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On treeings arising from HNN extensions

Yoshikata Kida

TL;DR

The paper investigates treeings for discrete measured groupoids arising from HNN extensions $G=ig\langle E,t \mid t a t^{-1}=\tau(a)\big\rangle$ and analyzes how a p.m.p. action, together with a normal subgroupoid $X\rtimes E$, yields a quotient groupoid whose components resemble Bass–Serre trees.Using Gaboriau’s induction on treeings and Maharam extensions, the author derives a treeing of the Maharam extension and studies the resulting cost and orbit-equivalence implications for kernels of modular homomorphisms.In the setting where $p=[E:E_-]>1$, $q=[E:E_+]>1$, and $p\neq q$, the induced treeing leads to the conclusion that the kernel ker$m$ is orbit equivalent to the free product structure $F_\infty\times \mathbb{Z}$, linking groupoid-theoretic costs to OE classifications.An explicit application to Baumslag–Solitar-type groups and a general splitting result for groupoid extensions provide a framework to deduce orbit-equivalence outcomes from cost arguments, with potential extensions to iterated amalgamated products.

Abstract

For certain HNN extensions including Baumslag-Solitar groups, a treeing is constructed from their certain probability-measure-preserving actions. This is a treeing of a quotient groupoid of the translation groupoid associated with their actions. As its application, for some of those HNN extensions, we show that the kernel of the modular homomorphism is measure equivalent to the direct product of the free group of infinite rank and Z.

On treeings arising from HNN extensions

TL;DR

The paper investigates treeings for discrete measured groupoids arising from HNN extensions $G=ig\langle E,t \mid t a t^{-1}=\tau(a)\big\rangle$ and analyzes how a p.m.p. action, together with a normal subgroupoid $X\rtimes E$, yields a quotient groupoid whose components resemble Bass–Serre trees.Using Gaboriau’s induction on treeings and Maharam extensions, the author derives a treeing of the Maharam extension and studies the resulting cost and orbit-equivalence implications for kernels of modular homomorphisms.In the setting where $p=[E:E_-]>1$, $q=[E:E_+]>1$, and $p\neq q$, the induced treeing leads to the conclusion that the kernel ker$m$ is orbit equivalent to the free product structure $F_\infty\times \mathbb{Z}$, linking groupoid-theoretic costs to OE classifications.An explicit application to Baumslag–Solitar-type groups and a general splitting result for groupoid extensions provide a framework to deduce orbit-equivalence outcomes from cost arguments, with potential extensions to iterated amalgamated products.

Abstract

For certain HNN extensions including Baumslag-Solitar groups, a treeing is constructed from their certain probability-measure-preserving actions. This is a treeing of a quotient groupoid of the translation groupoid associated with their actions. As its application, for some of those HNN extensions, we show that the kernel of the modular homomorphism is measure equivalent to the direct product of the free group of infinite rank and Z.
Paper Structure (15 sections, 42 theorems, 101 equations)

This paper contains 15 sections, 42 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Normal subgroupoids and quotient groupoids
  3. Preliminaries on groupoids
  4. Construction of quotient groupoids
  5. A criterion for quotient groupoids being p.m.p.
  6. Useful criteria for normality
  7. An example
  8. The Radon-Nikodym cocycle
  9. Treeings and the induction
  10. HNN extensions and treeings
  11. Being a treeing
  12. Maharam extensions and induced treeings
  13. Splitting of groupoid-extensions
  14. A general splitting result
  15. An application

Key Result

Theorem 1.1

Let $G$ be the above HNN extension. Suppose that $E$ is finitely generated, free abelian, and suppose also $p>1$, $q>1$ and $p\neq q$. Let $\bm{m}\colon G\to \mathbb{Q}^*_+$ be the modular homomorphism associated to $E$ defined by for $g\in G$ (see Subsection subsec-rn). Then $\ker \bm{m}$ is orbit equivalent to $F_\infty \times \mathbb{Z}$, where $F_\infty$ is the free group of countably infinit

Theorems & Definitions (92)

  • Theorem 1.1
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof : Proof of Theorem \ref{['thm-quotient']}
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 82 more