Virtually free pro-p groups
Herfort, Wolfgang1 ; Zalesskii, Pavel2
1 University of Technology at Vienna Vienna Austria
2 Department of Mathematics, University of Brasilia Brasilia, DF Brazil
Publications Mathématiques de l'IHÉS, Tome 118 (2013), pp. 193-211.

We prove that in the category of pro-p groups any finitely generated group G with a free open subgroup splits either as an amalgamated free product or as an HNN-extension over a finite p-group. From this result we deduce that such a pro-p group is the pro-p completion of a fundamental group of a finite graph of finite p-groups.

DOI : 10.1007/s10240-013-0051-4
Herfort, Wolfgang 1 ; Zalesskii, Pavel 2

1 University of Technology at Vienna Vienna Austria
2 Department of Mathematics, University of Brasilia Brasilia, DF Brazil 
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Herfort, Wolfgang; Zalesskii, Pavel. Virtually free pro-p groups. Publications Mathématiques de l'IHÉS, Tome 118 (2013), pp. 193-211. doi : 10.1007/s10240-013-0051-4. http://archive.numdam.org/articles/10.1007/s10240-013-0051-4/

[1.] Chatzidakis, Z. M. Some remarks on profinite HNN extensions, Isr. J. Math., Volume 85 (1994), pp. 11-18 | DOI | MR | Zbl

[2.] Dicks, W. Groups, Trees and Projective Modules, Springer, Berlin, 1980 | MR | Zbl

[3.] Grunewald, F.; Zalesskii, P. A. Genus for groups, J. Algebra, Volume 326 (2011), pp. 130-168 | DOI | MR | Zbl

[4.] Heller, A.; Reiner, I. Representations of cyclic groups in rings of integers. I, Ann. Math., Volume 76 (1962), pp. 73-92 | DOI | MR | Zbl

[5.] Herfort, W.; Ribes, L. Subgroups of free pro-p-products, Math. Proc. Camb. Philos. Soc., Volume 101 (1987), pp. 197-206 | DOI | MR | Zbl

[6.] Herfort, W.; Zalesskii, P. A. Profinite HNN-constructions, J. Group Theory, Volume 10 (2007), pp. 799-809 | MR | Zbl

[7.] Herfort, W.; Zalesskii, P. A. Virtually free pro-p groups whose torsion elements have finite centralizers, Bull. Lond. Math. Soc., Volume 40 (2008), pp. 929-936 | DOI | MR | Zbl

[8.] Herfort, W.; Zalesskii, P. A. A virtually free pro-p group need not be the fundamental group of a profinite graph of finite groups, Arch. Math., Volume 94 (2010), pp. 35-41 | DOI | MR | Zbl

[9.] W. Herfort, P. A. Zalesskii, and T. Zapata, Splitting theorems for pro-p groups acting on pro-p trees and 2-generated subgroups of free pro-p products with procyclic amalgamations. | arXiv

[10.] Hillman, J.; Schmidt, A. Pro-p groups of positive deficiency, Bull. Lond. Math. Soc., Volume 40 (2008), pp. 1065-1069 | DOI | MR | Zbl

[11.] Karrass, A.; Pietrovski, A.; Solitar, D. Finite and infinite cyclic extensions of free groups, J. Aust. Math. Soc., Volume 16 (1973), pp. 458-466 | DOI | MR | Zbl

[12.] Korenev, A. A. Pro-p groups with a finite number of ends, Mat. Zametki, Volume 76 (2004), pp. 531-538 translation in Math. Notes 76 (2004), 490–496 | DOI | MR | Zbl

[13.] Ribes, L.; Zalesskii, P. A. Profinite Groups, Springer, Berlin, 2000 | DOI | MR | Zbl

[14.] Ribes, L.; Zalesskii, P. A. Pro-p trees and applications, Ser. Progress in Mathematics, Birkhäuser, Boston (2000) | MR | Zbl

[15.] Roman’kov, L. Infinite generation of automorphism groups of free pro-p groups, Sib. Math. J., Volume 34 (1993), pp. 727-732 | DOI | MR | Zbl

[16.] Scheiderer, C. The structure of some virtually free pro-p groups, Proc. Am. Math. Soc., Volume 127 (1999), pp. 695-700 | DOI | MR | Zbl

[17.] Serre, J.-P. Sur la dimension cohomologique des groupes profinis, Topology, Volume 3 (1965), pp. 413-420 | DOI | MR | Zbl

[18.] Stallings, J. R. Applications of Categorical Algebra, Proc. Sympos. Pure Math., 124, American Mathematical Society, Providence (1970), pp. 124-129 (vol. XVII) | DOI

[19.] Wilson, J. S. Profinite Groups, London Math. Soc. Monographs, Clarendon Press, Oxford (1998) | MR | Zbl

[20.] Zalesskii, P. A.; Mel’nikov, O. V. Subgroups of profinite groups acting on trees, Math. USSR Sb., Volume 63 (1989), pp. 405-424 | DOI | MR | Zbl

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