Stallings automata for free-times-abelian groups: intersections and index

Article Sidebar

pdf (English)

Main Article Content

Jordi Delgado
Universidad del País Vasco. Departamento de Matemáticas
https://orcid.org/0000-0002-8365-8929
Enric Ventura
Universitat Politècnica de Catalunya. Departament de Matemàtiques
https://orcid.org/0000-0003-3519-4135

We extend the classical Stallings theory (describing subgroups of free groups as automata) to direct products of free and abelian groups: after introducing enriched automata (i.e., automata with extra abelian labels), we obtain an explicit bijection between subgroups and a certain type of such enriched automata, which—as it happens in the free group—is computable in the finitely generated case. This approach provides a neat geometric description of (even non-(finitely generated)) intersections of finitely generated subgroups within this non-Howson family. In particular, we give a geometric solution to the subgroup intersection problem and the finite index problem, providing recursive bases and transversals, respectively.

Paraules clau
free group, free-abelian group, direct product, subgroup, intersection, Stallings, automata

Article Details

Com citar
Delgado, Jordi; Ventura, Enric. «Stallings automata for free-times-abelian groups: intersections and index». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 2, p. 789-30, https://raco.cat/index.php/PublicacionsMatematiques/article/view/402264.
Referències
L. Bartholdi and P. V. Silva, Rational subsets of groups, in: “Handbook of Automata Theory. Vol. II—Automata in Mathematics and Selected Applications”, EMS Press, Berlin, 2021, pp. 841–869.

B. Baumslag, Intersections of finitely generated subgroups in free products, J. London Math. Soc. 41 (1966), 673–679. DOI: 10.1112/jlms/s1-41.1.673

R. G. Burns and S.-M. Kam, On the intersection of double cosets in free groups, with an application to amalgamated products, J. Algebra 210(1) (1998), 165–193. DOI: 10.1006/jabr.1998.7411

A. Carvalho, On the dynamics of extensions of free-abelian times free groups endomorphisms to the completion, Preprint (2021). arXiv:2011.05205

J. Delgado, Extensions of free groups: algebraic, geometric, and algorithmic aspects, Thesis (Ph.D.)-Universitat Polit`ecnica de Catalunya (2017). Available on https://www.tesisenred.net/

J. Delgado, M. Roy, and E. Ventura, Intersection configurations in free times free-abelian groups, Preprint (2021). arXiv:2107.12426

J. Delgado, M. Roy, and E. Ventura, On universally quotientable groups, in preparation

J. Delgado and P. V. Silva, On the lattice of subgroups of a free group: complements and rank, J. Groups Complex. Cryptol. 12(1) (2020), Paper no. 1, 24 pp.

J. Delgado and E. Ventura, Algorithmic problems for free-abelian times free groups, J. Algebra 391 (2013), 256–283. DOI: 10.1016/j.jalgebra.2013.04.033

J. Delgado and E. Ventura, A list of applications of Stallings automata, Trans. Comb. 11(3) (2022), 181–235. DOI: 10.22108/toc.2021.130387.1905

J. Delgado and E. Ventura, Stallings automata for free extensions, in preparation.

J. Delgado and E. Ventura, Stallings automata for free-by-abelian groups, in preparation.

J. Delgado, E. Ventura, and A. Zakharov, Intersection problem for Droms RAAGs, Internat. J. Algebra Comput. 28(7) (2018), 1129–1162. DOI: 10.1142/S0218196718500509

J. Delgado, E. Ventura, and A. Zakharov, Relative order and spectrum in free and related groups, Preprint (2021). arXiv:2105.03798

W. Dicks, Simplified Mineyev, Preprint 2011. Available on https://mat.uab.cat/∼dicks/SimplifiedMineyev.pdf.

J. Friedman, Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks, Mem. Amer. Math. Soc. 233(1100) (2015), 106 pp. DOI: 10.1090/memo/1100

