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Homomorphic Images of Locally Compact Groups Acting on Trees and Buildings

Max Carter, George A. Willis

TL;DR

This work generalises Cartan-like decompositions from Lie theory to totally disconnected locally compact groups acting on trees and buildings by introducing the contraction group property. The authors prove that topologically simple groups with this property have the closed range property, and extend these ideas to broader classes of tree automorphism groups and to automorphism groups of semi-regular right-angled buildings, including universal groups. Consequently, a wide array of groups acting on trees or buildings have the feature that every continuous homomorphism has closed range, yielding rigidity-like consequences for homomorphic images. The results deepen the structure theory of t.d.l.c. groups and provide tools for understanding representations and morphisms in this setting, building on and extending CW20.

Abstract

We study analogues of Cartan decompositions of Lie groups for totally disconnected locally compact groups. It is shown using these decompositions that a large class of totally disconnected locally compact groups acting on trees and buildings have the property that every continuous homomorphic image of the group is closed.

Homomorphic Images of Locally Compact Groups Acting on Trees and Buildings

TL;DR

This work generalises Cartan-like decompositions from Lie theory to totally disconnected locally compact groups acting on trees and buildings by introducing the contraction group property. The authors prove that topologically simple groups with this property have the closed range property, and extend these ideas to broader classes of tree automorphism groups and to automorphism groups of semi-regular right-angled buildings, including universal groups. Consequently, a wide array of groups acting on trees or buildings have the feature that every continuous homomorphism has closed range, yielding rigidity-like consequences for homomorphic images. The results deepen the structure theory of t.d.l.c. groups and provide tools for understanding representations and morphisms in this setting, building on and extending CW20.

Abstract

We study analogues of Cartan decompositions of Lie groups for totally disconnected locally compact groups. It is shown using these decompositions that a large class of totally disconnected locally compact groups acting on trees and buildings have the property that every continuous homomorphic image of the group is closed.
Paper Structure (8 sections, 21 theorems, 9 equations)

This paper contains 8 sections, 21 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Contraction Group and Closed Range Properties
  4. Cartan-like Decompositions
  5. The Contraction Group Property
  6. The Closed Range Property
  7. Homomorphic Images of Tree Automorphism Groups
  8. Homomorphic Images of Automorphism Groups of Buildings

Key Result

Proposition 3.6

Let $G$ be a topological group. Suppose that there exists a compact open subgroup $K \leqslant G$ and a Cartan-like decomposition $G=KAK$ with the contraction group property. Then, for any compact open subgroup $L \leqslant G$, there exists a Cartan-like decomposition $G=LUL$ with the contraction gr

Theorems & Definitions (48)

  • Definition 3.1: Cartan-like Decomposition
  • Remark 3.2
  • Definition 3.3: Contraction Group
  • Definition 3.4: Contraction Group Property
  • Definition 3.5: Closed Range Property
  • Proposition 3.6
  • proof
  • Example 3.7
  • Proposition 3.8
  • proof
  • ...and 38 more