Concentration of maps and group actions (Q605065)

This paper studies actions of compact groups and Levy groups on metric spaces such as R-trees, doubling spaces, metric graphs and Hadamard manifolds from the viewpoint of the theory of concentration of maps. A metrizable group \(G\) is called a Levy group if it contains an increasing chain of compact subgroups \(G_1\subset G_2 \subset \cdots\) such that their union is dense in \(G\) and for some right-invariant compatible distance function \(d_G\) on \(G\), the groups \(G_n\) equipped with the normalized Haar measures and the restrictions of the distance function \(d_G\) form a Levy family. (A family of metric spaces \((X_n, d_n)\) with a normalized measure \(\mu_n\) is called a Levy family if, for every \(\varepsilon>0\), and any sequence of subset \(A_n\subset X_n\) with \(\mu_n(X_n-A_n)\) uniformly bounded from below, \(\mu_n(N_\varepsilon(A_n))\to 1\), where \(N_\varepsilon(A_n))\) is an \(\varepsilon\)-neighborhood of \(A_n\).) One result proved in this paper says: If a Levy group \(G\) acts boundedly on a metric space satisfying a certain property by uniform isomorphisms, then, for any compact subset \(K\subset G\) and any \(\varepsilon>0\), there exists a point \(x\in K\) such that the diameter of \(K x\) is less than \(\varepsilon\).