M. Hall, Jr., Subgroups of finite index in free groups, Canad. J. Math. 1 (1949), 187–190. DOI: 10.4153/cjm-1949-017-2

A. G. Howson, On the intersection of finitely generated free groups, J. London Math. Soc. 29 (1954), 428–434. DOI: 10.1112/jlms/s1-29.4.428

S. V. Ivanov, On the intersection of finitely generated subgroups in free products of groups, Internat. J. Algebra Comput. 9(5) (1999), 521–528. DOI: 10.1142/S021819679900031X

S. V. Ivanov, Intersecting free subgroups in free products of groups., Internat. J. Algebra Comput. 11(3) (2001), 281–290. DOI: 10.1142/S0218196701000267

A. Jaikin-Zapirain, Approximation by subgroups of finite index and the Hanna Neumann conjecture, Duke Math. J. 166(10) (2017), 1955–1987. DOI: 10.1215/00127094-0000015X

I. Kapovich and A. Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248(2) (2002), 608–668. DOI: 10.1006/jabr.2001.9033

I. Kapovich, R. Weidmann, and A. Miasnikov, Foldings, graphs of groups and the membership problem, Internat. J. Algebra Comput. 15(1) (2005), 95–128. DOI: 10.1142/S021819670500213X

O. Kharlampovich, A. Miasnikov, and P. Weil, Stallings graphs for quasiconvex subgroups, J. Algebra 488 (2017), 442–483. DOI: 10.1016/j.jalgebra.2017.05.037

S. Margolis, M. Sapir, and P. Weil, Closed subgroups in pro-V topologies and the extension problem for inverse automata, Internat. J. Algebra Comput. 11(4) (2001), 405–445. DOI: 10.1142/S0218196701000498

A. Miasnikov, E. Ventura, and P. Weil, Algebraic extensions in free groups, in: “Geometric Group Theory”, Trends Math., Birkh¨auser, Basel, 2007, pp. 225–253. DOI: 10.1007/978-3-7643-8412-8_12

I. Mineyev, Submultiplicativity and the Hanna Neumann conjecture, Ann. of Math. (2) 175(1) (2012), 393–414. DOI: 10.4007/annals.2012.175.1.11

H. Neumann, On the intersection of finitely generated free groups, Publ. Math. Debrecen 4 (1956), 186–189.

D. Puder, Primitive words, free factors and measure preservation, Israel J. Math. 201(1) (2014), 25–73. DOI: 10.1007/s11856-013-0055-2

A. Roig, E. Ventura, and P. Weil, On the complexity of the Whitehead minimization problem, Internat. J. Algebra Comput. 17(8) (2007), 1611–1634. DOI: 10.1142/S0218196707004244

M. Roy and E. Ventura, Fixed subgroups and computation of auto-fixed closures in free-abelian times free groups, J. Pure Appl. Algebra 224(4) (2020), 106210, 19 pp. DOI: 10.1016/j.jpaa.2019.106210

M. Roy and E. Ventura, Degrees of compression and inertia for free-abelian times free groups, J. Algebra 568 (2021), 241–272. DOI: 10.1016/j.jalgebra.2020.09.040

J.-P. Serre, “Trees”, Translated from the French by John Stillwell, Springer Verlag, Berlin-New York, 1980.

P. V. Silva, X. Soler-Escriva, and E. Ventura ` , Finite automata for Schreier graphs of virtually free groups, J. Group Theory 19(1) (2016), 25–54. DOI: 10.1515/jgth-2015-0028

P. V. Silva and P. Weil, On an algorithm to decide whether a free group is a free factor of another, Theor. Inform. Appl. 42(2) (2008), 395–414. DOI: 10.1051/ita:2007040

J. R. Stallings, Topology of finite graphs, Invent. Math. 71(3) (1983), 551–565. DOI: 10.1007/BF02095994

E. Ventura, On fixed subgroups of maximal rank, Comm. Algebra 25(10) (1997), 3361–3375. DOI: 10.1080/00927879708826057