On subgroups of R. Thompson’s group $F$
HTML articles powered by AMS MathViewer
- by Gili Golan and Mark Sapir PDF
- Trans. Amer. Math. Soc. 369 (2017), 8857-8878 Request permission
Abstract:
We provide two ways to show that the R. Thompson group $F$ has maximal subgroups of infinite index which do not fix any number in the unit interval under the natural action of $F$ on $(0,1)$, thus solving a problem by D. Savchuk. The first way employs Jones’ subgroup of the R. Thompson group $F$ and leads to an explicit finitely generated example. The second way employs directed 2-complexes and 2-dimensional analogs of Stallings’ core graphs and gives many implicit examples. We also show that $F$ has a decreasing sequence of finitely generated subgroups $F>H_1>H_2>\cdots$ such that $\cap H_i=\{1\}$ and for every $i$ there exist only finitely many subgroups of $F$ containing $H_i$.References
- 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 and Bronlyn Wassink, Finite index subgroups of R. Thompson’s group $F$, arXiv:0711.1014.
- 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, Quasi-isometrically embedded subgroups of Thompson’s group $F$, J. Algebra 212 (1999), no. 1, 65–78. MR 1670622, DOI 10.1006/jabr.1998.7618
- 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, 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
- Gili Golan, The generation problem in Thompson group $F$, arxiv:1608.02572.
- Gili Golan and Mark Sapir, On Jones’ subgroup of R. Thompson group $F$, accepted in Advanced Studies in Pure Mathematics. Proceedings of the MSJ-SI conference “Hyperbolic Geometry and Geometric Group Theory”, arXiv: 1501.00724.
- Gili Golan and Mark Sapir, Stabilizers of finite sets of points in R. Thompson’s group $F$, to appear in St. Petersburg Math. J.
- 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
- A. Ju. Ol′šanskiĭ, An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 2, 309–321, 479 (Russian). MR 571100
- 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
- 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. Sci. 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
- Yan Wu and Xiaoman Chen, Distortion of wreath products in Thompson’s group $F$, Chinese Ann. Math. Ser. B 35 (2014), no. 5, 801–816. MR 3246937, DOI 10.1007/s11401-014-0851-y
Additional Information
- Gili Golan
- Affiliation: Department of Mathematics, Bar-Ilan University, 5290002 Ramat-Gan, Israel
- MR Author ID: 1031418
- Mark Sapir
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 189574
- Received by editor(s): August 10, 2015
- Received by editor(s) in revised form: October 14, 2015, and April 22, 2016
- Published electronically: August 3, 2017
- Additional Notes: This research was partially supported by the NSF grant DMS-1500180. The paper was written while the second author was visiting the Max Planck Institute for Mathematics in Bonn.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 8857-8878
- MSC (2010): Primary 20F65, 20G07
- DOI: https://doi.org/10.1090/tran/6982
- MathSciNet review: 3710646