Thick embeddings into the Heisenberg group and coarse wirings into groups with polynomial growth

Thick embedding of a graph in a metric space is a drawing of the graph in the space, which maintains a minimal distance between those parts of the graph naturally meant to be distant. Its coarse-geometry analogue is a coarse wiring, which allows a controlled error term called the load. We present a conjecture of Itai Benjamini on thick embeddings into transitive graphs: the lower bound on embedding volume, arising from the separation profile of the ambient graph, is in fact sharp. We prove a weak form of this conjecture in two ambient-graph settings. In the case of groups with polynomial growth, we exhibit coarse wirings whose volume matches the lower bound, at the cost of logarithmic load. In the case of the three-dimensional discrete Heisenberg group, we construct thick embeddings but with volume which is optimal only up to a polylogarithmic factor.