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Immersions of complexes of groups

Jagerynn Ting Verano

TL;DR

The paper develops a functorial framework for complexes of groups by isolating local data into local complexes of groups and proving that these local structures are always developable and recoverable from their developments. It introduces a broader notion of immersions of complexes of groups and proves that locally isometric immersions into non-positively curved complexes of groups yield NPC complexes, with the induced map on fundamental groups $\pi_1$-injective and elevations between developments being isometric embeddings. This local-to-global perspective provides a Cartan–Hadamard-type control in the CoG setting and connects to group actions on CAT(0) spaces, including applications to hyperbolic and relatively geometric actions. The results establish a robust functorial developability criterion, enable a cohesive treatment of local and global structure, and extend the utility of complexes of groups in understanding symmetry and embedding phenomena in geometric group theory.

Abstract

Given a complex of groups, we construct a new class of complex of groups that records its local data and offer a functorial perspective on the statement that complexes of groups are locally developable. We also construct a new notion of an immersion of complexes of groups and establish that a locally isometric immersion of a complex of groups into a non-positively curved complex of groups is $π_1$-injective. Furthermore, the domain complex of groups is developable and the induced map on geometric realizations of developments is an isometric embedding.

Immersions of complexes of groups

TL;DR

The paper develops a functorial framework for complexes of groups by isolating local data into local complexes of groups and proving that these local structures are always developable and recoverable from their developments. It introduces a broader notion of immersions of complexes of groups and proves that locally isometric immersions into non-positively curved complexes of groups yield NPC complexes, with the induced map on fundamental groups -injective and elevations between developments being isometric embeddings. This local-to-global perspective provides a Cartan–Hadamard-type control in the CoG setting and connects to group actions on CAT(0) spaces, including applications to hyperbolic and relatively geometric actions. The results establish a robust functorial developability criterion, enable a cohesive treatment of local and global structure, and extend the utility of complexes of groups in understanding symmetry and embedding phenomena in geometric group theory.

Abstract

Given a complex of groups, we construct a new class of complex of groups that records its local data and offer a functorial perspective on the statement that complexes of groups are locally developable. We also construct a new notion of an immersion of complexes of groups and establish that a locally isometric immersion of a complex of groups into a non-positively curved complex of groups is -injective. Furthermore, the domain complex of groups is developable and the induced map on geometric realizations of developments is an isometric embedding.

Paper Structure

This paper contains 12 sections, 20 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Complexes of groups
  3. Small categories without loops
  4. The fundamental group of a complex of groups
  5. Developability
  6. The local development
  7. Metrizing a complex of groups
  8. Local complexes of groups
  9. The local development revisited
  10. A functorial version of the developability criterion
  11. Immersions of complexes of groups
  12. Acknowledgements

Key Result

Theorem 1.1

Let $G(\mathcal{Y})=(G_\sigma, \psi_a, g_{a,b})$ be a complex of groups over a scwol $\mathcal{Y}$, and let $\gamma\in V(\mathcal{Y})$. The local complex of groups $L(\mathcal{Y}(\gamma))$ over $\gamma$ has the following properties:

Figures (2)

  • Figure 1: The scwol $\mathcal{Y}(\gamma)$ (right) illustrated by all the types of morphisms it contains.
  • Figure 2: The local complex of groups $L(\mathcal{Y}(\gamma))$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 46 more