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Geometriae Dedicata

, Volume 146, Issue 1, pp 141–158 | Cite as

Maximal Schottky extension groups

  • Rubén A. Hidalgo
Original Paper

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.

Keywords

Schottky groups Automorphisms Riemann surfaces 

Mathematics Subject Classification (2000)

30F10 30F40 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Técnica Federico Santa MaríaValparaísoChile

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