AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
The Generation Problem in Thompson Group $F$
About this Title
Gili Golan Polak
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 292, Number 1451
ISBNs: 978-1-4704-6723-4 (print); 978-1-4704-7696-0 (online)
DOI: https://doi.org/10.1090/memo/1451
Published electronically: November 17, 2023
Keywords: Thompson’s group $F$,
decision problems,
diagram groups,
directed $2$-complexes,
the Stallings $2$-core,
closed subgroups,
maximal subgroups,
homeomorphisms of the interval
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries on $F$
- 3. The Stallings $2$-core of subgroups of diagram groups
- 4. Paths on the core of a subgroup $H\le F$
- 5. The closure of a subgroup $H\le F$
- 6. Transitivity of the action of $H$ on the set $\mathcal D$
- 7. The generation problem in $F$
- 8. The Tuples algorithm
- 9. $F$ is a cyclic extension of a subgroup $K$ which has a maximal elementary amenable subgroup
- 10. Computations related to $\mathcal {L}(H)$
- 11. Solvable subgroups of Thompson’s group $F$
- 12. Open problems
Abstract
We show that the generation problem in Thompson’s group $F$ is decidable, i.e., there is an algorithm which decides if a finite set of elements of $F$ generates the whole $F$. The algorithm makes use of the Stallings $2$-core of subgroups of $F$, which can be defined in an analogous way to the Stallings core of subgroups of a finitely generated free group. Further study of the Stallings $2$-core of subgroups of $F$ provides a solution to another algorithmic problem in $F$. Namely, given a finitely generated subgroup $H$ of $F$, it is decidable if $H$ acts transitively on the set of finite dyadic fractions $\mathcal D$. Other applications of the study include the construction of new maximal subgroups of $F$ of infinite index, among which, a maximal subgroup of infinite index which acts transitively on the set $\mathcal D$ and the construction of an elementary amenable subgroup of $F$ which is maximal in a normal subgroup of $F$.- James Belk and Francesco Matucci, Conjugacy and dynamics in Thompson’s groups, Geom. Dedicata 169 (2014), 239–261. MR 3175247, DOI 10.1007/s10711-013-9853-2
- James Belk, N. Hossain, Francesco Matucci, and Robert McGrail, Implementation of a Solution to the Conjugacy Problem in Thompson’s Group $F$, ACM Commun. Comput. Algebra 47, No. 3-4 (2014), 120–121.
- Collin Bleak and Bronlyn Wassink, Finite index subgroups of R. Thompson’s group $F$, arXiv:0711.1014, (2007).
- Collin Bleak, A geometric classification of some solvable groups of homeomorphisms, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 352–372. MR 2439629, DOI 10.1112/jlms/jdn017
- Collin Bleak, An algebraic classification of some solvable groups of homeomorphisms, J. Algebra 319 (2008), no. 4, 1368–1397. MR 2383051, DOI 10.1016/j.jalgebra.2007.11.012
- Collin Bleak, A minimal non-solvable group of homeomorphisms, Groups Geom. Dyn. 3 (2009), no. 1, 1–37. MR 2466019, DOI 10.4171/GGD/50
- Collin Bleak, Tara Brough, and Susan Hermiller, Determining solubility for finitely generated groups of PL homeomorphisms, Trans. Amer. Math. Soc. 374 (2021), no. 10, 6815–6837. MR 4315590, DOI 10.1090/tran/8421
- Matthew G. Brin, Elementary amenable subgroups of R. Thompson’s group $F$, Internat. J. Algebra Comput. 15 (2005), no. 4, 619–642. MR 2160570, DOI 10.1142/S0218196705002517
- Kenneth S. Brown, Finiteness properties of groups, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 45–75. MR 885095, DOI 10.1016/0022-4049(87)90015-6
- José Burillo, Francesco Matucci, and Enric Ventura, The conjugacy problem in extensions of Thompson’s group $F$, Israel J. Math. 216 (2016), no. 1, 15–59. MR 3556962, DOI 10.1007/s11856-016-1403-9
- J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438
- Sean Cleary, Distortion of wreath products in some finitely presented groups, Pacific J. Math. 228 (2006), no. 1, 53–61. MR 2263024, DOI 10.2140/pjm.2006.228.53
- Sean Cleary, Murray Elder, Andrew Rechnitzer, and Jennifer Taback, Random subgroups of Thompson’s group $F$, Groups Geom. Dyn. 4 (2010), no. 1, 91–126. MR 2566302, DOI 10.4171/GGD/76
- Gábor Elek and Nicolas Monod, On the topological full group of a minimal Cantor $\mathbf {Z}^2$-system, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3549–3552. MR 3080176, DOI 10.1090/S0002-9939-2013-11654-0
- Tsachik Gelander, Gili Golan, and Kate Juschenko, Invariable generation of Thompson groups, J. Algebra 478 (2017), 261–270. MR 3621672, DOI 10.1016/j.jalgebra.2017.01.019
- Gili Golan Polak, Random generation of Thompson group $F$, J. Algebra 593 (2022), 507–524. MR 4349287, DOI 10.1016/j.jalgebra.2021.11.024
- Gili Golan Polak, On maximal subgroups of Thompson’s group $F$, To appear in Groups Geom. Dyn., arXiv:2209.03244, (2022).
