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On residual finiteness of graphs of free groups with cyclic edge groups

Adrien Abgrall, Zachary Munro

Abstract

We characterize which groups splitting as finite graphs of free groups with cyclic edge groups are residually finite. Such a group $G$ is residually finite if and only if all its Baumslag-Solitar subgroups are residually finite. From a presentation of $G$, we construct a finite labeled graph $Γ$, and show that residual finiteness of $G$ is equivalent to an easily-detectable property of this graph. This characterization proves a conjecture of Wise.

On residual finiteness of graphs of free groups with cyclic edge groups

Abstract

We characterize which groups splitting as finite graphs of free groups with cyclic edge groups are residually finite. Such a group is residually finite if and only if all its Baumslag-Solitar subgroups are residually finite. From a presentation of , we construct a finite labeled graph , and show that residual finiteness of is equivalent to an easily-detectable property of this graph. This characterization proves a conjecture of Wise.

Paper Structure

This paper contains 11 sections, 31 theorems, 2 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorial complexes and words
  4. Graphs of groups
  5. Coverings and residual finiteness
  6. Small-cancellation theory
  7. Unbalanced elements and Baumslag-Solitar subgroups
  8. Detecting very unbalanced elements
  9. Residual finiteness
  10. Normalizing subwords
  11. Building the cover

Key Result

Theorem A

Suppose $G$ splits as a finite graph of free groups with cyclic edge groups. Then $G$ is residually finite if and only if the Baumslag-Solitar subgroups of $G$ are residually-finite.

Figures (6)

  • Figure 1: An example of a $\widehat{X}(n,m)$ cover for $BS(1,q)$.
  • Figure 2: A visual representation of one step of the normalization procedure, replacing the red subpath with the green.
  • Figure 3: The various spaces used in the construction of the cover $\widehat{X}\to P_G$. Squares commute and vertical arrows are coverings.
  • Figure 4: Two situations in which Lemma \ref{['lem:butterflyBalloon']} is applicable.
  • Figure 5: A relator and tube corresponding to some $t_j$.
  • ...and 1 more figures

Theorems & Definitions (92)

  • Theorem A
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Combinatorial covering condition
  • proof
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 82 more