Geometriae Dedicata

, Volume 117, Issue 1, pp 137–160 | Cite as

Roundness Properties of Groups

Article

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

Keywords

roundness metric invariant Cayley graph Baum-Connes conjecture 

Mathematics Subject Classification

Primary: 20F65 Secondary: 57M07, 46B20 

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematical ScienceBinghamton UniversityBinghamtonU.S.A
  2. 2.Department of Mathematics & StatisticsCanisius CollegeBuffaloU.S.A

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