- Gili Golan Polak, Thompson’s group $F$ is almost $\frac {3}{2}$-generated, Bull. Lond. Math. Soc., 55 (5), 2023.
- Gili Golan Polak and Mark Sapir, On closed subgroups of Thompson group $F$, To appear in Israel J. Math, arXiv:2105.00531, (2021).
- Gili Golan and Mark Sapir, On Jones’ subgroup of R. Thompson group $F$, J. Algebra 470 (2017), 122–159. MR 3565428, DOI 10.1016/j.jalgebra.2016.09.001
- Gili Golan Polak and Mark Sapir, On some generating set of Thompson’s group $F$, São Paulo J. Math. Sci. (2023), https://doi.org/10.1007/s40863-023-00360-0
- Gili Golan and Mark Sapir, On subgroups of R. Thompson’s group $F$, Trans. Amer. Math. Soc. 369 (2017), no. 12, 8857–8878. MR 3710646, DOI 10.1090/tran/6982
- G. Golan and M. Sapir, On the stabilizers of finite sets of numbers in the R. Thompson group $F$, Algebra i Analiz 29 (2017), no. 1, 70–110; English transl., St. Petersburg Math. J. 29 (2018), no. 1, 51–79. MR 3660685, DOI 10.1090/spmj/1482
- Victor Guba and Mark Sapir, Diagram groups, Mem. Amer. Math. Soc. 130 (1997), no. 620, viii+117. MR 1396957, DOI 10.1090/memo/0620
- V. S. Guba and M. V. Sapir, On subgroups of the R. Thompson group $F$ and other diagram groups, Mat. Sb. 190 (1999), no. 8, 3–60 (Russian, with Russian summary); English transl., Sb. Math. 190 (1999), no. 7-8, 1077–1130. MR 1725439, DOI 10.1070/SM1999v190n08ABEH000419
- V. S. Guba and M. V. Sapir, Diagram groups and directed 2-complexes: homotopy and homology, J. Pure Appl. Algebra 205 (2006), no. 1, 1–47. MR 2193190, DOI 10.1016/j.jpaa.2005.06.012
- Vaughan Jones, Some unitary representations of Thompson’s groups $F$ and $T$, J. Comb. Algebra 1 (2017), no. 1, 1–44. MR 3589908, DOI 10.4171/JCA/1-1-1
- Martin Kassabov and Francesco Matucci, The simultaneous conjugacy problem in groups of piecewise linear functions, Groups Geom. Dyn. 6 (2012), no. 2, 279–315. MR 2914861, DOI 10.4171/GGD/158
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 89, Springer-Verlag, Berlin-New York, 1977. MR 577064
- Andrés Navas, Quelques groupes moyennables de difféomorphismes de l’intervalle, Bol. Soc. Mat. Mexicana (3) 10 (2004), no. 2, 219–244 (2005) (French, with English and French summaries). MR 2135961
- E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45–47. MR 642423, DOI 10.1112/blms/14.1.45
- Mark V. Sapir, Combinatorial algebra: syntax and semantics, Springer Monographs in Mathematics, Springer, Cham, 2014. With contributions by Victor S. Guba and Mikhail V. Volkov. MR 3243545, DOI 10.1007/978-3-319-08031-4
- Mark Sapir, Personal communication.
- Dmytro Savchuk, Some graphs related to Thompson’s group $F$, Combinatorial and geometric group theory, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010, pp. 279–296. MR 2744025, DOI 10.1007/978-3-7643-9911-5_{1}2
- Dmytro Savchuk, Schreier graphs of actions of Thompson’s group $F$ on the unit interval and on the Cantor set, Geom. Dedicata 175 (2015), 355–372. MR 3323646, DOI 10.1007/s10711-014-9951-9
- Vladimir Shpilrain and Alexander Ushakov, Thompson’s group and public key cryptography, Lecture Notes Comp. Sc. 3531 (2005), 151–164.
- John R. Stallings, Foldings of $G$-trees, Arboreal group theory (Berkeley, CA, 1988) Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 355–368. MR 1105341, DOI 10.1007/978-1-4612-3142-4_{1}4
- Daniel T. Wise, A residually finite version of Rips’s construction, Bull. London Math. Soc. 35 (2003), no. 1, 23–29. MR 1934427, DOI 10.1112/S0024609302001406