Uller Spaces to Have Smallest Possible Inner Radii

If fR n g 1 n=1 is a sequence of hyperbolic Riemann surfaces such that for any r > 0 a hyperbolic ball of radius r embeds isometrically into R n for all n suuciently large, then as n ! 1 the inner radii of the associated Teichm uller spaces i(T(R n)) ! 2. A straightforward consequence of this is that if a Riemann surface R contains arbitrarily large hyperbolic balls then i(T(R)) = 2. Let ? be a Fuchsian group acting on the upper half plane H , and hence also on the lower half plane H , in the complex plane. Let Q 1 (?) denote the complex Banach space of bounded quadratic diierentials with respect to ? deened in H. The Teichm uller space T(?) of ? is realized via the Bers embedding as a bounded region in Q 1 (?). The inner radius i(?) of T(?) is the supremum of radii of balls in Q 1 (?) centered at the origin which are contained in T(?). It has long been known 2] that if H carries the hyperbolic metric of curvature ?4 then (1:1) i(?) 2 whenever T(?) 6 = f0g: If ? is an elementary group then we have equality in (1.1), 4]. On the other hand it can be shown, following Gehring and Pommerenke 3], that the inequality in (1.1) is strict for coonite ? ((6], 11]). We now state our main theorem: