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Semi-free subgroups of a profinite surface group. (English) Zbl 1437.20023

Summary: We show that every closed normal subgroup of infinite index in a profinite surface group \(\Gamma\) is contained in a semi-free profinite normal subgroup of \(\Gamma\). This answers a question of L. Bary-Soroker et al. [Math. Res. Lett. 18, No. 3, 459–471 (2011; Zbl 1252.20030)].

MSC:

20E18 Limits, profinite groups
20F65 Geometric group theory
20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth

Citations:

Zbl 1252.20030
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References:

[1] L. Bary-Soroker, D. Haran and D. Harbater, Permanence criteria for semi-free profinite groups, Math. Ann. 348 (2010), no. 3, 539-563. · Zbl 1256.20028
[2] L. Bary-Soroker, K. F. Stevenson and P. A. Zalesskii, Subgroups of profinite surface groups, Math. Res. Lett. 18 (2011), no. 3, 459-471. · Zbl 1252.20030
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[5] A. Pacheco, K. F. Stevenson and P. Zalesskii, Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic, Math. Ann. 343 (2009), no. 2, 463-486. · Zbl 1184.14046
[6] L. Ribes and P. Zalesskii, Profinite Groups, 2nd ed., Ergeb. Math. Grenzgeb. (3) 40, Springer, Berlin, 2010. · Zbl 1197.20022
[7] M. Shusterman, Free subgroups of finitely generated free profinite groups, J. Lond. Math. Soc. (2) 93 (2016), no. 2, 361-378. · Zbl 1403.20040
[8] P. A. Zalesskii, Profinite surface groups and the congruence kernel of arithmetic lattices in {\rm SL}_{2}(\mathbb{R}), Israel J. Math. 146 (2005), 111-123. · Zbl 1076.20020
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