Abstract
We show that the Stallings–Bieri groups, along with certain other Bestvina–Brady groups, have quadratic Dehn function.
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Abrams, A., Brady, N., Dani, P., Duchin, M., Young, R.: Pushing fillings in right-angled Artin groups. J. Lond. Math. Soc. 87(2), 663–688 (2013)
Amchislavska, M., Riley, T.: Lamplighters, metabelian groups, and horocyclic products of trees. To appear in L’Enseignement Mathématique. http://arxiv.org/abs/1405.1660
Baumslag, G., Bridson, M.R., Miller III, C.F., Short, H.: Finitely presented subgroups of automatic groups and their isoperimetric functions. J. Lond. Math. Soc. 56(2), 292–304 (1997)
Bestvina, M., Brady, N.: Morse theory and finiteness properties of groups. Invent. Math. 129, 445–470 (1997)
Bieri, R.: Homological dimension of discrete groups. Mathematics Department, Queen Mary College, London. Queen Mary College Mathematics Notes (1976)
Brady, N., Bridson, M.R., Forester, M., Shankar, K.: Snowflake groups, Perron-Frobenius eigenvalues and isoperimetric spectra. Geom. Topol. 13, 141–187 (2009)
Bridson, M.R.: personal communication
Bridson, M.R.: Doubles, finiteness properties of groups, and quadratic isoperimetric inequalities. J. Algebra 214, 652–667 (1999)
Bridson, M.R.: The geometry of the word problem. In: Invitations to Geometry and Topology, vol. 7 of Oxf. Grad. Texts Math, pp. 29–91. Oxford University Press, Oxford (2002)
Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)
Davis, M.W.: The geometry and topology of Coxeter groups. London Mathematical Society Monographs Series, vol. 32. Princeton University Press, Princeton (2008)
Dison, W.: An isoperimetric function for Bestvina-Brady groups. Bull. Lond. Math. Soc. 40, 384–394 (2008)
Dison, W., Elder, M., Riley, T.R., Young, R.: The Dehn function of Stallings’ group. Geom. Funct. Anal. 19, 406–422 (2009)
Druţu, C.: Filling in solvable groups and in lattices in semisimple groups. Topology 43, 983–1033 (2004)
Gersten, S.M.: Finiteness properties of asynchronously automatic groups, in Geometric group theory (Columbus, OH, 1992), vol. 3, pp. 121–133. Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin (1995)
Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric group theory, Vol. 2 (Sussex, 1991), vol. 182 of London Math. Soc. Lecture Note Ser., pp. 1–295 Cambridge University Press, Cambridge (1993)
Groves, D.: personal communication
Stallings, J.: A finitely presented group whose 3-dimensional integral homology is not finitely generated. Am. J. Math. 85, 541–543 (1963)
Young, R.: The Dehn function of \({\rm SL}(n;{\mathbb{Z}})\). Ann. Math. 177(2), 969–1027 (2013)
Acknowledgments
The authors are grateful to Noel Brady for many valuable discussions related to this work, and to the referee, for helpful comments that have improved the paper. The second author was partially supported by NSF Grant DMS-1105765.
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Carter, W., Forester, M. The Dehn functions of Stallings–Bieri groups. Math. Ann. 368, 671–683 (2017). https://doi.org/10.1007/s00208-016-1470-6
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DOI: https://doi.org/10.1007/s00208-016-1470-6