Generalizations of the Kolmogorov-Barzdin embedding estimates

We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.Comment: 45 page

Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes and Gromov's notion of geometric overlap. Using the work of Gunder and Wagner, we give an estimate for the combinatorial expansion and geometric overlap of random Linial-Meshulam complexes