A list of applications of Stallings automata

Document Type : Workshop on Graphs, Topology and Topological Groups, Cape Town, South Africa

Authors

1 Department of Mathematics, University of the Basque Country, Faculty of Science and Technology, Barrio Sarriena, s/n 48940 Leioa, Spain

2 Departament de Matemàtiques, Universitat Politècnica de Catalunya and Institut de Matemàtiques de la UPC-BarcelonaTech, Barcelona, Catalunya

Abstract

This survey is intended to be a fast (and reasonably updated) reference for the theory of Stallings automata and its applications to the study of subgroups of the free group, with the main accent on algorithmic aspects. Consequently, results concerning finitely generated subgroups have greater prominence in the paper. However, when possible, we try to state the results with more generality, including the usually overlooked non-(finitely-generated) case.

Keywords

Main Subjects


[1] Y. Antolin and A. Jaikin-Zapirain. The Hanna Neumann conjecture for surface groups (preprint). https://matematicas.uam.es/~andrei.jaikin/preprints/articulos/. 2021 (cit. on pp. 211, 222).
[2] Y. Antolín, A. Martino, and I. Schwabrow. “Kurosh rank of intersections of subgroups of free products of rightorderable groups”. Mathematical Research Letters 21.4 (2014), pp. 649–661 (cit. on p. 211).
[3] G. Arzhantseva. “A property of subgroups of infinite index in a free group”. Proceedings of the American Mathematical Society 128.11 (2000), pp. 3205–3210 (cit. on p. 225).
[4] G. N. Arzhantseva. “On the groups in which the subgroups with fixed number of generators are free”. Fundamental’naya i Prikladnaya Matematika 3.3 (1997), pp. 675–683 (cit. on p. 225).
[5] G. N. Arzhantseva and A. Y. Ol’shanskii. “Generality of the class of groups in which subgroups with a lesser number of generators are free”. Mat. Zametki 59.4 (1996), pp. 489–496, 638 (cit. on p. 225).
[6] G. N. Arzhantseva and A. Y. Ol’shanskii. “The class of groups all of whose subgroups with lesser number of generators are free is generic”. Mathematical Notes 59.4 (Apr. 1996), pp. 350–355 (cit. on pp. 182, 225).
[7] G. Arzhantseva. “Generic properties of finitely presented groups and Howson’s Theorem”. Communications in Algebra 26.11 (Jan. 1998), pp. 3783–3792 (cit. on p. 225).
[8] G. N. Arzhantseva. “Generic properties of finitely presented groups”. PhD thesis. Moscow Lomonosov State University, Dec. 1998 (cit. on p. 225).
[9] L. Bartholdi and P. V. Silva. “Rational subsets of groups”. arXiv:1012.1532 (Dec. 2010) (cit. on pp. 181, 182, 197).
[10] F. Bassino, A. Martino, C. Nicaud, E. Ventura, and P.Weil. “Statistical properties of subgroups of free groups”. Random Structures & Algorithms 42.3 (May 2013), pp. 349–373 (cit. on pp. 226, 227).
[11] F. Bassino, C. Nicaud, and P.Weil. “Random generation of finitely generated subgroups of a free group”. International Journal of Algebra and Computation 18.02 (Mar. 2008), pp. 375–405 (cit. on p. 226).
[12] F. Bassino, C. Nicaud, and P.Weil. “On the genericity of Whitehead minimality”. Journal of Group Theory 19.1 (2016), pp. 137–159 (cit. on p. 226).
[13] F. Bassino, C. Nicaud, and P.Weil. “Generic properties of subgroups of free groups and finite presentations”.In: Algebra and computer science. Vol. 677. Contemp. Math. Amer. Math. Soc., Providence, RI, 2016, pp. 1–43 (cit. on p. 226).
[14] G. Baumslag, A. Myasnikov, and V. Remeslennikov. “Malnormality is decidable in free groups”. International Journal of Algebra and Computation 9.6 (1999), pp. 687–692 (cit. on p. 212).
[15] B. Beeker and N. Lazarovich. “Stallings’ folds for cube complexes”. Israel Journal of Mathematics 227.1 (Aug.2018), pp. 331–363 (cit. on p. 183).
[16] M. Benois. “Parties rationnelles du groupe libre”. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A 269 (1969), pp. 1188–1190 (cit. on p. 186).
[17] M. A. Berger, A. Felzenbaum, and A. Fraenkel. “Remark on the multiplicity of a partition of a group into cosets”.Fundamenta Mathematicae 128 (1987), pp. 139–144 (cit. on p. 207).
[18] M. Bestvina and M. Feighn. “Bounding the complexity of simplicial group actions on trees”. Inventiones Mathematicae 103.3 (1991), pp. 449–469 (cit. on p. 182).
[19] M. Bestvina and M. Handel. “Train tracks and automorphisms of free groups”. Annals of Mathematics. Second Series 135.1 (1992), pp. 1–51 (cit. on p. 219).
[20] J.-C. Birget, S. Margolis, J. Meakin, and P.Weil. “PSPACE-complete problems for subgroups of free groups and inverse finite automata”. Theoretical Computer Science 242.1 (July 2000), pp. 247–281 (cit. on p. 215).
[21] O. Bogopolski. Introduction to Group Theory. Zurich, Switzerland: European Mathematical Society Publishing House, Feb. 2008 (cit. on p. 186).
[22] O. Bogopolski and O. Maslakova. “An algorithm for finding a basis of the fixed point subgroup of an automorphism of a free group”. International Journal of Algebra and Computation 26.1 (2016), pp. 29–67 (cit. on p. 221).
[23] O. Bogopolski and R.Weidmann. “On the uniqueness of factors of amalgamated products”. Journal of Group Theory 5.2 (2002), pp. 233–240 (cit. on p. 182).
[24] B. H. Bowditch. “Peripheral splittings of groups”. Transactions of the American Mathematical Society 353.10 (2001), pp. 4057–4082 (cit. on p. 182).
[25] C. Champetier. “Propriétés statistiques des groupes de présentation finie”. Advances in Mathematics 116.2 (Dec.1995), pp. 197–262 (cit. on p. 224).
[26] C. Champetier. “Propriétés génériques des groupes de présentation finie”. PhD thesis. Universite Lyon, 1991 (cit. on p. 224).
[27] F. Chouraqui. “The space of coset partitions of Fn and Herzog-Schönheim conjecture”. arXiv:1804.11103 [math] (Apr. 2018) (cit. on p. 207).
[28] F. Chouraqui. “About an extension of the Davenport-Rado result to the Herzog-Schonheim conjecture for free groups”. arXiv:1901.09898 [math] (Jan. 2019) (cit. on p. 207).
[29] F. Chouraqui. “The Herzog-Schönheim conjecture for finitely generated groups”. International Journal of Algebra and Computation 29.6 (2019), pp. 1083–1112 (cit. on pp. 207, 208).
[30] F. Chouraqui. “An approach to the Herzog-Schönheim conjecture using automata”. In: Developments in Language Theory. Ed. by N. Jonoska and D. Savchuk. Lecture Notes in Computer Science. Cham: Springer International Publishing, 2020, pp. 55–68 (cit. on p. 207).
[31] L. Ciobanu and W. Dicks. “Two examples in the Galois theory of free groups”. Journal of Algebra 305.1 (2006), pp. 540–547 (cit. on p. 221).
[32] D. J. Collins and E. C. Turner. “All automorphisms of free groups with maximal rank fixed subgroups”. Mathematical Proceedings of the Cambridge Philosophical Society 119.4 (May 1996), pp. 615–630 (cit. on p. 219).
[33] P. Dani and I. Levcovitz. “Subgroups of right-angled Coxeter groups via Stallings-like techniques”. Journal of Combinatorial Algebra 5.3 (Oct. 2021), pp. 237–295 (cit. on p. 183).
[34] S. Das and M. Mj. “Controlled Floyd separation and non relatively hyperbolic groups”. Journal of the Ramanujan Mathematical Society 30.3 (2015), pp. 267–294 (cit. on p. 212).
[35] J. Delgado. “Extensions of free groups: algebraic, geometric, and algorithmic aspects”. Ph.D. Thesis. Universitat Politècnica de Catalunya, Sept. 2017 (cit. on p. 183).
[36] J. Delgado and P. V. Silva. “On the lattice of subgroups of a free group: complements and rank”. journal of Groups, Complexity, Cryptology Volume 12, issue 1 (Mar. 2020) (cit. on p. 200).
[37] J. Delgado and E. Ventura. “Algorithmic problems for free-abelian times free groups”. Journal of Algebra 391 (Oct. 2013), pp. 256–283 (cit. on p. 183).
[38] J. Delgado and E. Ventura. “Stallings automata for free-times-abelian groups: intersections and index”. Publicacions Matemàtiques 66.2 (2022), pp. 789 –830 (cit. on p. 183).
[39] J. Delgado, E. Ventura, and A. Zakharov. “Relative order and spectrum in free and related groups”. Communications in Contemporary Mathematics (to appear) (cit. on pp. 212, 214).
[40] M. Delgado, S. Margolis, and B. Steinberg. “Combinatorial group theory, inverse monoids, automata, and global semigroup theory”. International Journal of Algebra and Computation 12.01n02 (Feb. 2002), pp. 179–211 (cit. on p. 182).
[41] W. Dicks. “Simplified Mineyev’s proof of Hanna Neumann conjecture”. https : / / mat . uab . cat / ~dicks / SimplifiedMineyev.pdf. May 2012 (cit. on p. 211).
[42] W. Dicks and M. J. Dunwoody. “On equalizers of sections”. Journal of Algebra 216.1 (1999), pp. 20–39 (cit. on p. 182).
[43] W. Dicks and E. Ventura. The group fixed by a family of injective endomorphisms of a free group. Vol. 195. Contemporary Mathematics. American Mathematical Society, 1996 (cit. on p. 222).
[44] M. J. Dunwoody. “Groups acting on protrees”. Journal of the London Mathematical Society. Second Series 56.1 (1997), pp. 125–136 (cit. on p. 182).
[45] M. J. Dunwoody. “Folding sequences”. In: The Epstein birthday schrift. Vol. 1. Geom. Topol. Monogr. Geom. Topol. Publ., Coventry, 1998, pp. 139–158 (cit. on p. 182).
[46] M. J. Dunwoody. “A small unstable action on a tree”. Mathematical Research Letters 6.5-6 (1999), pp. 697–710 (cit. on p. 182).
[47] J. L. Dyer and G. P. Scott. “Periodic automorphisms of free groups”. Communications in Algebra 3.3 (Jan. 1975), pp. 195–201 (cit. on p. 219).
[48] P. Erdös. “On integers of the form 2k + p and some related problems”. Summa Brasil. Math. 2 (1950) (cit. on p. 207).
[49] M. Feighn and M. Handel. “Algorithmic constructions of relative train track maps and CTs”. Groups, Geometry, and Dynamics 12.3 (Aug. 2018), pp. 1159–1238 (cit. on p. 221).
[50] J. Friedman. Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks. Vol. 233. Memoirs of the American Mathematical Society. American Mathematical Society, Jan. 2015 (cit. on p. 211).
[51] Y. Ginosar. “Tile the group”. Elemente der Mathematik 73.2 (2018), pp. 66–73 (cit. on p. 207).
[52] L. Greenberg. “Commensurable groups of Moebius transformations”. In: Discontinuous groups and Riemann surfaces (AM-79), volume 79: Proceedings of the 1973 conference at the university of maryland. (AM-79). Ed. By L. Greenberg. Princeton University Press, 1974, pp. 227–238 (cit. on p. 206).
[53] M. Gromov. “Hyperbolic Groups”. In: Essays in Group Theory. Mathematical Sciences Research Institute Publications. Springer, New York, NY, 1987, pp. 75–263 (cit. on pp. 223, 224).
[54] V. Guirardel. “Approximations of stable actions on R-trees”. Commentarii Mathematici Helvetici 73.1 (1998), pp. 89–121 (cit. on p. 182).
[55] V. Guirardel. “Reading small actions of a one-ended hyperbolic group on R-trees from its JSJ splitting”. American Journal of Mathematics 122.4 (2000), pp. 667–688 (cit. on p. 182).
[56] M. Hall Jr. “Coset representations in free groups”. Transactions of the American Mathematical Society 67 (1949), pp. 421–432 (cit. on pp. 204, 224).
[57] M. Hall Jr. “Subgroups of finite index in free groups”. Canadian Journal of Mathematics 1.2 (Apr. 1949), pp. 187–190 (cit. on p. 224).
[58] M. Hall Jr. “A topology for free groups and related groups”. Annals of Mathematics 52.1 (July 1950), pp. 127–139 (cit. on p. 205).
[59] M. Heusener and R.Weidmann. “A remark on Whitehead’s cut-vertex lemma”. Journal of Group Theory 22.1 (Jan. 2019), pp. 15–21 (cit. on p. 203).
[60] A. G. Howson. “On the intersection of finitely generated free groups”. Journal of the London Mathematical Society s1-29.4 (Oct. 1954), pp. 428–434 (cit. on pp. 208, 210).
[61] W. Imrich and E. C. Turner. “Endomorphisms of free groups and their fixed points”. Mathematical Proceedings of the Cambridge Philosophical Society 105.3 (May 1989), pp. 421–422 (cit. on p. 219).
[62] S. V. Ivanov. “On the intersection of finitely generated subgroups in free products of groups”. International Journal of Algebra and Computation 09.05 (Oct. 1999), pp. 521–528 (cit. on p. 183).
[63] S. V. Ivanov. “The intersection of subgroups in free groups and linear programming”. Mathematische Annalen 370.3 (Apr. 2018), pp. 1909–1940 (cit. on p. 222).
[64] A. Jaikin-Zapirain. “Approximation by subgroups of finite index and the Hanna Neumann conjecture”. Duke Mathematical Journal 166.10 (2017), pp. 1955–1987 (cit. on p. 211).
[65] T. Jitsukawa. “Malnormal subgroups of free groups”. In: Computational and statistical group theory (Las Vegas, NV/Hoboken, NJ, 2001). Vol. 298. Contemp. Math. Amer. Math. Soc., Providence, RI, 2002, pp. 83–95 (cit. on p. 225).
[66] D. L. Johnson. Presentations of Groups. London Mathematical Society Student Texts. Cambridge: Cambridge University Press, 1997 (cit. on p. 185).
[67] I. Kapovich and A. Myasnikov. “Stallings foldings and subgroups of free groups”. Journal of Algebra 248.2 (Feb.2002), pp. 608–668 (cit. on pp. 181, 182, 197, 215, 216).
[68] I. Kapovich and P. Schupp. “Genericity, the Arzhantseva-Ol’shanskii method and the isomorphism problem for one-relator groups”. Mathematische Annalen 331.1 (Jan. 2005), pp. 1–19 (cit. on p. 225).
[69] I. Kapovich and P. E. Schuppp. “Random quotients of the modular group are rigid and essentially incompressible”. Journal für die Reine und Angewandte Mathematik 628 (2009), pp. 91–119 (cit. on p. 225).
[70] I. Kapovich, R.Weidmann, and A. Myasnikov. “Foldings, graphs of groups and the membership problem”.International Journal of Algebra and Computation 15.01 (Feb. 2005), pp. 95–128 (cit. on p. 183).
[71] O. Kharlampovich, A. Miasnikov, and P.Weil. “Stallings graphs for quasi-convex subgroups”. Journal of Algebra 488 (Oct. 2017), pp. 442–483 (cit. on p. 183).
[72] O. Kharlampovich and P.Weil. “On the generalized membership problem in relatively hyperbolic groups”. In:Fields of Logic and Computation III. Ed. by A. Blass, P. Cégielski, N. Dershowitz, M. Droste, and B. Finkbeiner.Lecture Notes in Computer Science. Cham: Springer International Publishing, 2020, pp. 147–155 (cit. on p. 183).
[73] O. G. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov, and D. E. Serbin. “Subgroups of fully residually free groups: algorithmic problems”. In: Group theory, statistics, and cryptography. AMS special session combinatorial and statistical group theory, New York University, NY, USA, April 12–13, 2003. Providence, RI: American Mathematical Society (AMS), 2004, pp. 63–101 (cit. on p. 183).
[74] D. E. Knuth and P. B. Bendix. “Simple word problems in universal algebras”. In: Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967). Pergamon, Oxford, 1970, pp. 263–297 (cit. on p. 206).
[75] N. M. D. Kolodner. “On algebraic extensions and decomposition of homomorphisms of free groups”. Journal of Algebra 569 (Mar. 2021), pp. 595–615 (cit. on p. 218).
[76] M. Linton. “On the intersections of finitely generated subgroups of free groups: reduced rank to full rank”.arXiv:2108.10814 [cs, math] (Sept. 2021) (cit. on pp. 211, 214).
[77] R. C. Lyndon and P. E. Schupp. Combinatorial Group Theory. Springer, Mar. 2001 (cit. on pp. 198, 217).
[78] L. Margolis and O. Schnabel. “The Herzog-Schönheim conjecture for small groups and harmonic subgroups”. Beiträge zur Algebra und Geometrie 60.3 (2019), pp. 399–418 (cit. on p. 207).
