Abstract
The main aim of this paper is to describe the closure of a finitely generated subgroup of a finitely generated free group in the proabelian topology. Our approach depends heavily on the description of the abelian kernel of a finite inverse semigroup.
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References
Almeida, J., Shahzamanian, M.H., Steinberg, B.: The pro-nilpotent group topology on a free group. J. Algebra 480, 332–345 (2017)
Ash, C.J.: Inevitable graphs: a proof of the type ii conjecture and some related decision procedures. Int. J. Algebra Comput. 1(01), 127–146 (1991)
Auinger, K., Steinberg, B.: A constructive version of the Ribes-Zalesskiĭ product theorem. Math. Z 250(2), 287–297 (2005)
Ballester-Bolinches, A., Ezquerro, L.M.: Classes of Finite Groups, Mathematics and its Applications. Springer, New York (2006)
Ballester-Bolinches, A., Pérez-Calabuig, V.: The abelian kernel of an inverse semigroup. Mathematics 8(8), 1219 (2020)
Delgado, M.: Abelian pointlikes of a semigroup. Semigroup Forum 56, 339–361 (1998)
Hall, M.: A topology for free groups and related groups. Ann. Math. 52, 127–139 (1950)
Herwig, B., Lascar, D.: Extending partial automorphisms and the profinite topology on free groups. Trans. Amer. Math. Soc. 352(5), 1985–2021 (2000)
Margolis, S., Sapir, M., Weil, P.: Closed subgroups in pro-\( {V}\) topologies and the extension problem for inverse automata. Int. J. Algebra Comput. 11(4), 405–445 (2001)
Margolis, S.W.: Consequences of Ash’s proof of the Rhodes type \({\rm II}\) conjecture. In: Monash Conference on Semigroup Theory (Melbourne, 1990), pp. 180–205. World Sci. Publ., River Edge, NJ (1991)
Pin, J.-É., Reutenauer, C.: A conjecture on the Hall topology for the free group. Bull. London Math. Soc. 23, 356–362 (1991)
Rhodes, J., Tilson, B.R.: Improved lower bounds for the complexity of finite semigroups. J. Pure Appl. Algebra 2(1), 13–71 (1972)
Ribes, L., Zalesski, P.A.: On the profinite topology on a free group. Bull. London Math. Soc. 25(1), 37–43 (1993)
Ribes, L., Zalesski, P.A.: The pro-p topology of a free group and algorithmic problems in semigroups. Int. J. Algebra Comput. 4(03), 359–374 (1994)
Robinson, D.J.S.: A Course in the Theory of Groups. Springer-Verlag, New-York (1982)
Stallings, J.: Topology of finite graphs. Invent. Math. 71(3), 551–565 (1983)
Steinberg, B.: Monoid kernels and profinite topologies on the free abelian group. Bull. Austral. Math. Soc. 60(03), 391–402 (1999)
Steinberg, B.: Finite state automata: a geometric approach. Trans. Amer. Math. Soc. 353(9), 3409–3464 (2001)
Acknowledgements
The authors have been supported by the research grant PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovación y Universidades (Spanish Government), the Agencia Estatal de Investigación (Spain), and FEDER (European Union), and PROMETEO/2017/057 from the Generalitat (Valencian Community, Spain).
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Ballester-Bolinches, A., Pérez-Calabuig, V. Abelian kernels, profinite topologies and the extension problem. Ricerche mat 72, 91–106 (2023). https://doi.org/10.1007/s11587-021-00601-4
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DOI: https://doi.org/10.1007/s11587-021-00601-4