On minimal, strongly proximal actions of locally compact groups
Let \(G\) be a locally compact group acting continuously on a compact \(X\). The compact space \(P(X)\) of all probability measures on \(X\) with the weak\(*\)-topology has an induced continuous action of \(G\). \textit{H. Furstenberg} introduced the following notions [Ann. Math. (2) 77, 335--386 (1963; Zbl 0192.12704)]: the action is strongly proximal if the closure of every \(G\)-orbit in \(P(X)\) contains a Dirac measure; if the action is minimal and strongly proximal, then \(X\) is called a \(G\)-boundary (or the action is called a boundary action). This paper considers to what extent the topology of a \(G\)-boundary determines \(G\). If \(G\) has `no small subgroups' and \(X\) is a compact manifold which is a \(G\)-boundary, and \(G\) acts faithfully on \(X\), then it is shown that the action splits into the direct product of a boundary action of a discrete group and a boundary action of a semisimple Lie group. The Hilbert-Smith conjecture says that a locally compact group acting faithfully on a topological manifold \(X\) has no small subgroups, and the author discusses the implications of the main result where the conjecture is known to be true (when \(X\) is a circle, for example).
scientific article; zbMATH DE number 2053318
scientific article