New Parameters for Fuchsian Groups of Genus 2

We give a new real-analytic embedding of the Teichmüller space of closed Riemann surfaces of genus 2 into R6. The parameters are explicitly defined in terms of the underlying hyperbolic geometry. The embedding is accomplished by writing down four matrices in PSL(2, R), where the entries in these matrices are explicit algebraic functions of the parameters. Explicit inequalities are given to define the image of the embedding; the four matrices corresponding to a point in this image generate a fuchsian group representing a closed Riemann surface of genus 2. In this note we introduce a new, canonical, real-analytic embedding of the Teichmüller space T2, of closed Riemann surfaces of genus 2, onto an explicitly defined region R ⊂ R. The embedding is defined in terms of the underlying hyperbolic geometry; in particular, the parameters are elementary functions of lengths of simple closed geodesics, and angles and distances between simple closed geodesics. We start with a specific marked Riemann surface S0, and a specific set of normalized (non-standard) generators, a0, b0, c0, d0 ∈ PSL(2,R), for the fuchsian group G0 representing S0. Then we can realize a point in T2 as a set of appropriately normalized generators a, b, c, d ∈ PSL(2,R) for the fuchsian group G representing a deformation S of S0. We write the entries in the generators, a, . . . , d, as elementary functions of eight parameters, all defined in terms of the underlying hyperbolic geometry, and we write down explicit formulae expressing two of these parameters as functions of the other six. Three of our six parameters are necessarily positive; we give two additional inequalities to obtain necessary and sufficient conditions for the group G ⊂ PSL(2,R), generated by a, . . . , d, to be an appropriately normalized quasiconformal deformation of G0. There is a related embedding in [3], where the parameters are fixed points of hyperbolic elements of G. As in [3], we identify the Teichmüller space with DF , the identity component of the space of discrete faithful representations of π1(S0) into PSL(2,R) modulo conjugation. The main differences between these two embeddings is that in [3] the four matrices do not have unit determinant and the parameters there are defined in terms of fixed points of elements of the group; here our matrices do have unit determinant, and the parameters are defined in terms of the underlying hyperbolic geometry. Received by the editors October 20, 1997 and, in revised form, February 20, 1998. 1991 Mathematics Subject Classification. Primary 30F10; Secondary 32G15. Research supported in part by NSF Grant DMS 9500557. c ©1999 American Mathematical Society