  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.

Metric Geometry;Group Theory