Inertia of retracts in Demushkin groups
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Henrique Souza
Abstract
Exploring inequalities involving the rank and relation gradients of pro-đť‘ť modules and building upon recent results of Y. AntolĂn, A. Jaikin-Zapiran and M. Shusterman, we prove that every retract of a Demushkin group is inert in the sense of the Dicks–Ventura Inertia Conjecture.
Award Identifier / Grant number: PID2020-114032GB-I00
Award Identifier / Grant number: CEX2019-000904-S4
Funding statement: This paper is partially supported by the Spanish MINECO through the grants PID2020-114032GB-I00, and the “Severo Ochoa” programs for Centres of Excellence CEX2019-000904-S4.
Acknowledgements
The author would like to thank Andrei Jaikin-Zapirain, Pavel Zalesskii, Theo Zapata and the anonymous referee for helpful comments. He also thanks Andrei Jaikin-Zapirain for introducing him to the problem of inertia of retracts. The contents of this paper form a part of the author’s M.Sc. dissertation at the University of Brasilia, presented under the advice of Theo Zapata.
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Communicated by: Benjamin Klopsch
References
[1] Y. AntolĂn and A. Jaikin-Zapirain, The Hanna Neumann conjecture for surface groups, Compos. Math. 158 (2022), no. 9, 1850–1877. 10.1112/S0010437X22007709Suche in Google Scholar
[2] W. Dicks and E. Ventura, The Group Fixed by a Family of Injective Endomorphisms of a Free Group, Contemp. Math. 195, American Mathematical Society, Providence, 1996. 10.1090/conm/195Suche in Google Scholar
[3] W. Herfort and L. Ribes, Subgroups of free pro-𝑝-products, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 2, 197–206. 10.1017/S0305004100066548Suche in Google Scholar
[4] A. Jaikin-Zapirain and M. Shusterman, The Hanna Neumann conjecture for Demushkin groups, Adv. Math. 349 (2019), 1–28. 10.1016/j.aim.2019.04.013Suche in Google Scholar
[5] A. Lubotzky, Combinatorial group theory for pro-𝑝-groups, J. Pure Appl. Algebra 25 (1982), no. 3, 311–325. 10.1016/0022-4049(82)90086-XSuche in Google Scholar
[6] L. Ribes, Profinite Graphs and Groups, Ergeb. Math. Grenzgeb. (3) 66, Springer, Cham, 2017. 10.1007/978-3-319-61199-0Suche in Google Scholar
[7] L. Ribes and P. Zalesskii, Profinite Groups, 2nd ed., Ergeb. Math. Grenzgeb. (3) 40, Springer, Berlin, 2010. 10.1007/978-3-642-01642-4Suche in Google Scholar
[8] J.-P. Serre, Galois Cohomology, Springer, Berlin, 1997. 10.1007/978-3-642-59141-9Suche in Google Scholar
[9] M. Shusterman and P. Zalesskii, Virtual retraction and Howson’s theorem in pro-𝑝 groups, Trans. Amer. Math. Soc. 373 (2020), no. 3, 1501–1527. 10.1090/tran/7784Suche in Google Scholar
[10] E. C. Turner, Test words for automorphisms of free groups, Bull. Lond. Math. Soc. 28 (1996), no. 3, 255–263. 10.1112/blms/28.3.255Suche in Google Scholar
[11] E. Ventura, Fixed subgroups in free groups: A survey, Combinatorial and Geometric Group Theory, Contemp. Math. 296, American Mathematical Society, Providence (2002), 231–255. 10.1090/conm/296/05077Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Fusion systems realizing certain Todd modules
- Diagonal embeddings of finite alternating groups
- Vertex-transitive graphs with local action the symmetric group on ordered pairs
- On the Schur multiplier of finite đť‘ť-groups of maximal class
- A classification of skew morphisms of dihedral groups
- On closed subgroups of precompact groups
- Inertia of retracts in Demushkin groups
- On arithmetic properties of solvable Baumslag–Solitar groups
Artikel in diesem Heft
- Frontmatter
- Fusion systems realizing certain Todd modules
- Diagonal embeddings of finite alternating groups
- Vertex-transitive graphs with local action the symmetric group on ordered pairs
- On the Schur multiplier of finite đť‘ť-groups of maximal class
- A classification of skew morphisms of dihedral groups
- On closed subgroups of precompact groups
- Inertia of retracts in Demushkin groups
- On arithmetic properties of solvable Baumslag–Solitar groups
