Abstract
Let F be a free non-abelian group. We show that for any group word w the set w[F] of all values of w in F is rational in F if and only if w[F]=1 or w[F]=F. We generalize this to a wide class of free products of groups.
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Anissimow, A.W., Seifert, F.D.: Zur algebraischen Charakteristik der durch kontext-freie Sprachen definierten Gruppen. Elektron. Informationsverarb. Kybern. 11(10–12), 695–702 (1975)
Bazhenova, G.A.: Closure of one class of groups under free products. Sib. Math. J. 41, 611–613 (2000)
Bazhenova, G.A.: On rational sets in finitely generated nilpotent groups. Algebra Log. 39, 215–223 (2000)
Baumslag, B.: Intersections of finitely generated subgroups in free products. J. Lond. Math. Soc. 41, 673–679 (1966)
Birman, J.: Braids, Links and Mapping Class Groups. Annals of Math. Studies. Princeton University Press, Princeton (1974)
Calegari, D.: SCL, vol. 20. Math. Soc. Japan Mem., Tokyo (2009)
Calegari, D.: Quasimorphisms and laws. Algebr. Geom. Topol. 10, 215–217 (2010)
Eilenberg, S., Schützenberger, M.P.: Rational sets in commutative monoids. J. Algebra 13, 173–191 (1969)
Gersten, S.: Cohomological lower bounds for isoperimetric functions on groups. Topology 37, 1031–1072 (1998)
Gilman, R.H.: Formal languages and infinite groups. In: Geometric and Computational Perspectives of Infinite Groups, Minneapolis, MN and New Brunswick, NJ, 1994. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 25, pp. 27–51. Amer. Math. Soc, Providence (1996)
Gromov, M.: Volume and bounded cohomology. Publ. Math. IHES 56, 5–99 (1982)
Gromov, M.: Asymptotic invariants of infinite groups. In: London Math. Soc. Lect. Notes Ser., vol. 182. Cambridge University Press, Cambridge (1993)
Larsen, M., Shalev, A.: Word maps and Waring type problems. J. Am. Math. Soc. 22, 437–466 (2009)
Liebeck, M., O’Brian, E., Shalev, A., Tiep, P.: The Ore conjecture. J. Eur. Math. Soc. 12, 939–1008 (2010)
Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. Wiley Interscience, New York (1968)
Makanin, G.S.: Equations in a free group. Izv. Akad. Nauk SSSR, Ser. Mat. 46, 1199–1273 (1982) (Russian)
Neumann, H.: Varieties of Groups. Springer, New York (1967)
Ore, O.: Some remarks on commutators. Proc. Am. Math. Soc. 2, 307–314 (1951)
Razborov, A.: On the parametrization of solutions for equations in free groups. Int. J. Algebra Comput. 3, 251–273 (1993)
Rhemtulla, A.H.: A problem of bounded expressability in free products. Proc. Camb. Philos. Soc. 64, 573–584 (1968)
Rhemtulla, A.H.: Commutators of certain finitely generated soluble groups. Can. J. Math. 21, 1160–1164 (1969)
Roman’kov, V.A.: Width of verbal subgroups in solvable groups. Algebra Log. 21, 41–49 (1982)
Segal, D.: Words: Notes on Verbal Width in Groups. London Math. Soc. Lect. Notes Ser., vol. 361. Cambridge University Press, Cambridge (2009)
Shalev, A.: Word maps, conjugacy classes, and a noncommutative Waring-type theorem. Ann. Math. 170, 1383–1416 (2009)
Stroud, P.W.: Topics in the theory of verbal subgroups. PhD Thesis, University of Cambridge, Cambridge (1966)
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We thank the anonymous referees for their very useful comments.
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A. Myasnikov’s research supported in part by NSF DMS under Project 0914773.
V. Roman’kov’s research supported in part by Russian Fund of Basic Research under Project 10-01-00383a.
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Myasnikov, A., Roman’kov, V. On Rationality of Verbal Subsets in a Group. Theory Comput Syst 52, 587–598 (2013). https://doi.org/10.1007/s00224-012-9394-3
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DOI: https://doi.org/10.1007/s00224-012-9394-3