Roundness properties of groups

Details Roundness properties of groups Roundness properties of groups

2004-12-10

arxiv.org/abs/math/0412219v1

Geom. Dedicata 117 (2006), pgs. 137-160

Mathematics

Metric Geometry

22 pages, 2 figures

Scientific paper

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 $\geq 2$ 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.

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