Maximal virtual schottky groups: explicit constructions (2010)
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 $g \in \{0,1\}$. Moreover, for each $g \in \{0,1\}$ 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 $\mathcal{S}_{g}$, is a finite dimensional complex manifold that parametrizes quasiconformal deformations of Schottky groups of rank $g$. If $g ≥ 2$, then $\mathcal{S}_{g}$ 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 $\mathcal{S}_{g}$, 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.