Annales Academi½ Scientiarum Fennic½ Mathematica Volumen 26, 2001, 267{304 MATRICES FOR FENCHEL{NIELSEN COORDINATES Bernard Maskit The University at Stony Brook, Mathematics Department Stony Brook NY 11794-3651, U.S.A.;
[email protected] Abstract. We give an explicit construction of matrix generators for ¯nitely generated Fuch- sian groups, in terms of appropriately de¯ned Fenchel{Nielsen (F-N) coordinates. The F-N coordi- nates are de¯ned in terms of an F-N system on the underlying orbifold; this is an ordered maximal set of simple disjoint closed geodesics, together with an ordering of the set of complementary pairs of pants. The F-N coordinate point consists of the hyperbolic sines of both the lengths of these geodesics, and the lengths of arc de¯ning the twists about them. The mapping from these F-N coordinates to the appropriate representation space is smooth and algebraic. We also show that the matrix generators are canonically de¯ned, up to conjugation, by the F-N coordinates. As a corol- lary, we obtain that the TeichmullerÄ modular group acts as a group of algebraic di®eomorphisms on this Fenchel{Nielsen embedding of the TeichmullerÄ space. 1. Introduction There are several di®erent ways to describe a closed Riemann surface of genus at least 2; these include its representation as an algebraic curve; its representation as a period matrix; its representation as a Fuchsian group; its representation as a hyperbolic manifold, in particular, using Fenchel{Nielsen (F-N) coordinates; its representation as a Schottky group; etc. One of the major problems in the overall theory is that of connecting these di®erent visions.