Properness, Cauchy indivisibility and the Weil completion of a group of isometries (Q692400)

A continuous action of a topological group \(G\) on a metric space \((X,d)\) is said to be proper if there do not exists nets \(\{ g_i\} \subset G\) and \(\{ x_i\}\subset X\) with both \(\{ x_i\}\) and \(\{ g_ix_i\}\) convergent in \(X\), while \(\{ g_i\}\) has no convergent subnets in \(G\). The paper introduces a related notion, that of a ``Cauchy indivisible'' action, defined by the condition that whenever a net \(\{ g_i\} \subset G\) has no convergent subnets, if \(\{ g_iy\}\) is Cauchy for some \(y\in X\), then \(\{ g_ix\}\) is Cauchy for each \(x\in X\). It is shown that in general these two notions are independent, but that when \(X\) is either complete or locally compact, they are equivalent. Other results deal with the situation where \(X\) is separable, \(G\) is the group of isometries on \(X\), with the obvious action, and this action is Cauchy indivisible. In this setting a subset \(Y\) of the completion of \(X\) is defined such that \(X\subset Y\) and the action of the Ellis semigroup \(E\) on \(Y\) is proper. The equivalence of three conditions is established: (i) the Ellis semigroup \(E\) being a group, (ii) the existence of a Weil completion for \(G\), (iii) the equality of \(Y\) from the previous sentence with an a priori larger subset of the completion of \(X\). Connections are established between the existence of Borel sections of Cauchy indivisible actions (Borel sets in \(X\) that meet each orbit of \(G\) in a single point), and what are called fundamental sets for proper actions.