Isoperimetric Profiles and Regular Embeddings of locally compact groups

In this article we extend the notion of $L^p$-measure subgroups couplings, a quantitative asymmetric version of measure equivalence that was introduced by Delabie, Koivisto, Le Ma\^itre and Tessera for finitely generated groups, to the setting of locally compact compactly generated unimodular groups. As an example of these couplings; using ideas from Bader and Rosendal, we prove a "dynamical criteria" for the existence of regular embeddings between amenable locally compact compactly generated unimodular groups, namely the existence of an $L^\infty$-measure subgroup coupling that is coarsely $m$-to-$1$. We also prove that the existence of an $L^p$-measure subgroup that is coarsely $m$-to-$1$ implies the monotonicity of the $L^p$-isoperimetric profile, as well as sublinear version of this result. As a corollary we obtain that the $L^p$-isoperimetric profile is monotonous under regular embeddings, as well as coarse embeddings, between amenable unimodular locally compact compactly generated groups.