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Fixed Points of Automorphisms of Free Pro-p Groups of Rank 2

Published online by Cambridge University Press:  20 November 2018

Wolfgang N. Herfort ,
Luis Ribes  and
Pavel A. Zalesskii
Wolfgang N. Herfort
Affiliation:
Institute fi Angew. und Numer. Mathematik Technische Universität Wien A-1040 Wien Austria e–mail: herfort@uranus.tuwien.ac.at
Luis Ribes
Affiliation:
Institute fi Angew. und Numer. Mathematik Technische Universität Wien A-1040 Wien Austria e–mail: herfort@uranus.tuwien.ac.at
Pavel A. Zalesskii
Affiliation:
Institute of Techn. Cybernetics Academy of Sciences 220605 Minsk Byelorussia e–mail: mahaniok%bas10.basnet.minsk.by@demos.su
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Abstract

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Let p be a prime number, and let F be a free pro-p group of rank two. Consider an automorphism α of F of finite order m, and let FixF(α) = {x ∈ F | α(x) = x} be the subgroup of F consisting of the elements fixed by α. It is known that if m is prime to p and α = idF, then the rank of FixF(α) is infinite. In this paper we show that if m is a finite power pr of p, the rank of FixF(α) is at most 2. We conjecture that if the rank of F is n and the order of a is a power of α, then rank (FixF(α)) ≤ n.

Keywords

20E1820E3622C05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 383 - 404
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Bestvina, M. and Handelm, M., Train tracks and automorphisms of free groups, Ann. of Math. 135(1992), 1–51.Google Scholar
2. Binz, D., Neukirch, J. and Wenzel, G.H., y4 subgroup theorem for free products ofproftnite groups, J. Algebra 19(1971), 104–109.Google Scholar
3. Dyer, J.L. and Scott, G.P., Periodic automorphisms of free groups, Comm. Algebra (3) 3(1975), 195–201.Google Scholar
4. Fried, M.D. and Jarden, M., Field Arithmetic, Springer-Verlag, Berlin, 1986.Google Scholar
5. Gersten, S.M., Fixed points of automorphisms of free groups, Invent. Math. 84(1986), 91–119.Google Scholar
6. Gildenhuys, D. and Lim, C.K., Free pro-C groups, Math. Z. 125(1972), 233–254.Google Scholar
7. Gildenhuys, D. and Ribes, L., A Kurosh subgroup theorem for free pro-C groups, Trans Amer. Math. Soc. 186(1973), 30^-329.Google Scholar
8. Gildenhuys, D., Profinite groups and Boolean graphs, J. Pure Appl. Algebra 12(1987), 21–47.Google Scholar
9. Herfort, W. and Ribes, L., Torsion elements and centralizers in free products of profinite groups, J. Reine Angew. Math. 358(1985), 155–182.Google Scholar
10. Herfort, W., Frobenius subgroups of free products ofprosolvable groups, Monatsh. Math. 108(1989), 165–182.Google Scholar
11. Herfort, W., On automorphisms offreepro-p groups I, Proc. Amer. Math. Soc. 108(1989) 287-295.Google Scholar
12. Lazard, M., Groupes analytiques p-adiques, Publ. Math. IHES 26(1965).Google Scholar
13. Lubotzky, A., Combinatorial Group Theory for Pro-p Groups, J. Pure Appl. Algebra 25(1982), 311–325.Google Scholar
14. Ribes, L., Introduction to profinite groups and Galois cohomology, Queen's Papers in Pure and Appl. Math., Queen's Univ., Kingston, Ontario, 1970.Google Scholar
15. Ribes, L., The Cartesian subgroup of a free product of profinite groups, Contemp. Math. 109(1990), 147–158.Google Scholar
16. Serre, J-P., Sur la dimension cohomologique des groupes profinis, Topology 3(1965), 413–420.Google Scholar
17. Serre, J-P., Cohomologie Galoisienne, Springer-Verlag, Berlin, 1965.Google Scholar