Dynamics of actions of automorphisms of discrete groups \(G\) on \(\mathrm{Sub}_G\) and applications to lattices in Lie groups (Q2694831)

Summary: For a locally compact Hausdorff group \(G\) and the compact space \(\mathrm{Sub}_G\) of closed subgroups of \(G\) endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of \(G\) on \(\mathrm{Sub}_G\) in terms of distality and expansivity. We prove that an infinite discrete group \(G\), which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on \(\mathrm{Sub}_G^c\), the space of cyclic subgroups of \(G\), while only the finite order automorphisms of \(G\) act distally on \(\mathrm{Sub}_G^c\). For an automorphism \(T\) of a connected Lie group \(G\) which keeps a lattice \(\Gamma\) invariant, we compare the behaviour of the actions of \(T\) on \(\mathrm{Sub}_G\) and \(\mathrm{Sub}_{\Gamma}\) in terms of distality. Under certain necessary conditions on the Lie group \(G\), we show that \(T\) acts distally on \(\mathrm{Sub}_G\) if and only if it acts distally on \(\mathrm{Sub}_{\Gamma}\). We also obtain certain results about the structure of lattices in a connected Lie group.