Large parameter spaces of quasiconformally distinct hyperbolic structures

Abstract

SupposeS is a surface of infinite type with pants decomposition ⌆. We construct a real analytic embedding of an infinite-dimensional parameter space into the Fenchel-Nielsen space ofS with respect to ⌆, whose image is made up of topologically conjugate Fuchsian groups for which no two groups in the image are quasiconformally conjugate. Moreover, all of the Fuchsian groups in this parameter space have the same Fenchel—Nielsen twist parameters. As a consequence, arbitrarily close (in the Fenchel-Nielsen topology) to a hyperbolic structure forS there is an infinite-dimensional set of disjoint Teichmuller spaces.