On a Roelcke-precompact Polish group that cannot act transitively on a complete metric space (Q1650009)

\textit{J. Melleray} [J. Symb. Log. 75, No. 4, 1359--1365 (2010; Zbl 1218.03031)] asked whether there is a Polish group that admits no transitive and continuous action on a non-trivial complete metric space and whether such a group can be Roelcke-precompact. Let \(\mu\) denote the Lebesgue measure on \([0,1]\). Let Aut\(^*(\mu)\) be the group of bijections \(\varphi\) between full measure subsets of \([0,1]\) such that both \(\varphi\) and \(\varphi^{-1}\) are measurable where such functions are identified up to the equality almost everywhere with respect to the null ideal. Let Homeo\(^+[0,1]\) denote the group of increasing self-homeomorphisms of the interval \([0,1]\) equipped with the topology of uniform convergence. The author shows that the Roelcke-precompact groups Aut\(^*(\mu)\) and Homeo\(^+[0,1]\) have the property described in Melleray's question. The author also gives a general characterization of continuous isometric transitive actions of a Roelcke-precompact Polish group \(G\) on a complete metric space. In particular, if \(\varphi:G\to B\) is a Bohr compactification of \(G\), then \(\varphi\) is surjective.