The geometry of one-relator groups satisfying a polynomial isoperimetric inequality
Authors:
Giles Gardam and Daniel J. Woodhouse
Journal:
Proc. Amer. Math. Soc. 147 (2019), 125-129
MSC (2010):
Primary 20F65; Secondary 20F67, 20E06, 20F05
DOI:
https://doi.org/10.1090/proc/14238
Published electronically:
October 18, 2018
MathSciNet review:
3876736
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For every pair of positive integers we construct a one-relator group
whose Dehn function is
where
. The group
has no subgroup isomorphic to a Baumslag-Solitar group
with
, but it is not automatic, not CAT(0), and cannot act freely on a CAT(0) cube complex. This answers a long-standing question on the automaticity of one-relator groups and gives counterexamples to a conjecture of Wise.
- [1] Gilbert Baumslag, Some open problems, Summer School in Group Theory in Banff, 1996, CRM Proc. Lecture Notes, vol. 17, Amer. Math. Soc., Providence, RI, 1999, pp. 1–9. MR 1653682
- [2] Aldo A. Bernasconi, On HNN-extensions and the complexity of the word problem for one-relator groups, ProQuest LLC, Ann Arbor, MI, 1994. Thesis (Ph.D.)–The University of Utah. MR 2691093
- [3] N. Brady and M. R. Bridson, There is only one gap in the isoperimetric spectrum, Geom. Funct. Anal. 10 (2000), no. 5, 1053–1070. MR 1800063, https://doi.org/10.1007/PL00001646
- [4] Martin R. Bridson, The geometry of the word problem, Invitations to geometry and topology, Oxf. Grad. Texts Math., vol. 7, Oxford Univ. Press, Oxford, 2002, pp. 29–91. MR 1967746
- [5] Martin R. Bridson and Laurence Reeves, On the absence of automaticity of certain free-by-cyclic groups, 2006.
- [6] Jack O. Button, Tubular groups and non positive curvature, 2017. Available at https://arxiv.org/abs/1712.00290.
- [7] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694
- [8] Giles Gardam, Encoding and detecting properties in finitely presented groups, DPhil thesis, 2017. Available at https://ora.ox.ac.uk/objects/uuid:0c8a7009-7e04-4f66-911b-298ad87061fb.
- [9] S. M. Gersten, Dehn functions and 𝑙₁-norms of finite presentations, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 195–224. MR 1230635, https://doi.org/10.1007/978-1-4613-9730-4_9
- [10] S. M. Gersten, Problems on automatic groups, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 225–232. MR 1230636, https://doi.org/10.1007/978-1-4613-9730-4_10
- [11] S. M. Gersten, The automorphism group of a free group is not a 𝐶𝐴𝑇(0) group, Proc. Amer. Math. Soc. 121 (1994), no. 4, 999–1002. MR 1195719, https://doi.org/10.1090/S0002-9939-1994-1195719-9
- [12] Frédéric Haglund, Isometries of CAT(0) cube complexes are semi-simple, 2007. Available at https://arxiv.org/abs/0705.3386.
- [13] Roger C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2) 52 (1950), 650–665. MR 47046, https://doi.org/10.2307/1969440
- [14] Alexei Myasnikov, Alexander Ushakov, and Dong Wook Won, The word problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable, J. Algebra 345 (2011), 324–342. MR 2842068, https://doi.org/10.1016/j.jalgebra.2011.07.024
- [15] Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, 2011. Available at http://www.math.mcgill.ca/wise/papers.html.
- [16] Daniel T. Wise, Cubular tubular groups, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5503–5521. MR 3240932, https://doi.org/10.1090/S0002-9947-2014-06065-0
- [17] Daniel J. Woodhouse, A generalized axis theorem for cube complexes, Algebr. Geom. Topol. 17 (2017), no. 5, 2737–2751. MR 3704240, https://doi.org/10.2140/agt.2017.17.2737
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20F65, 20F67, 20E06, 20F05
Retrieve articles in all journals with MSC (2010): 20F65, 20F67, 20E06, 20F05
Additional Information
Giles Gardam
Affiliation:
Department of Mathematics, Technion, Haifa, Israel
Email:
gilesgar@technion.ac.il
Daniel J. Woodhouse
Affiliation:
Department of Mathematics, Technion, Haifa, Israel
Email:
woodhouse.da@technion.ac.il
DOI:
https://doi.org/10.1090/proc/14238
Keywords:
One-relator groups,
Dehn functions,
automatic groups,
CAT(0) spaces,
CAT(0) cube complexes,
Baumslag--Solitar subgroups
Received by editor(s):
December 8, 2017
Received by editor(s) in revised form:
May 3, 2018
Published electronically:
October 18, 2018
Additional Notes:
The first author was supported by the Israel Science Foundation (grant 662/15).
The second author was supported by the Israel Science Foundation (grant 1026/15).
Communicated by:
David Futer
Article copyright:
© Copyright 2018
American Mathematical Society