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Cubulating one-relator groups with torsion

Published online by Cambridge University Press:  25 July 2013

JOSEPH LAUER  and
DANIEL T. WISE
JOSEPH LAUER
Affiliation:
Dept. of Mathematics & Statistics, McGill University, Montreal, QC, Canada. e-mail: lauer@math.mit.edu, wise@math.mcgill.ca
DANIEL T. WISE
Affiliation:
Dept. of Mathematics & Statistics, McGill University, Montreal, QC, Canada. e-mail: lauer@math.mit.edu, wise@math.mcgill.ca
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Abstract

Let 〈a1, . . ., am ∣ wn〉 be a presentation of a group G, where n ≥ 2. We define a system of codimension-1 subspaces in the universal cover, and invoke Sageev's construction to produce an action of G on a CAT(0) cube complex. We show that the action is proper and cocompact when n ≥ 4. A fundamental tool is a geometric generalization of Pride's C(2n) small-cancellation result. We prove similar results for staggered groups with torsion.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 411 - 429
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Brodskiĭ, S. D.Equations over groups, and groups with one defining relation. Sibirsk. Mat. Zh. 25 (2) (1984), 84–103.Google Scholar
[2]Fischer, J., Karrass, A. and Solitar, D.On one-relator groups having elements of finite order. Proc. Amer. Math. Soc. 33 (1972), 297–301.CrossRefGoogle Scholar
[3]Gromov, M.Hyperbolic groups. In Essays in Group Theory, vol. 8 Math. Sci. Res. Inst. Publ. (Springer, New York, 1987), pages 75–263.CrossRefGoogle Scholar
[4]Howie, J.On pairs of 2-complexes and systems of equations over groups. J. Reine Angew. Math. 324 (1981), 165–174.Google Scholar
[5]Howie, J.On locally indicable groups. Math. Z. 180 (4) (1982), 445–461.CrossRefGoogle Scholar
[6]Howie, J.How to generalize one-relator group theory. In Gersten, S. M. and Stallings, J. R., editors, Combinatorial group theory and topology (Princeton, N.J., 1987), pages 53–78. Princeton University Press.CrossRefGoogle Scholar
[7]Hruska, G. C. and Wise, D. T. Finiteness properties of cubulated groups. Compositio Math. pp. 1–58, to appear.Google Scholar
[8]Hruska, G. C. and Wise, D. T.Towers, ladders and the B. B. Newman spelling theorem. J. Aust. Math. Soc. 71 (1) (2001), 53–69.CrossRefGoogle Scholar
[9]Lyndon, R. C. and Schupp, P. E.Combinatorial group theory. Springer-Verlag, Berlin, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89.Google Scholar
[10]Magnus, W.Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz). J. Reine Angew. Math. 163 (1930), 141–165.CrossRefGoogle Scholar
[11]Newman, B. B.Some results on one-relator groups. Bull. Amer. Math. Soc. 74 (1968), 568–571.CrossRefGoogle Scholar
[12]Pride, S. J.Small cancellation conditions satisfied by one-relator groups. Math. Z. 184 (2) (1983), 283–286.CrossRefGoogle Scholar
[13]Sageev, M.Ends of group pairs and non-positively curved cube complexes. Proc. London Math. Soc. (3) 71 (3) (1995), 585–617.CrossRefGoogle Scholar
[14]Sageev, M.Codimension-1 subgroups and splittings of groups. J. Algebra 189 (2) (1997), 377–389.CrossRefGoogle Scholar
[15]Short, H.Quasiconvexity and a theorem of Howson's. In Ghys, É., Haefliger, A. and Verjovsky, A., editors, Group theory from a geometrical viewpoint (Trieste, 1990) (World Scientific Publishing, River Edge, NJ, 1991), pages 168–176.Google Scholar
[16]Weinbaum, C. M.On relators and diagrams for groups with one defining relation. Illinois J. Math. 16 (1972), 308–322.CrossRefGoogle Scholar
[17]Wise, D. T.Cubulating small cancellation groups. GAFA, Geom. Funct. Anal. 14 (1) (2004), 150–214.CrossRefGoogle Scholar