Abstract
Roundness of metric spaces was introduced by Per Enflo as a tool to study uniform structures of linear topological spaces. The present paper investigates geometric and topological properties detected by the roundness of general metric spaces. In particular, we show that geodesic spaces of roundness 2 are contractible, and that a compact Riemannian manifold with roundness >1 must be simply connected. We then focus our investigation on Cayley graphs of finitely generated groups. One of our main results is that every Cayley graph of a free Abelian group on ⩾ 2 generators has roundness =1. We show that if a group has no Cayley graph of roundness =1, then it must be a torsion group with every element of order 2,3,5, or 7
Similar content being viewed by others
References
Bridson M.R., Haefliger A. (1999). Metric Spaces of Non-positive Curvature. Grundlehren Math Wiss 319, Springer-Verlag, Berlin
Carlsson G., Pedersen E.K. (1995). Controlled algebra and the Novikov conjectures for K- and L-theory. Topology 34:731–758
de la Harpe, P. and Valette, A.: La propriété (T) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque No. 175 (1989), 158 pp.
Delorme P. (1977). 1-cohomologie des représentations unitaires des groupes de Lie semisimples et résolubles. Bull. Soc. Math. France 105:281–336
Deza M.M., Laurent M. (1997). Geometry of Cuts and Metrics. Algorithms Combin 15, Springer-Verlag, Berlin
Enflo P. (1969). On the non-existence of uniform homeomorphisms between Lp-spaces. Ark. Mat 8:103–105
Enflo P. (1969). On a problem of Smirnov. Ark. Mat 8:107–109
Enflo P. (1970). Uniform structures and square roots in topological groups I. Israel J. Math 8:230–252
Enflo P. (1970). Uniform structures and square roots in topological groups, II. Israel J. Math 8:253–272
Faraut J., Harzallah K. (1974). Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier (Grenoble) 24:171–217
Farrell F.T., Lafont J.-F. (2005). EZ-structures and topological applications. Comment. Math. Helv. 80:103–121
Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, (English summary) Progr. in Math. 152, Birkhäuser, Boston, 1999
Gromov M. (2003). Random walk in random groups. Geom. Funct. Anal 13:73–146
Guntner, E., Higson, N. and Weinberger, A.: The Novikov Conjecture for Linear Groups, preprint.
Guentner E., Kaminker J. (2002). Exactness and the Novikov conjecture. Topology 41:411–418
Lennard C.J., Tonge A.M., Weston A. (1997). Generalized roundness and negative type. Michigan Math. J 44:37–45
Naor A., Schechtman G. (2002). Remarks on non linear type and Pisier’s inequality. J. Reine Angew. Math 552:213–236
Nowak P.W. (2005). Coarse embeddings of metric spaces into Banach spaces. Proc. Amer. Math. Soc. 133:2589–2596
Prassidis E., Weston A. (2004). Uniform Banach groups and structures. C. R. Math. Acad. Sci. Soc. R. Can 26:25–32
Schoenberg I.J. (1938). Metric spaces and positive definite functions. Trans. Amer. Math. Soc 44:522–536
Sela Z. (1992). Uniform embeddings of hyperbolic groups in Hilbert spaces. Israel J. Math 80:171–181
Whitehead J.H.C. (1932). Convex regions in the geometry of paths. Quart. J. Math. Oxford, Ser 3:33–42
Yu G. (2000). The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math 139:201–240
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by a Canisius College Summer Research Grant
Rights and permissions
About this article
Cite this article
Lafont, JF., Prassidis, S. Roundness Properties of Groups. Geom Dedicata 117, 137–160 (2006). https://doi.org/10.1007/s10711-005-9019-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-005-9019-y