Topological overlap profiles are introduced as higher-dimensional analogues of the separation profile, with new calculations and coarse geometric obstructions.
Embedding products of trees into higher rank
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abstract
We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$ copies of the $3$-regular tree $T_3$ into any thick Euclidean building of rank $n$ with co-compact affine Weyl group. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings.
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Topological expanders, coarse geometry and thick embeddings of complexes
Topological overlap profiles are introduced as higher-dimensional analogues of the separation profile, with new calculations and coarse geometric obstructions.