Enlighten Publications

In this section

Embedding Q into a finitely presented group

Belk, J. , Hyde, J. and Matucci, F. (2022) Embedding Q into a finitely presented group. Bulletin of the American Mathematical Society, 59, pp. 561-567. (doi: 10.1090/bull/1762)

Text
280390.pdf - Accepted Version
251kB

Abstract

We observe that the group of all lifts of elements of Thompson's group <i>T</i> to the real line is finitely presented and contains the additive group Q of the rational numbers. This gives an explicit realization of the Higman embedding theorem for Q, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.

Item Type:	Articles
Additional Information:	The first author was partially supported by EPSRC grant EP/R032866/1 as well as the National Science Foundation under Grant No. DMS-1854367 during the creation of this paper. The third author is a member of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM) and gratefully acknowledges the support of the Funda¸c˜ao para a Ciˆencia e a Tecnologia (CEMAT-Ciˆencias FCT projects UIDB/04621/2020 and UIDP/04621/2020) and of the Universit`a degli Studi di Milano–Bicocca (FA project ATE-2016-0045 “Strutture Algebriche”).
Status:	Published
Refereed:	Yes
Glasgow Author(s) Enlighten ID:	Belk, Dr Jim
Authors:	Belk, J., Hyde, J., and Matucci, F.
Subjects:	Q Science > QA Mathematics
College/School:	College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:	Bulletin of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0273-0979
ISSN (Online):	1088-9485
Published Online:	11 August 2022
Copyright Holders:	Copyright © 2022 American Mathematical Society
First Published:	First published in Bulletin of the American Mathematical Society 59: 561-567
Publisher Policy:	Reproduced in accordance with the publisher copyright policy
Related URLs:	arXiv

University Staff: Request a correction | Enlighten Editors: Update this record

References

J. Belk, J. Hyde, and F. Matucci, QInTbar.nb, GitHub repository, 2021. https://github.com/jimbelk/QInTbar. C. Bleak, M. Kassabov, and F. Matucci, Structure theorems for groups of homeomorphisms of the circle. International Journal of Algebra and Computation 21.06 (2011): 1007–1036. doi:10.1142/S0218196711006571 J. Burillo and S. Cleary, The automorphism group of Thompson’s group F: subgroups and metric properties. Revista Matemática Iberoamericana 29.3 (2013): 809–828. doi:10.4171/RMI/741. M. Brin, The chameleon groups of Richard J. Thompson: Automorphisms and dynamics. Publications Mathématiques de l'IHÉS 84 (1996): 5–33. doi:10.1007/BF02698834. J. Cannon, W. Floyd, and W. Parry, Introductory notes on Richard Thompson’s groups. L'Enseignement mathématique 42 (1996): 215–256. doi:10.5169/seals-87877. P. Freyd and A. Heller, Splitting homotopy idempotents II. Journal of Pure and Applied Algebra 89.1–2 (1993): 93–106. doi:10.1016/0022-4049(93)90088-B. L. Funar and C. Kapoudjian, The braided Ptolemy-Thompson group is finitely presented. Geom. Topol. 12.1 (2008), 475–530. doi:10.2140/gt.2008.12.475. E. Ghys, Groups acting on the circle. L'Enseignement mathématique 47.3/4 (2001): 329–408. doi:10.5169/seals-65441. E. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle. Commentarii Mathematici Helvetici 62.1 (1987): 185–239. doi:10.1007/BF02564445. P. Hall, On the finiteness of certain soluble groups. Proceedings of the London Mathematical Society 3.4 (1959): 595–622. doi:10.1112/plms/s3-9.4.595. G. Higman, Subgroups of finitely presented groups. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 262.1311 (1961): 455–475. doi:10.1098/rspa.1961.0132. G. Higman, Finitely presented infinite simple groups. Notes on Pure Mathematics 8 (1974), Australian National University, Canberra. D. Holt and B. Eick and E. A. O’Brien, Handbook of computational group theory. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005. xvi+514 pp A. Ould Houcine, Embeddings in finitely presented groups which preserve the center. Journal of Algebra 307.1 (2007): 1–23. doi:10.1016/j.jalgebra.2006.07.015. B. Hurley, Small Cancellation Theory Over Groups Equipped with an Integer-Valued Length Function. Studies in Logic and the Foundations of Mathematics 95 (1980): 157–214. doi:10.1016/S0049-237X(08)71337-5. D. L. Johnson, Embedding some recursively presented groups. Groups St. Andrews 1997 in Bath, II, 410–416, London Math. Soc. Lecture Note Ser., 261, Cambridge Univ. Press, Cambridge, 1999. P. Lochak and L. Schneps, The universal Ptolemy-Teichmüller groupoid. Geometric Galois actions, 2, 325–347, London Math. Soc. Lecture Note Ser. 243, Cambridge Univ. Press, Cambridge, 1997. V. Mazurov and E. Khukhro, Unsolved problems in group theory: The Kourovka notebook, No. 14. Sobolev Institute of Mathematics, 1999. arXiv:1401.0300. V. Mikaelian, On a problem on explicit embeddings of the group Q. International Journal of Mathematics and Mathematical Sciences 2005.13 (2005): 2119–2123. doi:10.1155/IJMMS.2005.2119. V. Mikaelian, The Higman operations and embeddings of recursive groups. arXiv:2002.09728 R. Thompson, Notes on three groups of homeomorphisms. Unpublished but widely circulated handwritten notes (1965): 1–11.

Deposit and Record Details

ID Code:	280390
Depositing User:	Dr Jim Belk
Datestamp:	26 Sep 2022 13:56
Last Modified:	27 Sep 2022 01:31
Date of acceptance:	26 October 2021
Date of first online publication:	11 August 2022
Date Deposited:	26 September 2022
Data Availability Statement:	No