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JSJ Decompositions of pro-p Groups

Pavel Zalesskii

TL;DR

This work develops a comprehensive JSJ framework for pro-p groups by extending Bass-Serre theory to pro-p trees, defining universally elliptic splittings, and constructing canonical deformation spaces via trees of cylinders. It proves existence of JSJ decompositions over broad edge families for finitely generated pro-p groups and analyzes their vertices as rigid or flexible, with canonical, automorphism-invariant realizations. The paper also provides extensive examples (free, virtually free, RAAGs, PD pro-p groups) and builds a robust toolkit—incidence structures, relative finite presentability, and relative refinements—to study vertex-group decompositions and compatibility. It concludes with a deep treatment of automorphisms, tree-of-cylinders constructions, and the compatibility JSJ deformations, delivering a unified, applicable framework for pro-p splittings and their algebraic consequences.

Abstract

We develop JSJ decomposition theory of pro-p groups.

JSJ Decompositions of pro-p Groups

TL;DR

This work develops a comprehensive JSJ framework for pro-p groups by extending Bass-Serre theory to pro-p trees, defining universally elliptic splittings, and constructing canonical deformation spaces via trees of cylinders. It proves existence of JSJ decompositions over broad edge families for finitely generated pro-p groups and analyzes their vertices as rigid or flexible, with canonical, automorphism-invariant realizations. The paper also provides extensive examples (free, virtually free, RAAGs, PD pro-p groups) and builds a robust toolkit—incidence structures, relative finite presentability, and relative refinements—to study vertex-group decompositions and compatibility. It concludes with a deep treatment of automorphisms, tree-of-cylinders constructions, and the compatibility JSJ deformations, delivering a unified, applicable framework for pro-p splittings and their algebraic consequences.

Abstract

We develop JSJ decomposition theory of pro-p groups.

Paper Structure

This paper contains 41 sections, 131 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Trees
  4. G-trees
  5. $(\mathcal{A},\mathcal{K})$-trees
  6. Graph of pro-$p$ groups
  7. Accessibility
  8. Standard refinements
  9. Universally elliptic pro-$p$ trees
  10. JSJ splittings
  11. Examples of JSJ-splittings
  12. Free pro-$p$ groups
  13. Virtually free pro-$p$ groups
  14. Free splittings
  15. Splittings over finite groups
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $\mathcal{E}$ be a continuous family of subgroups of a pro-$p$ group $G$. If $G$ is finitely generated or $\mathcal{E}$-accessible, then JSJ pro-$p$$\mathcal{E}$-tree $T$ exists.

Theorems & Definitions (291)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Proposition 2.2
  • ...and 281 more