Measurable Imbeddings, Free Products, and Graph Products

TL;DR

This paper investigates a generalized form of measure-based group embedding called Measurable Imbeddability, demonstrating how such embeddings behave under free and graph product constructions.

Contribution

It establishes that Measurable Imbeddability is preserved under free and graph product operations given certain conditions, extending previous understanding of measure-theoretic group relations.

Findings

Measurable Imbeddability is preserved under free products with specific assumptions.

The result extends to graph products of groups.

Provides new techniques for analyzing measure-theoretic group embeddings.

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 $\Lambda_1$ measurably imbeds into $\Gamma_1$, and $\Lambda_2$ measurably imbeds into $\Gamma_2$ under an additional assumption that lets the corresponding fundamental domains to be arranged in a special way, then $\Lambda_1 * \Lambda_2$ measurably imbeds into $\Gamma_1 * \Gamma_2$. Building upon the techniques we used, we show that the analogous result holds for graph products of groups.