Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub$_G$ and Applications to Lattices in Lie Groups

TL;DR

This paper investigates the dynamics of automorphisms on the space of subgroups of discrete groups and applies these findings to the structure of lattices in Lie groups, revealing conditions for distality and expansivity.

Contribution

It characterizes the dynamics of automorphisms on subgroup spaces and establishes criteria linking the behavior on Lie groups and their lattices, including new structural insights.

Findings

The maximal solvable normal subgroup of a lattice is polycyclic.

Sub$^c_ extGamma$ is closed in Sub$_ extGamma$.

Automorphisms of certain groups do not act expansively or distally on subgroup spaces.

Abstract

For a discrete group $G$ and the compact space Sub$_G$ of (closed) subgroups of $G$ endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of $G$ on Sub$_G$ in terms of distality and expansivity. We also study the structure and properties of lattices $\Gamma$ in a connected Lie group. In particular, we show that the unique maximal solvable normal subgroup of $\Gamma$ is polycyclic and the corresponding quotient of $\Gamma$ is either finite or admits a cofinite subgroup which is a lattice in a connected semisimple Lie group with certain properties. We also show that Sub$^c_\Gamma$, the set of cyclic subgroups of $\Gamma$, is closed in Sub$_\Gamma$. We prove that an infinite discrete group $\Gamma$ which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on Sub$^c_\Gamma$, while only the finite order automorphisms of $\Gamma$ act distally on Sub$^c_\Gamma$. For an automorphism $T$ of a connected Lie group $G$ and a $T$-invariant lattice $\Gamma$ in $G$, we compare the behaviour of the actions of $T$ on Sub$_G$ and Sub$_\Gamma$ in terms of distality. We put certain conditions on the structure of the Lie group $G$ under which we show that $T$ acts distally on Sub$_G$ if and only if it acts distally on Sub$_\Gamma$. We construct counter examples to show that this does not hold in general if the conditions on the Lie group are relaxed.