Free products in R. Thompson’s group $V$
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- by Collin Bleak and Olga Salazar-Díaz PDF
- Trans. Amer. Math. Soc. 365 (2013), 5967-5997 Request permission
Abstract:
We investigate some product structures in R. Thompson’s group $V$, primarily by studying the topological dynamics associated with $V$’s action on the Cantor set $\mathfrak {C}$. We draw attention to the class $\mathcal {D}_{(V,\mathfrak {C})}$ of groups which have embeddings as demonstrative subgroups of $V$ whose class can be used to assist in forming various products. Note that $\mathcal {D}_{(V,\mathfrak {C})}$ contains all finite groups, the free group on two generators, and $\mathbf {Q}/\mathbf {Z}$, and is closed under passing to subgroups and under taking direct products of any member by any finite member. If $G\leq V$ and $H\in \mathcal {D}_{(V,\mathfrak {C})}$, then $G\wr H$ embeds into $V$. Finally, if $G$, $H\in \mathcal {D}_{(V,\mathfrak {C})}$, then $G*H$ embeds in $V$.
Using a dynamical approach, we also show the perhaps surprising result that $Z^2*Z$ does not embed in $V$, even though $V$ has many embedded copies of $Z^2$ and has many embedded copies of free products of various pairs of its subgroups.
References
- Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
- Nathan Barker, Simultaneous conjugacy in R. Thompson’s group ${V}$, Dissertation, University of Newcastle-upon-Tyne, Newcastle, England, 2011.
- J.M. Belk and F. Matucci, Conjugacy in Thompson’s groups, submitted (2008), \verb+arXiv:math.GR/0708.4250v1+.
- —, Dynamics in Thompson’s group $F$, submitted (2008), \verb+arXiv:math.GR/0710.3633v1+.
- 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, Some questions about the dimension of a group action, Bull. Lond. Math. Soc. 40 (2008), no. 5, 770–776. MR 2439642, DOI 10.1112/blms/bdn060
- Collin Bleak, Hannah Bowman, Alison Gordon, Garrett Graham, Jacob Hughes, Francesco Matucci, and Eugenia Sapir, Centralizers in the R. Thompson group $v_n$, submitted (2011), 1–32.
- Collin Bleak, Martin Kassabov, and Francesco Matucci, Structure theorems for groups of homeomorphisms of the circle, Internat. J. Algebra Comput. 21 (2011), no. 6, 1007–1036. MR 2847521, DOI 10.1142/S0218196711006571
- Matthew G. Brin, Higher dimensional Thompson groups, Geom. Dedicata 108 (2004), 163–192. MR 2112673, DOI 10.1007/s10711-004-8122-9
- 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
- Matthew G. Brin and Craig C. Squier, Groups of piecewise linear homeomorphisms of the real line, Invent. Math. 79 (1985), no. 3, 485–498. MR 782231, DOI 10.1007/BF01388519
- José Burillo, Sean Cleary, Melanie Stein, and Jennifer Taback, Combinatorial and metric properties of Thompson’s group $T$, Trans. Amer. Math. Soc. 361 (2009), no. 2, 631–652. MR 2452818, DOI 10.1090/S0002-9947-08-04381-X
- Danny Calegari, Denominator bounds in Thompson-like groups and flows, Groups Geom. Dyn. 1 (2007), no. 2, 101–109. MR 2319453, DOI 10.4171/GGD/6
- 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
- M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), no. 3, 449–457. MR 807066, DOI 10.1007/BF01388581
- Étienne Ghys and Vlad Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987), no. 2, 185–239 (French). MR 896095, DOI 10.1007/BF02564445
- É Ghys and V. Sergiescu, personal communication (2011).
- Graham Higman, Finitely presented infinite simple groups, Notes on Pure Mathematics, No. 8, Australian National University, Department of Pure Mathematics, Department of Mathematics, I.A.S., Canberra, 1974. MR 0376874
- Derek F. Holt, Sarah Rees, Claas E. Röver, and Richard M. Thomas, Groups with context-free co-word problem, J. London Math. Soc. (2) 71 (2005), no. 3, 643–657. MR 2132375, DOI 10.1112/S002461070500654X
- Felix Klein, Neue Beiträge zur Riemann’schen Functionentheorie, Math. Ann. 21 (1883), no. 2, 141–218 (German). MR 1510193, DOI 10.1007/BF01442920
- Marc Krasner and Léo Kaloujnine, Produit complet des groupes de permutations et problème d’extension de groupes. I, Acta Sci. Math. (Szeged) 13 (1950), 208–230 (French). MR 49890
- Marc Krasner and Léo Kaloujnine, Produit complet des groupes de permutations et problème de groupes. II, Acta Sci. Math. (Szeged) 14 (1951), 39–66 (French). MR 49891
- Marc Krasner and Léo Kaloujnine, Produit complet des groupes de permutations et problème d’extension de groupes. III, Acta Sci. Math. (Szeged) 14 (1951), 69–82 (French). MR 49892
- J. Lehnert and P. Schweitzer, The co-word problem for the Higman-Thompson group is context-free, Bull. Lond. Math. Soc. 39 (2007), no. 2, 235–241. MR 2323454, DOI 10.1112/blms/bdl043
- David E. Muller and Paul E. Schupp, Groups, the theory of ends, and context-free languages, J. Comput. System Sci. 26 (1983), no. 3, 295–310. MR 710250, DOI 10.1016/0022-0000(83)90003-X
- David E. Muller and Paul E. Schupp, The theory of ends, pushdown automata, and second-order logic, Theoret. Comput. Sci. 37 (1985), no. 1, 51–75. MR 796313, DOI 10.1016/0304-3975(85)90087-8
- Claas Röver, Subgroups of finitely presented simple groups, Ph.D. thesis, Pembroke College, University of Oxford, 1999.
- Matatyahu Rubin, Locally moving groups and reconstruction problems, Ordered groups and infinite permutation groups, Math. Appl., vol. 354, Kluwer Acad. Publ., Dordrecht, 1996, pp. 121–157. MR 1486199
- Olga Patricia Salazar-Díaz, Thompson’s group $V$ from a dynamical viewpoint, Internat. J. Algebra Comput. 20 (2010), no. 1, 39–70. MR 2655915, DOI 10.1142/S0218196710005534
- Elizabeth A. Scott, A construction which can be used to produce finitely presented infinite simple groups, J. Algebra 90 (1984), no. 2, 294–322. MR 760011, DOI 10.1016/0021-8693(84)90172-8
- Takashi Tsuboi, Group generated by half transvections, Kodai Math. J. 28 (2005), no. 3, 463–482. MR 2194538, DOI 10.2996/kmj/1134397761
Additional Information
- Collin Bleak
- Affiliation: School of Mathematics and Statistics, Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland
- MR Author ID: 831679
- ORCID: 0000-0001-5790-1940
- Email: collin@mcs.st-and.ac.uk
- Olga Salazar-Díaz
- Affiliation: Escuela de Matemáticas, Universidad Nacional de Colombia, Medellín, Colombia
- Email: opsalaza@yahoo.com
- Received by editor(s): November 13, 2009
- Received by editor(s) in revised form: February 22, 2011, and March 7, 2012
- Published electronically: June 19, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5967-5997
- MSC (2010): Primary 20F65, 37C85, 20E07, 20E32
- DOI: https://doi.org/10.1090/S0002-9947-2013-05823-0
- MathSciNet review: 3091272