Roundness properties of groups. (Q2494092)

Let \((X,d)\) be a metric space, \(p\in[1,\infty]\). The roundness of \((X,d)\) is \(p\) if \(p\) is the supremum of all \(q\) such that: for any four points \(x,y,z,t\) in \(X\), \(d(x,t)^q+d(y,z)^q\leq d(x,y)^q+d(x,z)^q+d(t,y)^q+d(t,z)^q\). The generalized roundness of \((X,d)\) is the supremum of all \(q\) such that: for every \(n\geq 2\) and any collection of \(2n\) points \(\{a_i\}^n_{i=1}\), \(\{b_i\}^n_{i=1}\), we have that \[ \sum_{1\leq i< j\leq n}(d(a_i,a_j)^q+ d(b_i,b_j)^q)\leq\sum_{1\leq i< j\leq n}d(a_i,b_j)^q. \] For example, the triangle inequality implies that any metric space has roundness at least 1. In this interesting paper, the authors prove the following results: 1. Every CAT(0)-space has roundness 2, 2. Proper geodesic spaces that have roundness 2 are contractible, 3. A nonsimply connected, compact, Riemannian manifold has roundness 1. Moreover, in the paper are made calculations for the Cayley graphs of finitely generated groups. The roundness spectrum of a group is defined as the collection of the roundnesses of all the Cayley graphs of the group (for different sets of generators). In this context the authors prove: 1. The roundness spectrum of a finitely generated free Abelian group on more than one generator is \(\{1\}\), 2. If the roundness spectrum of the group \(G\) does not contain 1, then \(G\) is a purely torsion group in which every element has order 2, 3, 5 or 7. In the last part of the paper are given some applications to Kazhdan groups and the Baum-Connes conjecture.