Measurable Imbeddings, Free Products, and Graph Products

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.