A note on relative amenability (Q2357404)

A convex compact \(G\)-space is a convex compact subset of a locally convex topological vector space such that the subset has a continuous affine \(G\)-action. Let \(G\) be a locally compact group. A closed subgroup \(H\leq G\) is relatively amenable if \(H\) fixes a point in every non-empty convex compact \(G\)-space. The class \(X\) is the collection of locally compact groups for which every relatively amenable closed subgroup is amenable. It is well-known that \(X\) contains discrete groups, and all groups amenable at infinity. Motivated by the work of \textit{P.-E. Caprace} and \textit{N. Monod} [Math. Proc. Camb. Philos. Soc. 150, No. 1, 97--128 (2011; Zbl 1218.22002); Groups Geom. Dyn. 8, No. 3, 747--774 (2014; Zbl 1306.43003)], the author studies a particular collection of \(X\) in this paper. More precisely, let \(Y\) be the smallest collection of locally compact groups with the following properties: {\parindent=0.7cm\begin{itemize}\item[(i)] \(Y\) contains all compact groups, discrete groups, and connected groups; \item[(ii)] \(Y\) is closed under directed unions of open subgroups; \item[(iii)] \(Y\) is closed under group extensions. \end{itemize}} As a main result of this work we mention the following: Theorem. The class \(Y\) is contained in \(X\) and has the following properties: {\parindent=0.7cm\begin{itemize} \item[(i)] \(Y\) is closed under taking closed subgroups; \item[(ii)] \(Y\) is closed under taking quotients by closed normal subgroups. \end{itemize}}