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

  • JÜRGEN MÜLLER (a1) and SIDDHARTHA SARKAR (a2)

Abstract

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.

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1. Breuer, T., Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series 280 (Cambridge University Press, Cambridge, UK, 2000).
2. Douady, A. and Douady, R., Algèbre et théories galoisiennes, volume 2 : Théories galoisiennes (CEDIC, Paris, 1979).
3. The GAP Group, GAP – groups, algorithms, programming – a system for computational discrete algebra, version 4.8.6 (2016), Available at: http://www.gap-system.org
4. Harvey, W. J., Cyclic groups of automorphisms of a compact Riemann surface, Q. J. Math. Oxf. Ser. (2) 17 (1966), 8697.
5. Hurwitz, A., Über algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1893), 403442.
6. Kulkarni, R., Symmetries of surfaces, Topology 26 (1987), 195203.
7. Kulkarni, R. and Maclachlan, C., Cyclic p-groups of symmetries of surfaces, Glasgow Math. J. 33 (1991), 213221.
8. Maclachlan, C., Abelian groups of automorphisms of compact Riemann surfaces, Proc. Lond. Math. Soc. 3–15 (1965), 699712.
9. Maclachlan, C. and Talu, Y., p-groups of symmetries of surfaces, Mich. Math. J. 45 (1998), 315332.
10. McCullough, D. and Miller, A., A stable genus increment for group actions on closed 2-manifolds, Topology 31 (1992), 367397.
11. Sah, C., Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 1342.
12. Sarkar, S., On the genus spectrum for p-groups of exponent p and p-groups of maximal class, J. Group Theory 12 (2009), 3954.
13. Talu, Y., Abelian p-groups of symmetries of surfaces, Taiwan. J. Math. 15 (3) (2011), 11291140.

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