[Paper Review] Dynamics of flat actions on totally disconnected, locally compact groups

Colin D. Reid|arXiv (Cornell University)|Mar 6, 2015

Geometric and Algebraic Topology34 references3 citations

TL;DR

This paper investigates the dynamics of flat groups of automorphisms on totally disconnected, locally compact (t.d.l.c.) groups, introducing and analyzing four key subgroups: the intersection of all tidy subgroups, the intersection of all $H$-invariant open subgroups, the smallest $H$-invariant quotient with no accumulating orbits, and the group generated by closures of contraction groups. The central result establishes that if all contraction groups are closed, then the group is anisotropic—meaning no nontrivial elements have contracting dynamics—thereby characterizing the absence of non-closed contraction groups in terms of the nub and relative Tits core structures.

ABSTRACT

Let $G$ be a totally disconnected, locally compact group and let $H$ be a virtually flat (for example, polycyclic) group of automorphisms of $G$. We study the structure of, and relationships between, various subgroups of $G$ defined by the dynamics of $H$. In particular, we consider the following four subgroups: the intersection of all tidy subgroups for $H$ on $G$ (in the case that $H$ is flat); the intersection of all $H$-invariant open subgroups of $G$; the smallest closed $H$-invariant subgroup $D$ such that no $H$-orbit on $G/D$ accumulates at the trivial coset; and the group generated by the closures of contraction groups of elements of $H$ on $G$.

Motivation & Objective

• To understand the structural interplay between flat groups of automorphisms and the dynamics of their actions on totally disconnected, locally compact (t.d.l.c.) groups.
• To identify and analyze four key subgroups defined by the dynamics of a flat group $H$ acting on a t.d.l.c. group $G$.
• To characterize when contraction groups are closed, particularly in relation to the nub and relative Tits core of the action.
• To generalize results from cyclic actions to flat group actions, especially regarding the role of the relative Tits core and the nub in controlling dynamical behavior.
• To establish conditions under which a t.d.l.c. group is anisotropic, i.e., has no nontrivial contracting elements.

Proposed method

• Uses tidy theory and scale theory to analyze the action of flat groups $H$ on a t.d.l.c. group $G$, focusing on the structure of contraction groups and their closures.
• Introduces the relative Tits core $G^{lat}_A$ as the closed subgroup generated by the closures of contraction groups of elements in $A \cup A^{-1}$, where $A$ is a set of automorphisms.
• Applies the concept of the nub $\mathrm{nub}_G(\alpha)$, the intersection of all compact open subgroups stabilized by $\alpha$, to study dynamical rigidity.
• Constructs reduced envelopes $S = \overline{G^{lat}_\alpha} \rtimes \langle \alpha \rangle$ to analyze the dynamics of individual automorphisms $\alpha$ and their interaction with the relative Tits core.
• Employs the global centralizer lattice and weakly decomposable group actions to study the structure of $S$, particularly the role of the quasi-cyclic center $\mathrm{QZ}(S)$.
• Uses the Mautner phenomenon and eigenfactor analysis to study subgroups of finite covolume and their invariance under flat group actions.

Experimental results

Research questions

• RQ1 Under what conditions is the contraction group of an automorphism $\alpha$ on a t.d.l.c. group $G$ closed?
• RQ2 How do the relative Tits core and the nub interact in the context of flat group actions on $G$?
• RQ3 What structural properties does the group $\overline{G^{lat}_\alpha} \rtimes \langle \alpha \rangle$ possess when $\mathrm{rnub}_G(\alpha) = \{1\}$?
• RQ4 When does a t.d.l.c. group $G$ admit a nontrivial compact normal subgroup that is invariant under a flat group of automorphisms?
• RQ5 Can the absence of non-closed contraction groups be characterized purely in terms of the nub and relative Tits core?

Key findings

• If all contraction groups in $G$ are closed, then $G$ is anisotropic, meaning no nontrivial element has a nontrivial contraction group.
• For a flat group $H$ acting on $G$, the relative Tits core $G^{lat}_H$ is the closed subgroup generated by the closures of the contraction groups of elements in $H \cup H^{-1}$.
• When $\mathrm{rnub}_G(\alpha) = \{1\}$, the reduced envelope $S = \overline{G^{lat}_\alpha} \rtimes \langle \alpha \rangle$ is compactly generated and has no nontrivial compact normal subgroups.
• If $\mathrm{rnub}_G(\alpha) = \{1\}$, then there exists $g \in S$ such that $\mathrm{nub}_S(g)$ is nontrivial, implying the existence of a nontrivial nub in the relative Tits core.
• The quasi-cyclic center $\mathrm{QZ}(S)$ of the reduced envelope $S$ is discrete and torsion-free, and intersects trivially with the relative Tits core $T = \overline{G^{lat}_\alpha}$.
• If $G$ is not anisotropic, then there exists $g \in G$ such that $\mathrm{con}(g)$ is not closed, which implies $\mathrm{nub}_G(g) \neq \{1\}$, leading to a contradiction if all contraction groups are assumed closed.