Maximal Schottky extension groups

Abstract

A Schottky extension group is a Kleinian group K containing a Schottky group G of rank g ≥  2 as a normal subgroup. It is well known that the index of G in K is at most 12(g − 1); if the index is 12(g − 1), then we say that K is a maximal Schottky extension group. A structural description of the maximal Schottky extension groups using 2-dimensional arguments, internal to Riemann surfaces and classical Kleinian groups in spirit, is provided. As a consequence, we re-obtain Zimmermann’s result which states that a maximal Schottky extension group is isomorphic to one of the following groups

$$D_{2}*_{{\mathbb Z}_{2}} D_{3}, \; D_{3}*_{{\mathbb Z}_{3}} {\mathcal A}_{4}, \;D_{4}*_{{\mathbb Z}_{4}} {\mathfrak S}_{4}, \; D_{5}*_{{\mathbb Z}_{5}} {\mathcal A}_{5},$$

where D _r is the dihedral group of order ${2r, {\mathcal A}_{r}}$ is the alternating group in r letters and ${{\mathfrak S}_{4}}$ is the symmetric group in 4 letters. The methods used by Zimmermann are from combinatorial group theory (finite extensions of free groups) and also dimension three, so our arguments are different.