Abstract
We prove that the R. Thompson groups F, T, V have linear divergence functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Recall the standard relation on the set of functions \({\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\): \(f\preceq _C g\) if \(f(x)\le Cg(Cx)+Cx+C\) for some \(C>1\) and all x. This defines the known equivalence relation on the set of functions \({\mathbb {R}}_+\rightarrow {\mathbb {R}}_+\): \(f\equiv _C g\) if \(f\preceq _C g\) and \(g\preceq _C f\).
\(p\equiv q\) denotes letter-by-letter equality of words p, q.
References
Birget, J.C.: The groups of Richard Thompson and complexity. Int. J. Algebra Comput. 14, 569–626 (2004)
Bleak, C., Bowman, H., Gordon, A., Graham, G., Hughes, J., Matucci, F., Sapir, J.: Centralizers in the R. Thompson group \(V_n\). Groups Geom. Dyn. 7(4), 821–865 (2013)
Bleak, C., Salazar-Daz, O.: Free products in R. Thompson’s group \(V\). Trans. Am. Math. Soc. 365(11), 5967–5997 (2013)
Brin, M.G.: Higher dimensional Thompson groups. Geom. Dedicata 108, 163–192 (2004)
Brin, M.G.: The algebra of strand splitting. I. A braided version of Thompson’s group \(V\). J. Group Theory 10(6), 757–788 (2007)
Burillo, J.: Quasi-isometrically embedded subgroups of Thompson’s group F. J. Algebra 212(1), 65–78 (1999)
Burillo, J., Cleary, S., Stein, M.I.: Metrics and embeddings of generalizations of Thompson’s group \(F\). Trans. Am. Math. Soc. 353(4), 1677–1689 (2001)
Burillo, J., Cleary, S., Stein, M., Taback, J.: Combinatorial and metric properties of Thompson’s group \(T\). Trans. Am. Math. Soc. 361, 631–652 (2009)
Cannon, J., Floyd, W., Parry, W.: Introductory notes on Richard Thompson’s groups. L’Enseignement Mathematique 42, 215–256 (1996)
Druţu, C., Mozes, S., Sapir, M.: Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Am. Math. Soc. 362(5), 2451–2505 (2010)
Druţu, C., Sapir, M.: Tree-graded spaces and asymptotic cones ofgroups. Topology 44, 959–1058 (2005). (with an appendix by D. Osinand M. Sapir)
Farley, D.S.: Actions of picture groups on CAT(0) cubical complexes. Geom. Dedicata 110, 221–242 (2005)
Gersten, S.: Quadratic divergence of geodesics in CAT(0) spaces. Geom. Funct. Anal. 4(1), 37–51 (1994)
Golan, G.: The generation problem in Thompson group \(F\). arXiv:1608.02572. Accessed 1 July 2017
Golan, G., Sapir, M.: On Jones’ subgroup of R. Thompson group \(F\). J. Algebra 470, 122–159 (2017)
Guba, V.: The Dehn function of Richard Thompson’s group \(F\) is quadratic. Invent. Math. 163(2), 313–342 (2006)
Guba, V., Sapir, M.: Diagram groups. Mem. Am. Math. Soc. 130(620), 1–117 (1997)
Jones, V.: Some unitary representations of Thompson’s groups \(F\) and \(T\). J. Comb. Algebra 1(1), 1–44 (2017)
Juschenko, K., Monod, N.: Cantor systems, piecewise translations and simple amenable groups. Ann. Math. (2) 178(2), 775–787 (2013)
Macura, N.: Detour functions and quasi-isometries. Q. J. Math. 53(2), 207–239 (2002)
Lodha, Y., Moore, J.T.: A nonamenable finitely presented group of piecewise projective homeomorphisms. (English summary) Groups Geom. Dyn. 10(1), 177–200 (2016)
Lubotzky, A., Mozes, S., Raghunathan, M.S.: The word and Riemannian metrics on lattices of semisimple groups. IHES Sci. Publ. Math. 91, 5–53 (2001)
Acknowledgements
The authors are grateful to the referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first author was supported in part by a Fulbright grant and a post-doctoral scholarship from Bar-Ilan University, the research of the second author was supported in part by the NSF Grant DMS-1500180.
Rights and permissions
About this article
Cite this article
Golan, G., Sapir, M. Divergence functions of Thompson groups. Geom Dedicata 201, 227–242 (2019). https://doi.org/10.1007/s10711-018-0390-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-018-0390-x