Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The structure of some virtually
free pro-$p$ groups

Author: Claus Scheiderer
Journal: Proc. Amer. Math. Soc. 127 (1999), 695-700
MSC (1991): Primary 20E18; Secondary 20E34, 20E36, 20E06
MathSciNet review: 1487337
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove two conjectures on pro-$p$ groups made by Herfort, Ribes and Zalesskii. The first says that a finitely generated pro-$p$ group which has an open free pro-$p$ subgroup of index $p$ is a free pro-$p$ product $H_0*(S_1\times H_1)*\cdots*(S_m\times H_m)$, where the $H_i$ are free pro-$p$ of finite rank and the $S_i$ are cyclic of order $p$. The second says that if $F$ is a free pro-$p$ group of finite rank and $S$ is a finite $p$-group of automorphisms of $F$, then $\operatorname{Fix}(S)$ is a free factor of $F$. The proofs use cohomology, and in particular a ``Brown theorem'' for profinite groups.

References [Enhancements On Off] (What's this?)

  • 1. J. L. Dyer, G. P. Scott, Periodic automorphisms of free groups, Comm. Algebra 3, 195-201 (1975). MR 51:5762
  • 2. D. Gildenhuys, L. Ribes, Profinite groups and boolean graphs, J. Pure Appl. Algebra 12, 21-47 (1978). MR 81g:20059
  • 3. W. N. Herfort, L. Ribes, On automorphisms of free pro-$p$-groups I, Proc. Am. Math. Soc. 108, 287-295 (1990). MR 90d:20048
  • 4. W. N. Herfort, L. Ribes, P. A. Zalesskii, Fixed points of automorphisms of free pro-$p$ groups of rank 2, Canad. J. Math. 47, 383-404 (1995). MR 96k:20053
  • 5. W. Herfort, L. Ribes, P. A. Zalesskii, Finite extensions of free pro-$p$ groups of rank at most two, Preprint 1996, to appear in Israel J. Math.
  • 6. J. Neukirch, Freie Produkte pro-endlicher Gruppen und ihre Kohomologie, Arch. Math. 22, 337-357 (1971). MR 50:490
  • 7. C. Scheiderer, Real and Étale Cohomology, Lect.Notes Math. 1588, Springer, Berlin 1994. MR 96c:14018
  • 8. C. Scheiderer, Farrell cohomology and Brown theorems for profinite groups, Manuscr. math. 91, 247-281 (1996). MR 97j:20050
  • 9. J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3, 413-420 (1965). MR 31:4853
  • 10. J.-P. Serre, Cohomologie Galoisienne, Cinquième édition. Lect. Notes Math. 5, Springer, Berlin, 1994. MR 96b:12010
  • 11. P. A. Zalesskii, O. V. Mel'nikov, Subgroups of profinite groups acting on trees, Math. USSR Sbornik 63, 405-424 (1989) (English translation). MR 90f:20041

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20E18, 20E34, 20E36, 20E06

Retrieve articles in all journals with MSC (1991): 20E18, 20E34, 20E36, 20E06

Additional Information

Claus Scheiderer
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

Keywords: Pro-$p$ groups, virtually free groups, group cohomology, Brown theorem
Received by editor(s): July 1, 1997
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1999 American Mathematical Society