Monodromy groups : the minimum genus problem

Silver, Stephen Andrew (1990) Monodromy groups : the minimum genus problem. University of Southampton, Doctoral Thesis.

Record type: Thesis (Doctoral)

Abstract

Every finite group is isomorphic to the monodromy group of some Riemann surface. In this thesis the minimum genus of those Riemann surfaces with monodromy group isomorphic to the finite group G is investigated for various G. A method is found for using groups of minimum genus 0 to produce wreath products of the same minimum genus. The minimum genus of many Frobenius groups with cyclic Frobenius subgroups, including all those in which the Frobenius kernel is also cyclic, is evaluated. The minimum genus of all dicyclic groups is found. For various p-groups the minimum genus is again evaluated. Many simple groups are investigated. In particular the minimum genus is evaluated for all Suzuki groups, for all PSL₂(q) Hurwitz groups and for all PSL₂(q) groups in which q is a power of 2 or 3, although in this last case the calculation of the minimum genus for a particular q depends on at least a partial factorization of a certain integer. PSL₂(41) has the unusual property of being a Hurwitz group in which the minimum genus is not attained by a (2, 3, 7) generating triple. It is found to be the only such PSL₂(q) Hurwitz group. An investigation is carried out with the A_n Hurwitz groups and all but finitely many are proved not to have the property.

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