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Measurable Imbeddings, Free Products, and Graph Products

Özkan Demir

Abstract

We study Measurable Imbeddability between groups, which is an order-like generalization of Measure Equivalence that allows the imbedded group to have an infinite measure fundamental domain. We prove if $Λ_1$ measurably imbeds into $Γ_1$, and $Λ_2$ measurably imbeds into $Γ_2$ under an additional assumption that lets the corresponding fundamental domains to be arranged in a special way, then $Λ_1 * Λ_2$ measurably imbeds into $Γ_1 * Γ_2$. Building upon the techniques we used, we show that the analogous result holds for graph products of groups.

Measurable Imbeddings, Free Products, and Graph Products

Abstract

We study Measurable Imbeddability between groups, which is an order-like generalization of Measure Equivalence that allows the imbedded group to have an infinite measure fundamental domain. We prove if measurably imbeds into , and measurably imbeds into under an additional assumption that lets the corresponding fundamental domains to be arranged in a special way, then measurably imbeds into . Building upon the techniques we used, we show that the analogous result holds for graph products of groups.
Paper Structure (4 sections, 12 theorems, 37 equations)

This paper contains 4 sections, 12 theorems, 37 equations.

Table of Contents

  1. Introduction and Statement of the Main Results
  2. Definitions, Remarks, and Relation to Randomorphisms
  3. Free Products
  4. Graph Products

Key Result

Theorem A

If $\Lambda_1 \overset{\mathrm{SMI}}{\rightarrowtail} \Gamma_1$ and $\Lambda_2 \overset{\mathrm{SMI}}{\rightarrowtail} \Gamma_2$, then $\Lambda_1 * \Lambda_2 \overset{\mathrm{SMI}}{\rightarrowtail} \Gamma_1 * \Gamma_2$ Moreover, for given $(\Lambda_i\rightarrowtail\Gamma_i)$-couplings $\Sigma_i$ wit

Theorems & Definitions (31)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2
  • Example 1
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Remark
  • ...and 21 more