Group theory
On the profinite rigidity of surface groups and surface words
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 119-122.

Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.121
Henry Wilton 1

1 DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK.
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Henry Wilton. On the profinite rigidity of surface groups and surface words. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 119-122. doi : 10.5802/crmath.121. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.121/

[1] Benjamin Baumslag Residually free groups, Proc. Lond. Math. Soc., Volume 17 (1967), pp. 402-418 | DOI | MR | Zbl

[2] Gilbert Baumslag On generalised free product, Math. Z., Volume 78 (1962), pp. 423-438 | DOI | MR | Zbl

[3] Gilbert Baumslag Residually finite groups with the same finite images, Compos. Math., Volume 29 (1974), pp. 249-252 | Numdam | MR | Zbl

[4] Martin R. Bridson; David B. McReynolds; Alan W. Reid; R. Spitler On the profinite rigidity of triangle groups (2004) (https://arxiv.org/abs/2004.07137)

[5] Martin R. Bridson; David B. McReynolds; Alan W. Reid; R. Spitler Absolute profinite rigidity and hyperbolic geometry, Ann. Math., Volume 192 (2020) no. 3, pp. 679-719 | DOI | MR | Zbl

[6] Fritz J. Grunewald; Andrei Jaikin-Zapirain; Pavel A. Zalesskii Cohomological goodness and the profinite completion of Bianchi groups, Duke Math. J., Volume 144 (2008) no. 1, pp. 53-72 | DOI | MR | Zbl

[7] Liam Hanany; Chen Meiri; Doron Puder Some orbits of free words that are determined by measures on finite groups, J. Algebra, Volume 555 (2020), pp. 305-324 | DOI | MR | Zbl

[8] Olga Kharlampovich; Alexei Myasnikov Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra, Volume 200 (1998) no. 2, pp. 472-516 | DOI | MR | Zbl

[9] Michael Magee; Doron Puder Surface words are determined by word measures on groups (2019) (https://arxiv.org/pdf/1902.04873.pdf, to appear in Israel Journal of Mathematics)

[10] Guennadi A. Noskov; Vladimir N. Remeslennikov; Vitaly A. Roman’kov Infinite groups, Itogi Nauki Tekh., Ser. Algebra, Topologiya, Geom., Volume 17 (1979), pp. 65-157 | Zbl

[11] Doron Puder; Ori Parzanchevski Measure preserving words are primitive, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 63-97 | DOI | MR | Zbl

[12] Zlil Sela Diophantine geometry over groups. I. Makanin–Razborov diagrams, Publ. Math., Inst. Hautes Étud. Sci., Volume 93 (2001), pp. 31-105 | DOI | Numdam | MR | Zbl

[13] Jean-Pierre Serre Galois cohomology, Springer Monographs in Mathematics, Springer, 1997 (Translated from the French by Patrick Ion and revised by the author) | Zbl

[14] Henry Wilton Hall’s Theorem for limit groups, Geom. Funct. Anal., Volume 18 (2008) no. 1, pp. 271-303 | DOI | MR | Zbl

[15] Henry Wilton Essential surfaces in graph pairs, J. Am. Math. Soc., Volume 31 (2018) no. 4, pp. 893-919 | DOI | MR | Zbl

[16] Henry Wilton; Pavel Zalesskii Distinguishing geometries using finite quotients, Geom. Topol., Volume 21 (2017) no. 1, pp. 345-384 | DOI | MR | Zbl

[17] Henry Wilton; Pavel A. Zalesskii Profinite detection of 3-manifold decompositions, Compos. Math., Volume 155 (2019) no. 2, pp. 246-259 | DOI | MR | Zbl

[18] Daniel T. Wise Subgroup separability of graphs of free groups with cyclic edge groups, Q. J. Math, Volume 51 (2000) no. 1, pp. 107-129 | DOI | MR | Zbl

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