The topology of Schottky spaces in higher dimensions

Abstract

The marked Schottky space records, up to conjugacy, all actions of a free group of fixed rank as a Schottky group on hyperbolic space of fixed dimension. In dimension three it is the classical Schottky space covering the moduli space of Riemann surfaces, studied complex-analytically. In higher dimensions each generator gains a rotational parameter, a special orthogonal transformation of the directions normal to its axis, with no classical analogue. Our main theorem treats the borderline dimension, twice the rank: there a dense open part of the space has fundamental group a product of cyclic groups of order two, one per generator, yet the whole space is simply connected, since each such loop contracts through the most degenerate configurations. As a consequence, any two Schottky groups of the same rank in this borderline dimension are quasiconformally isotopic, partially answering a question of Kapovich. We also show that a rotationally symmetric core is a strong deformation retract in every dimension, that this dense open part is homotopy equivalent to a product of special orthogonal groups, and that the analogous locus one dimension below has two connected components.