Topological expanders, coarse geometry and thick embeddings of complexes

We quantify the topological expansion properties of bounded degree simplicial complexes in terms of a family of sublinear functions, in analogy with the separation profile of Benjamini-Schramm-Tim\'ar for classical expansion of bounded degree graphs. We prove that, like the separation profile, these new invariants are monotone under regular maps between complexes satisfying appropriate higher connectivity assumptions. In the dimension $1$ case, we recover the cutwidth profile of Huang-Hume-Kelly-Lam. We also prove the seemingly new result that any $1$-dimensional topological expander necessarily contains a graphical expander. In higher dimensions, we give full calculations of these new invariants for Euclidean spaces, which are natural analogues of waist and width-volume inequalities due to Gromov and Guth respectively. We present several other methods of obtaining upper bounds including na\"ive (yet useful) direct product and fibring theorems, and show how lower bounds can be obtained via thick embeddings of complexes, in analogy with previous work of Barrett-Hume. Using this, we find lower bounds for $k$-expansion of $(k+1)$-fold horocyclic products of trees, and for rank $k$ symmetric spaces of non-compact type. As a further application, we prove that for every $k\geq 2$ there is no coarse embedding (and more generally, no regular map) from the $k$-fold horocyclic product of $3$-regular trees to either any product $(\mathbb{H}^2)^{k-2}\times H \times D$ where $\mathbb{H}^2$ is the real hyperbolic plane, $H$ is a bounded degree hyperbolic graph and $D$ is a doubling metric space, or to any symmetric space whose non-compact factor has corank (dimension minus rank) is strictly less than $k$.