[79] S Margolis, M Sapir, and PWeil. “Closed subgroups in pro-V topologies and the extension problem for inverse automata”. International Journal of Algebra and Computation 11 (2001), pp. 405–446 (cit. on pp. 182, 216, 219).
[80] S. W. Margolis and J. C. Meakin. “Free inverse monoids and graph immersions”. International Journal of Algebra and Computation 03.01 (Mar. 1993), pp. 79–99 (cit. on p. 182).
[81] L. Markus-Epstein. “Stallings foldings and subgroups of amalgams of finite groups”. International Journal of Algebra and Computation 17.8 (2007), pp. 1493–1535 (cit. on p. 183).
[82] A. Martino. “Intersections of automorphism fixed subgroups in the free group of rank three.” Algebraic & Geometric Topology 4 (2004), pp. 177–198 (cit. on p. 220).
[83] A. Martino and E. Ventura. “On automorphism-fixed subgroups of a free group”. Journal of Algebra 230.2 (2000), pp. 596–607 (cit. on p. 220).
[84] A. Martino and E. Ventura. “Fixed subgroups are compressed in free groups”. Communications in Algebra 32.10 (2004), pp. 3921–3935 (cit. on p. 222).
[85] J. McCool. “Some finitely presented subgroups of the automorphism group of a free group”. Journal of Algebra 35.1 (June 1975), pp. 205–213 (cit. on p. 221).
[86] A. Miasnikov, E. Ventura, and P.Weil. “Algebraic extensions in free groups”. In: Geometric Group Theory. Ed. By G. N. Arzhantseva, J. Burillo, L. Bartholdi, and E. Ventura. Trends in Mathematics. Birkhäuser Basel, Jan. 2007, pp. 225–253 (cit. on pp. 216, 217).
[87] S. Mijares and E. Ventura. “Onto extensions of free groups”. journal of Groups, Complexity, Cryptology Volume 13, issue 1 (Apr. 2021) (cit. on pp. 217, 218).
[88] I. Mineyev. “Submultiplicativity and the Hanna Neumann Conjecture”. Annals of Mathematics 175.1 (Jan. 2012), pp. 393–414 (cit. on p. 211).
[89] A. G. Myasnikov, V. N. Remeslennikov, and D. E. Serbin. “Fully residually free groups and graphs labeled by infinite words”. International Journal of Algebra and Computation 16.04 (Aug. 2006), pp. 689–737 (cit. on p. 183).
[90] H. Neumann. “On the intersection of finitely generated free groups”. Publicationes Mathematicae 4 (1956), pp. 186–189 (cit. on p. 210).
[91] W. D. Neumann. “On intersections of finitely generated subgroups of free groups”. In: Groups—Canberra 1989. Ed. by L. G. Kovács. Vol. 1456. Springer Berlin Heidelberg, 1990, pp. 161–170 (cit. on p. 211).
[92] A. Nikolaev and D. Serbin. “Membership problem in groups acting freely on Zn-trees”. Journal of Algebra 370 (Nov. 2012), pp. 410–444 (cit. on p. 183).
[93] A. V. Nikolaev and D. E. Serbin. “Finite index subgroups of fully residually free groups”. International Journal of Algebra and Computation 21.04 (June 2011), pp. 651–673 (cit. on p. 183).
[94] A. Y. Ol’shanskii. “Almost every group is hyperbolic”. International Journal of Algebra and Computation 02.01 (Mar. 1992), pp. 1–17 (cit. on p. 224).
[95] O. Parzanchevski and D. Puder. “Stallings graphs, algebraic extensions and primitive elements in F2”. Mathematical Proceedings of the Cambridge Philosophical Society 157.1 (July 2014), pp. 1–11 (cit. on p. 217).
[96] D. Puder. “Primitive words, free factors and measure preservation”. Israel Journal of Mathematics 201.1 (Jan. 2014), pp. 25–73 (cit. on pp. 200, 217).
[97] E. Rips and Z. Sela. “Cyclic splittings of finitely presented groups and the canonical JSJ deccomposition”. Annals of Mathematics. Second Series 146.1 (1997), pp. 55–109 (cit. on p. 182).
[98] D. J. S. Robinson. A Course in the Theory of Groups. Vol. 80. Graduate Texts in Mathematics. New York, NY: Springer New York, 1996 (cit. on p. 203).
[99] A. Roig, E. Ventura, and P.Weil. “On the complexity of the Whitehead minimization problem”. International Journal of Algebra and Computation 17.8 (2007), pp. 1611–1634 (cit. on p. 217).
