Metrizability of proper \(G\)-spaces and their orbit spaces (Q820621)

In this paper, a \textit{\(G\)-space} \(X\) is a completely regular Hausdorff space together with a fixed continuous action \(G\times X\to X\) defined by \((g,x)\mapsto gx\) of a locally compact group \(G\) on \(X\). Recall that a \(G\)-space \(X\) is called \textit{proper} in the sense of [\textit{R. S. Palais}, Ann. Math. (2) 73, 295--323 (1961; Zbl 0103.01802)], if each point of \(X\) has a neighborhood \(V\) such that for every point of \(X\) there is a neighborhood \(U\) with the property that the set \(\langle U,V\rangle=\{g\in G: gU\cap V\neq\emptyset\}\) has compact closure in \(G\). A compatible metric \(\rho\) on a metrizable \(G\)-space \(X\) is called \textit{\(G\)-invariant}, if \(\rho(gx,gy)=\rho(x,y)\) for all \(g\in G\) and \(x,y\in X\). \par In this paper, the authors address to the following Conjecture which goes back to [\textit{R. S. Palais}, Ann. Math. (2) 73, 295--323 (1961; Zbl 0103.01802)] and Problem posed in [\textit{S. A. Antonyan} and \textit{H. Juárez-Anguiano}, Topology Appl. 158, No. 15, 1910--1919 (2011; Zbl 1229.54041), Question 2]. \begin{itemize} \item[Conjecture:] Let \(G\) be a locally compact group and \(X\) a metrizable proper \(G\)-space. Then there is a compatible \(G\)-invariant metric. \item[Problem:] Let \(G\) be a locally compact group and \(X\) a proper \(G\)-space such that the \(G\)-orbit space \(X/G\) is metrizable. Does there exist a compact subgroup \(H\) of \(G\) such that the \(H\)-orbit space \(X/H\) is metrizable, too? \end{itemize} Several positive partial answers to Conjecture stated above were obtained, for instance, the following results were proved. If \(G\) is a Lie group then each separable metrizable proper \(G\)-space admits a compatible \(G\)-invariant metric [\textit{R. S. Palais}, Ann. Math. (2) 73, 295--323 (1961; Zbl 0103.01802)], and the same holds true for \(G\) an arbitrary metrizable group [\textit{J. de Vries}, Proc. Steklov Inst. Math. 154, 57--74 (1984; Zbl 0561.22004); translation from Tr. Mat. Inst. Steklova 154, 53--70 (1983)]. There is a compatible \(G\)-invariant metric on each locally compact metrizable proper \(G\)-space for a connected group \(G\) [\textit{J. L. Koszul}, Lectures on groups of transformations. Notes by R. R. Simha and R. Sridharan. Springer, Berlin; Tata Inst. of Fundamental Research, Bombay (1965; Zbl 0195.04605)]. If a group \(G\) is either separable or almost connected, then there is a compatible \(G\)-invariant metric on every locally separable metrizable proper \(G\)-space \(X\) [\textit{S. Antonyan} and \textit{S. de Neymet}, Acta Math. Hung. 98, No. 1--2, 59--69 (2003; Zbl 1026.22021)]. In Section 3 of this paper, they prove that if a group \(G\) is almost connected, then there is a compatible \(G\)-invariant metric on each strongly metrizable proper \(G\)-space, which generalizes almost all previous partial answers stated above. In Section 4 of this paper, they give a complete positive answer to the above Problem. In fact, they prove the following key result: Let \(X\) be a proper \(G\)-space such that the \(G\)-orbit space \(X/G\) is metrizable. Then, for every open subgroup \(U\) of \(G\), the \(U\)-orbit space \(X/U\) is metrizable too. Furthermore, they show that if \(X\) is a proper \(G\)-space such that the \(G\)-orbit space \(X/G\) is a paracompact \(p\)-space, then \(X\) is a paracompact \(p\)-space, too.