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Adapted almost invariant sets and splittings of groups

Peter Scott, Gadde Swarup

Abstract

We prove relative versions of many earlier results about almost invariant sets and splittings of groups. In particular, we prove a relative version of the algebraic torus theorem, and we prove the existence and uniqueness of relative versions of algebraic regular neighbourhoods and of JSJ decompositions.

Adapted almost invariant sets and splittings of groups

Abstract

We prove relative versions of many earlier results about almost invariant sets and splittings of groups. In particular, we prove a relative version of the algebraic torus theorem, and we prove the existence and uniqueness of relative versions of algebraic regular neighbourhoods and of JSJ decompositions.

Paper Structure

This paper contains 33 sections, 118 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ends
  4. Equivalence
  5. Enclosing
  6. Cubings
  7. Adapted almost invariant sets
  8. Group systems and Cayley graphs
  9. Extensions and Crossing
  10. Good position and very good position
  11. Algebraic regular neighbourhoods
  12. Coends
  13. Crossing of sets over $VPC$ groups
  14. Adapted ends of subsets of $G$
  15. Applications
  16. ...and 18 more sections

Key Result

Lemma 2.1

Let $G$ be a group with subgroups $S,H,H_{1},\ldots,H_{n}$.

Figures (1)

  • Figure 1:

Theorems & Definitions (228)

  • Lemma 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Definition 2.6
  • Remark 2.7
  • Corollary 2.8
  • Lemma 2.9
  • Definition 3.1
  • ...and 218 more