Maximal Virtual Schottky Groups: Explicit Constructions

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 sg, is a finite dimensional complex manifold that parametrizes quasiconformal deformations of schottky groups of rank g. if g ≥ 2, then sg 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 sg, 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.