Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Growing trees from compact subgroups

Pierre-Emmanuel Caprace, Timothée Marquis, Colin D. Reid

Abstract

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups $G$, finding a sufficient condition for $G$ to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group $G/N$ on a tree with faithful micro-supported action on the boundary, where $N$ is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of $G/N$ with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group $G$, for all direct factors of open subgroups of $G$ to be trivial or open.

Growing trees from compact subgroups

Abstract

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups , finding a sufficient condition for to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group on a tree with faithful micro-supported action on the boundary, where is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group , for all direct factors of open subgroups of to be trivial or open.

Paper Structure

This paper contains 9 sections, 23 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local structure theory and dynamics on Stone spaces
  4. Cayley--Abels graphs, ends and trees
  5. Local finiteness properties
  6. Arc-geometric actions of t.d.l.c. groups on trees
  7. TMS subgroups
  8. A sufficient condition for a micro-supported action on a tree
  9. Sources of TMS subgroups

Key Result

Proposition 1.2

Let $G$ be a compactly generated t.d.l.c. group acting faithfully and arc-geometrically on a leafless tree $T$. Suppose that for some half-tree $T'$ of $T$, the pointwise fixator $K$ of $T'$ in $G$ fixes only finitely many arcs of $T \setminus T'$. Then $K$ is a TMS subgroup of $G$.

Theorems & Definitions (42)

  • Definition 1.1
  • Proposition 1.2: See Proposition \ref{['prop:smooth_tree']}
  • Theorem 1.3: See Theorem \ref{['cpctend']}
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1: CRWpart1
  • Theorem 2.2: See CRWpart1
  • Lemma 2.3: See e.g. CRWpart2
  • Definition 2.4
  • Lemma 2.5
  • ...and 32 more