V we state the theorem of Bers in preparation for the actual construction in Section V1. Finally, in Section VII we use the constructed fundamental domain to prove a conjecture of Bets in [2] for the case of Riemann surfaces of genus 2.