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A new test for asphericity and diagrammatic reducibility of group presentations

Part of: Special aspects of infinite or finite groups Low-dimensional topology

Published online by Cambridge University Press:  26 January 2019

Jonathan Ariel Barmak  and
Elias Gabriel Minian
Jonathan Ariel Barmak
Affiliation:
Departamento de Matemática–IMAS FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina ([email protected]; [email protected])
Elias Gabriel Minian
Affiliation:
Departamento de Matemática–IMAS FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina ([email protected]; [email protected])
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Abstract

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.

Keywords

AsphericityDR presentationslabelled oriented treesweight testlocally indicable groupsAdian presentations

MSC classification

Primary: 57M20: Two-dimensional complexes
Secondary: 20F05: Generators, relations, and presentations 20F06: Cancellation theory; application of van Kampen diagrams 57M05: Fundamental group, presentations, free differential calculus
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 871 - 895
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Adams, J. F.. A new proof of a theorem of W. H. Cockcroft. J. London Math. Soc. 30 (1955), 482–488.CrossRefGoogle Scholar
2Bartels, A., Lück, W. and Weinberger, S.. On hyperbolic groups with spheres as boundary. J. Diff. Geom. 86 (2010), 1–16.CrossRefGoogle Scholar
3Bogley, W.. J.H.C. Whitehead's asphericity question. In Two-dimensional homotopy and combinatorial group theory (eds. Hog-Angeloni, C., Metzler, W. and Sieradski, A.J.). London Mathematical Society Lecture Note Series 197 (Cambridge: Cambridge University Press, 1993).Google Scholar
4Brick, S.. Normal-convexity and equations over groups. Invent. Math. 94 (1988), 81–104.CrossRefGoogle Scholar
5Brick, S.. A note on coverings and Kervaire complexes. Bull. Austral. Math. Soc. 46 (1992), 1–21.CrossRefGoogle Scholar
6Chiswell, I., Collins, D. and Huebschmann, J.. Aspherical group presentations. Math. Z. 178 (1981), 1–36.CrossRefGoogle Scholar
7Clay, A. and Rolfsen, D.. Ordered groups and topology. Preprint (2015) Available at http://arxiv.org/abs/1511.05088Google Scholar
8Cockcroft, W. H.. On two-dimensional aspherical complexes. Proc. London Math. Soc. (3) 4 (1954), 375–384.CrossRefGoogle Scholar
9Corson, J. M. and Trace, B.. Diagrammatically reducible complexes and Haken manifolds. J. Austral. Math. Soc. (Series A) 69 (2000), 116–126.CrossRefGoogle Scholar
10Gersten, S. M.. Reducible diagrams and equations over groups. Essays in group theory, Math. Sci. Res. Ins. Publ. 8 (Springer Verlag), 1987.CrossRefGoogle Scholar
11Gersten, S. M.. Branched coverings of 2-complexes and diagrammatic reducibility. Trans. Amer. Math. Soc. 303 (1987), 689–706.Google Scholar
12Gersten, S. M.. Some remarks on subgroups of hyperbolic groups. Preprint (1999) Available at http://www.math.utah.edu/sg/Papers/sgs.pdfGoogle Scholar
13Gersten, S. M.. Asphericity for certain groups of cohomological dimension 2. Preprint (2015) Available at http://arxiv.org/abs/1501.06875Google Scholar
14Gromov, M.. Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.CrossRefGoogle Scholar
15Howie, J.. On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math. 324 (1981), 165–174.Google Scholar
16Howie, J.. On locally indicable groups. Math. Z. 180 (1982), 445–451.CrossRefGoogle Scholar
17Howie, J.. Some remarks on a problem of J. H. C. Whitehead. Topology 22 (1983), 475–485.CrossRefGoogle Scholar
18Howie, J.. The solution of length three equations over groups. Proc. Edinburgh Math. Soc. 26 (1983), 89–96.CrossRefGoogle Scholar
19Howie, J.. On the asphericity of ribbon disc complements. Trans. Amer. Math. Soc. 289 (1985), 281–302.CrossRefGoogle Scholar
20Howie, J.. Minimal Seifert manifolds for higher ribbon knots. The Epstein birthday schrift, 261–293 (electronic). Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry, 1998.Google Scholar
21Huck, G. and Rosebrock, S.. Weight tests and hyperbolic groups. In Combinatorial and Geometric Group Theory (eds. Howie, J., Duncan, A. and Gilbert, N.). London Math. Soc. Lecture Note Ser, Vol. 204, pp. 174–183 (London: Cambridge University Press, 1995).Google Scholar
22Ivanov, S. V.. An asphericity conjecture and Kaplansky problem on zero divisors. J. Algebra 216 (1999), 13–19.CrossRefGoogle Scholar
23Klyachko, A. and Thom, A.. New topological methods to solve equations over groups. Algebr. Geom. Topol. 17 (2017), 331–353.CrossRefGoogle Scholar
24Krstić, S.. Systems of equations over locally p-indicable groups. Invent. Math. 81 (1985), 373–378.CrossRefGoogle Scholar
25Lück, W.. Aspherical manifolds. Bull. Manifold Atlas (2012) 1–17.Google Scholar
26Lyndon, R. and Schupp, P.. Combinatorial group theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, xiv+339 pp. (Berlin-New York, Springer-Verlag, 1977).Google Scholar
27Pride, S.. Star-complexes. and the dependence problems for hyperbolic complexes. Glasgow Math. J. 30 (1988), 155–170.CrossRefGoogle Scholar
28Rourke, C. P.. On dunce hats and the Kervaire conjecture. Papers presented to Christopher Zeeman, pp. 221–230 (University of Warwick, 1988).Google Scholar
29Sieradski, A. J.. A coloring test for asphericity. Quart. J. Math. Oxford (2) 34 (1983), 97–106.CrossRefGoogle Scholar
30Sieradski, A. J.. Algebraic topology for two dimensional complexes. In Two-dimensional homotopy and combinatorial group theory (eds. Hog-Angeloni, C., Metzler, W. and Sieradski, A.J.). London Mathematical Society Lecture Note Series 197 (Cambridge: Cambridge University Press, 1993).CrossRefGoogle Scholar
31Steenbock, M.. Rips-Segev torsion-free groups without the unique product property. J. Algebra 438 (2015), 337–378.CrossRefGoogle Scholar
32Whitehead, J. H. C.. On adding relations to homotopy groups. Ann. of Math. (2) 42 (1941), 409–428.CrossRefGoogle Scholar