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A STRUCTURED DESCRIPTION OF THE GENUS SPECTRUM OF ABELIAN p-GROUPS

Part of: Abelian groups Riemann surfaces Other groups of matrices

Published online by Cambridge University Press:  10 July 2018

JÜRGEN MÜLLER  and
SIDDHARTHA SARKAR
JÜRGEN MÜLLER
Affiliation:
Arbeitsgruppe Algebra und Zahlentheorie, Bergische Universität Wuppertal, Gauß-Straße 20D-42119 Wuppertal, Germany e-mail: juergen.mueller@math.uni-wuppertal.de
SIDDHARTHA SARKAR
Affiliation:
Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Indore-Bypass Road, Bhauri, Bhopal 462066, India e-mail: sidhu@iiserb.ac.in
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Abstract

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The genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the Abelian p-groups. Motivated by earlier work of Talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of Abelian p-groups, including the reduced minimum genus. In particular, we determine the complete genus spectrum for a large subclass, namely, those having ‘large’ defining invariants. With our method we construct infinitely many counterexamples to a conjecture of Talu, which states that an Abelian p-group is recoverable from its genus spectrum. Finally, we give a series of examples of our method, in the course of which we prove, for example, that almost all elementary Abelian p-groups are uniquely determined by their minimum genus, and that almost all Abelian p-groups of exponent p2 are uniquely determined by their minimum genus and Kulkarni invariant.

MSC classification

Primary: 20H10: Fuchsian groups and their generalizations
Secondary: 20K01: Finite abelian groups 30F35: Fuchsian groups and automorphic functions 57S25: Groups acting on specific manifolds
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 2 , May 2019 , pp. 381 - 423
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

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