[100] M. Roy and E. Ventura. “Degrees of compression and inertia for free-abelian times free groups”. Journal of Algebra 568 (2021), pp. 241–272 (cit. on pp. 222, 223).
[101] J. Sakarovitch. Elements of Automata Theory. Cambridge University Press, Oct. 2009 (cit. on p. 184).
[102] Z. Sela. “Acylindrical accessibility for groups”. Inventiones Mathematicae 129.3 (1997), pp. 527–565 (cit. on p. 182).
[103] Z. Sela. “Diophantine geometry over groups. I: Makanin-Razborov diagrams”. Publications Mathématiques 93 (2001), pp. 31–105 (cit. on p. 182).
[104] J.-P. Serre. Arbres, amalgames, SL2. Paris, 1977 (cit. on p. 181). [105] P. V. Silva, X. Soler-Escrivà, and E. Ventura. “Finite automata for Schreier graphs of virtually free groups”.
Journal of Group Theory 19.1 (2016), pp. 25–54 (cit. on p. 183).
[106] P. V. Silva and P.Weil. “On an algorithm to decide whether a free group is a free factor of another”. RAIRO. Theoretical Informatics and Applications 42.2 (2008), pp. 395–414 (cit. on p. 217).
[107] J. R. Stallings. “Topology of finite graphs”. Inventiones Mathematicae 71 (Mar. 1983), pp. 551–565 (cit. on pp. 181,182, 195–197, 206).
[108] J. R. Stallings. “Foldings of G-trees”. In: Arboreal Group Theory. Ed. by R. C. Alperin. Mathematical Sciences Research Institute Publications 19. Springer New York, 1991, pp. 355–368 (cit. on p. 182).
[109] J. R. Stallings. “Whitehead graphs on handlebodies”. In: Geometric group theory down under (Canberra, 1996).de Gruyter, Berlin, 1999, pp. 317–330 (cit. on p. 202).
[110] J. R. Stallings and A. R.Wolf. The Todd-Coxeter process, using graphs. 1987 (cit. on p. 206).
[111] B. Steinberg. “Inverse automata and profinite topologies on a free group”. Journal of Pure and Applied Algebra 167.2 (Feb. 2002), pp. 341–359 (cit. on p. 182).
[112] M. Takahasi. “Note on chain conditions in free groups”. Osaka Journal of Mathematics 3.2 (1951), pp. 221–225 (cit. on p. 216).
[113] J. A. Todd and H. S. M. Coxeter. “A practical method for enumerating cosets of a finite abstract group”. Proceedings of the Edinburgh Mathematical Society. Series II 5 (1936), pp. 26–34 (cit. on p. 206).
[114] N. W. M. Touikan. “A fast algorithm for Stallings’ folding process”. International Journal of Algebra and Computation 16.06 (Dec. 2006), pp. 1031–1045 (cit. on p. 197).
[115] E. Ventura. “Fixed subgroups in free groups: a survey”. In: Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001). Vol. 296. Contemp. Math. Providence, RI: Amer. Math. Soc., 2002, pp. 231–255 (cit. on p. 219).
[116] E. Ventura. “On fixed subgroups of maximal rank”. Communications in Algebra 25.10 (1997), pp. 3361–3375 (cit. on pp. 216, 219, 220).
[117] E. Ventura. “Computing fixed closures in free groups”. Illinois Journal of Mathematics 54.1 (2010), pp. 175–186 (cit. on p. 221).
[118] J. H. C.Whitehead. “On certain sets of elements in a free group”. Proceedings of the London Mathematical Society s2-41.1 (Jan. 1936), pp. 48–56 (cit. on p. 202).
[119] Q. Zhang, E. Ventura, and J.Wu. “Fixed subgroups are compressed in surface groups”. International Journal of Algebra and Computation 25.05 (Aug. 2015), pp. 865–887 (cit. on p. 222).
[120] v. Znám. “On exactly covering systems of arithmetic sequences”. In: Number Theory (Colloq., János Bolyai Math. Soc., Debrecen, 1968). North-Holland, Amsterdam, 1970, pp. 221–225 (cit. on p. 207). 
Volume 11, Issue 3 - Serial Number 3
Introduction to the Proceedings of WGTTG2021
September 2022
Pages 181-235
  • Receive Date: 01 September 2021
  • Revise Date: 19 November 2021
  • Accept Date: 22 November 2021
  • Published Online: 01 September 2022