Abstract

HIDALGO, RUBÉN A.. Maximal Virtual Schottky Groups: Explicit Constructions. Rev.colomb.mat. [online]. 2010, vol.44, n.1, pp.41-57. ISSN 0034-7426.

A Schottky group of rank g is a purely loxodromic Kleinian group, with non-empty region of discontinuity, isomorphic to the free group of rank g. A virtual Schottky group is a Kleinian group K containing a Schottky group Γ as a finite index subgroup. In this case, let g be the rank of Γ. The group K is an elementary Kleinian group if and only if g ∈ {0,1}. Moreover, for each g ∈ {0,1} and for every integer n ≥ 2, it is possible to find K and Γ as above for which the index of Γ in K is n. If g ≥ 2, then the index of Γ in K is at most 12(g-1). If K contains a Schottky subgroup of rank g ≥ 2 and index 12(g-1), then K is called a maximal virtual Schottky group. We provide explicit examples of maximal virtual Schottky groups and corresponding explicit Schottky normal subgroups of rank g ≥ 2 of lowest rank and index 12(g-1). Every maximal Schottky extension Schottky group is quasiconformally conjugate to one of these explicit examples. Schottky space of rank g, denoted by S_g, is a finite dimensional complex manifold that parametrizes quasiconformal deformations of Schottky groups of rank g. If g ≥ 2, then S_g has dimension 3(g-1). Each virtual Schottky group, containing a Schottky group of rank g as a finite index subgroup, produces a sublocus in S_g, called a Schottky strata. The maximal virtual Schottky groups produce the maximal Schottky strata. As a consequence of the results, we see that the maximal Schottky strata is the disjoint union of properly embedded quasiconformal deformation spaces of maximal virtual Schottky groups.

Keywords : Schottky groups; Kleinian groups; Automorphisms; Riemann surfaces.

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