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The fundamentals of cubical isometry groups

Merlin Incerti-Medici

TL;DR

This work develops a comprehensive framework to study cubical isometry groups of CAT$(0)$ cube complexes with special, compact quotients by encoding cubical isometries as rooted tree isomorphisms through $\Lambda$-invariant cubical edge-labelings and portraits. It establishes a semi-local characterization for portraits that induce cubical isometries and shows how to extend partial data from a reduced subtree to full isometries, enabling explicit constructions and topological generators for stabilizers and $\mathrm{Aut}(X)$. Existence of such edge-labelings is linked to freeness of the action and the quotient being special, connecting to lattice envelopes and universal groups in the non-discrete tdlc setting. The framework unifies automorphism theory across trees, buildings, and Salvetti complexes, and yields concrete finite-generation results for automorphism groups in key cases, including RAAG-related geometries.

Abstract

We develop the fundamental theory to study cubical isometry groups as totally disconnected, locally compact groups. We show how cubical isometries are determined by their local actions and how this can be applied in explicit constructions. These results are closely related to some of the authors recent work on cubical isometries. We reformulate and generalize these previous results in a way that is necessary and more suited for upcoming applications.

The fundamentals of cubical isometry groups

TL;DR

This work develops a comprehensive framework to study cubical isometry groups of CAT cube complexes with special, compact quotients by encoding cubical isometries as rooted tree isomorphisms through -invariant cubical edge-labelings and portraits. It establishes a semi-local characterization for portraits that induce cubical isometries and shows how to extend partial data from a reduced subtree to full isometries, enabling explicit constructions and topological generators for stabilizers and . Existence of such edge-labelings is linked to freeness of the action and the quotient being special, connecting to lattice envelopes and universal groups in the non-discrete tdlc setting. The framework unifies automorphism theory across trees, buildings, and Salvetti complexes, and yields concrete finite-generation results for automorphism groups in key cases, including RAAG-related geometries.

Abstract

We develop the fundamental theory to study cubical isometry groups as totally disconnected, locally compact groups. We show how cubical isometries are determined by their local actions and how this can be applied in explicit constructions. These results are closely related to some of the authors recent work on cubical isometries. We reformulate and generalize these previous results in a way that is necessary and more suited for upcoming applications.

Paper Structure

This paper contains 14 sections, 21 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. History and motivation
  3. Rooted tree isomorphisms of self-similar trees
  4. Cubical isometries
  5. Related works
  6. Trees
  7. Self-similar trees
  8. Isomorphisms between self-similar trees
  9. Describing cubical isometries as tree isomorphisms
  10. Edge paths and cubical edge-labelings
  11. Cubical portraits
  12. Characterising $\mathop{\mathrm{CAT(0)}}\nolimits$ cube complexes with cubical edge-labelings
  13. Topological generators for stabilizer subgroups
  14. Proof of Theorem \ref{['thm:characterization.of.cubical.portraits']} and Theorem \ref{['thm:canonical.portrait.extension']}

Key Result

Theorem 2.3

Let $(\prescript{}{v}{ \sigma})_{v \in \mathcal{L}_A}$ be a portrait. The map $\sigma$ induces an injective morphism of rooted trees $T_A \rightarrow T'_A$ if and only if for every $v \in \Sigma_v$, we have that $\prescript{}{v}{ \sigma}( \Sigma_v ) \subseteq \Sigma'_{\sigma(v)}$. Furthermore, $\sig

Figures (1)

  • Figure 1: The pair $s_i, s_j$ depicted here is an innermost cancellation of the word/edge-path $s_1 \dots s_n$. One can inductively show that the hyperplanes $\hat{h}_k$ for $i < k < j$ all intersect with the hyperplane $\hat{h}_i$. This yields a chain of squares as depicted in the figure. From this chain of squares, we can conclude that $s_j = s_i^{-1}$ and that $[s_i, s_k] = 1$ for all $i < k < j$.

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Remark 3.1
  • Definition 3.2
  • ...and 48 more