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 ${\mathrm{\Lambda }}_1$ measurably imbeds into ${\mathrm{\Gamma }}_1$, and ${\mathrm{\Lambda }}_2$ measurably imbeds into ${\mathrm{\Gamma }}_2$ under an additional assumption that lets the corresponding fundamental domains to be arranged in a special way, then ${\mathrm{\Lambda }}_1\ast {\mathrm{\Lambda }}_2$ measurably imbeds into ${\mathrm{\Gamma }}_1\ast {\mathrm{\Gamma }}_2$. Building upon the techniques we used, we show that the analogous result holds for graph products of groups.