CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 4, Pages 97–107 (August 23, 2000) S 1088-4173(00)00054-0 MATRIX REPRESENTATIONS AND THE TEICHM ¨ ULLER SPACE OF THE TWICE PUNCTURED TORUS J. O. BUTTON Abstract. We realise the Teichm¨ uller space of the twice-punctured torus as a set of triples of matrices that are suitably normalised. As a consequence, we see the space as a simple open subset of R 4 which is obtained directly from the matrix entries. We also discuss the connection between this representation and the one in terms of the traces of elements. 0. Introduction The Teichm¨ uller space of a surface is an important object, and as evidence of its importance we see in the literature many definitions, seemingly different but all ultimately the same. For instance, topologists would define it in terms of diffeomorphisms isotopic to the identity (see [19]), and complex analysts would define it in terms of quasi-conformal mappings (see [1]). However we do this though, as a topological space it is homeomorphic to R n for the appropriate value of n, and a lot of effort goes into parametrising it, namely embedding it into R n as a connected open subset, where the coordinates of a point in Teichm¨ uller space can be suitably interpreted in terms of the original definition. Of course different definitions will yield different subsets, and there will be variations depending on what properties we wish this embedding to have. Many approaches are based on considering hyperbolic structures on the surface, along with a marking of the fundamental group. This is the approach taken by Bers (see [2]) where the points are bounded quadratic differentials, and by Thurston using projective measured laminations (see [6]), and in both these cases Teichm¨ uller space will be embedded into R n as a bounded domain, thus giving rise to a compactification which allows interpretation of points on the boundary. Another approach using the hyperbolic structure is to find parameters that relate to the surface as closely as possible, for instance if the surface has a hyperbolic structure, then we can use the lengths of certain closed geodesics and twisting angles. See [7] for this approach. However, in this paper we take the view that a surface with a hyperbolic struc- ture is given to us concretely by a Fuchsian subgroup, namely the discrete subgroup of PSL(2, R) by which we quotient the upper half plane to obtain this structure, which is determined by this subgroup up to conjugacy. Thus to study the group is to study the surface, and we use an equivalent definition of Teichm¨ uller space as isomorphisms of a base group into PSL(2, R) up to conjugacy that preserve para- bolic elements and whose image is discrete. This embeds naturally into the space of all representations of the base group into PSL(2, R) which preserve parabolics. Received by the editors August 16, 1999 and, in revised form, July 10, 2000. 2000 Mathematics Subject Classification. Primary 20H10; Secondary 32G15. c 2000 American Mathematical Society 97