LX annos nato COMPARISON OF HYPERBOLIC AND EXTREMAL LENGTHS

Let S be a hyperbolic Riemann surface of finite type (that is, S: (JIG, whete Uis the upper half-plane and G is a flnitely generated, torsionfree Fuchsian group), and let w be a hyperbolic simple loop on S (that is, w is a simple loop on S, and w is represented by a hyperbolic element A in G). There are two natural notions of length for such a loop: first, there is the hyperbolic length / ofthe shortest geodesic freely homotopic to w on S, and second, there is the extremal length m of the family of loops freely homotopic to w on S. The purpose of this note is to give some com-parisons between these two notions of length. When we need to emphasize the dependence of say / on w, ot A, or,S, we will write /(w), or l(A), or /(w, S). The proofs all take place in the context of a non-elementary finitely generated Fuchsian group; the group may have torsion. All the results are easily seen to be equally valid for elementary Fuchsian groups. In general, we say that a set Xc U is precisely intsariant under the element A<G if A(X):y, and,B(X)n X:0 for all B in G which are not powers of l. we say that the hyperbolic element A€G is strictly simple if the axis Ln of A is precisely invariant under I in G. In particular, if I is strictly simple, then there are no fixed points of elliptic elements of G lying on Ln. For any hyperbolic element A of G, we define l:l(A) to be the geodesic length of A; that is,,4 is conjugate in PSL(2, R) to a unique element of the form z*etz, />0; equivalently, ltr1,l11:2cosh(tl2). If G is torsion free, this definition agrees with that of the first paragraph. We let w be the projection of Ln on UlG, so that w is a geodesic- Let U ' be U with all fixed points of elliptic elements of G deleted, and let S':U'lG. Then m: m(A) is the extremal length of the family of loops freely homotopic to w on,S'. We normalize G so that A(z):etz, l=0. We denote the projection from U to,S, or from U'to S', by p: U*5. A topological collar aboat w is a subsurface So of S', containing w, where 56 is topologically an (open) annulus