Thick embeddings of graphs into symmetric spaces via coarse geometry

We prove estimates for the optimal volume of thick embeddings of finite graphs into symmetric spaces, generalising results of Kolmagorov-Barzdin and Gromov-Guth for embeddings into Euclidean spaces. We distinguish two very different behaviours depending on the rank of the non-compact factor. For rank at least 2, we construct thick wirings of N-vertex graphs with volume CNln(N) and prove that this is optimal. For rank at most 1 we prove lower bounds of the form cNa for some (explicit) a>1 which depends on the dimension of the Euclidean factor and the conformal dimension of the boundary of the non-compact factor. The key ingredient is a coarse geometric analogue of a thick embedding called a coarse wiring, with the key property that the minimal volume of a thick embedding is comparable to the minimal volume of a coarse wiring for symmetric spaces of dimension at least 3.