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

Abstract

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.