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Classification of p-groups of automorphisms of Riemann surfaces and their lower central series

  • Reza Zomorrodian (a1)

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In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local form

The restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence about the finite nilpotent groups.

This suggests a new problem of determining for each prime p, the “p-group” analogue of Hurwitz theorem. It turns out, as often happens in questions of this nature, that p = 2 and p = 3 are indeed quite exceptional and harder to deal with while computing their lower central series than other primes. Actually, p = 3 is the most difficult, but all the other primes p ≥ 5 can be dealt with at once.

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References

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1.Burnside, W., Theory of groups of finite order (New York, 1955).
2.Johnson, D. L., Presentation of groups, London Mathematical Society Lecture Note Series No. 22.
3.Hoare, A. H. M., Karrass, A. and Solitar, D., Subgroups of infinite index in Fuchsian groups, Math. Z. 125 (1972), 5969.
4.Hurwitz, A., Ueber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. 41 (1892), 403–42.
5.Macbeath, A. M., Residual nilpotency of Fuchsian groups,Illinois J. Math. (2) 28 (1984), 299311.
6.Maclachlan, C., Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. (3) 15 (1965), 699712.
7.Zomorrodian, R., Nilpotent automorphism groups of Riemann surfaces, Trans. Amer. Math. Soc. 288 (1985), 241255.

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