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}^c_G , the space of cyclic subgroups of G , while only the finite order automorphisms of G act distally on \mathrm{Sub}^c_G . 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.