Orbits of distal actions on locally compact groups (Q2892870)

The notion of distality was introduced by Hilbert and it can be given as follows in the case of a semigroup \(\Gamma\) acting continuously by endomorphisms on a locally compact group \(G\). The action of \(\Gamma\) on \(G\) is distal, if the neutral element of \(G\) does not belong to the closure \(\overline{\Gamma x}\) of the orbit \(\Gamma x\), for every non-identical element \(x\) of \(G\). Moreover, the action of \(\Gamma\) on \(G\) is said to have MOC (i.e., minimal orbit closures), if \(y\in \overline{\Gamma x}\) implies \(\overline{\Gamma x}=\overline{\Gamma y}\) for any pair of elements \(x\) and \(y\) of \(G\). A distal action necessarily has MOC. It is known that the converse implication holds in several cases, while it is an open problem whether having MOC implies distality in case \(\Gamma\) is a group acting continuously by automorphisms on a locally compact group \(G\).NEWLINENEWLINEThe authors prove this equivalence between distality and having MOC, when \(\Gamma\) is a compactly generated locally compact abelian group acting continuously by automorphisms on a locally compact group \(G\). Moreover, assuming that \(G\) is also totally disconnected, they prove the same equivalence, relaxing the hypothesis on \(\Gamma\) to being a generalized \(FC^-\)-group.NEWLINENEWLINEFurthermore, the MOC property is considered with respect to passing to invariant subgroups and quotients. Indeed, it is proved that, when \(\Gamma\) is a compactly generated locally compact abelian group acting continuously by automorphisms on a locally compact group \(G\), and \(H\) is a normal closed invariant subgroup of \(G\), then the action of \(\Gamma\) on \(G\) has MOC precisely when the actions of \(\Gamma\) on both \(H\) and \(G/H\) have MOC. The same conclusion is achived in this paper assuming that \(G\) is a locally compact group, \(\Gamma\) is a subsemigroup of \(\mathrm{Aut}(G)\) and \(K\) is a compact metrizable invariant subgroup of \(G\).