Embedding ℚ into a finitely presented group

Belk, James; Hyde, James; Matucci, Francesco

doi:10.1090/bull/1762

Embedding $\mathbb {Q}$ into a finitely presented group
HTML articles powered by AMS MathViewer

by James Belk, James Hyde and Francesco Matucci HTML | PDF

Bull. Amer. Math. Soc. 59 (2022), 561-567 Request permission

Abstract:

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

References

J. Belk, J. Hyde, and F. Matucci, QInTbar.nb, GitHub repository, 2021. https://github.com/jimbelk/QInTbar.
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
José Burillo and Sean Cleary, The automorphism group of Thompson’s group $F$: subgroups and metric properties, Rev. Mat. Iberoam. 29 (2013), no. 3, 809–828. MR 3090138, DOI 10.4171/RMI/741
Matthew G. Brin, The chameleon groups of Richard J. Thompson: automorphisms and dynamics, Inst. Hautes Études Sci. Publ. Math. 84 (1996), 5–33 (1997). MR 1441005, DOI 10.1007/BF02698834
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
Peter Freyd and Alex Heller, Splitting homotopy idempotents. II, J. Pure Appl. Algebra 89 (1993), no. 1-2, 93–106. MR 1239554, DOI 10.1016/0022-4049(93)90088-B
Louis Funar and Christophe Kapoudjian, The braided Ptolemy-Thompson group is finitely presented, Geom. Topol. 12 (2008), no. 1, 475–530. MR 2390352, DOI 10.2140/gt.2008.12.475
Étienne Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no. 3-4, 329–407. MR 1876932
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
P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595–622. MR 110750, DOI 10.1112/plms/s3-9.4.595
G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. London Ser. A 262 (1961), 455–475. MR 130286, DOI 10.1098/rspa.1961.0132
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
Abderezak Ould Houcine, Embeddings in finitely presented groups which preserve the center, J. Algebra 307 (2007), no. 1, 1–23. MR 2278040, DOI 10.1016/j.jalgebra.2006.07.015
Bernard M. Hurley, Small cancellation theory over groups equipped with an integer-valued length function, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Studies in Logic and the Foundations of Mathematics, vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 157–214. MR 579944
D. L. Johnson, Embedding some recursively presented groups, Groups St. Andrews 1997 in Bath, II, London Math. Soc. Lecture Note Ser., vol. 261, Cambridge Univ. Press, Cambridge, 1999, pp. 410–416. MR 1676637, DOI 10.1017/CBO9780511666148.004
Pierre Lochak and Leila Schneps, The universal Ptolemy-Teichmüller groupoid, Geometric Galois actions, 2, London Math. Soc. Lecture Note Ser., vol. 243, Cambridge Univ. Press, Cambridge, 1997, pp. 325–347. MR 1653018, DOI 10.1017/CBO9780511666124.014
V. D. Mazurov and E. I. Khukhro (eds.), The Kourovka notebook, Fourteenth augmented edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 1999. Unsolved problems in group theory. MR 1733915
Vahagn H. Mikaelian, On a problem on explicit embeddings of the group $\Bbb Q$, Int. J. Math. Math. Sci. 13 (2005), 2119–2123. MR 2177699, DOI 10.1155/IJMMS.2005.2119
V. Mikaelian, The Higman operations and embeddings of recursive groups, arXiv:2002.09728, 2020.-
R. Thompson, Notes on three groups of homeomorphisms, Unpublished but widely circulated handwritten notes, 1965, pp. 1–11.