The authors study several ways to measure the geometric complexity of an embedding of a simplicial complex into Euclidean space. The main theme is the connection between the effect of the topological complexity of the embedding and its geometrical complexity. The starting point is a result of Kolmogorov and Barzdin from the 1960s about the embedding of graphs in \(\mathbb R^3\), for which they introduced a notion of “expander”. Roughly stated, the result of Kolmogorov and Barzdin measures the difficulty of embedding expanders.
In the paper under review, the authors generalize the work of Kolmogorov and Barzdin in several directions. The first direction is a higher-dimensional analogue which deals with embeddings of \(k\)-dimensional simplicial complexes (generalizing embeddings of graphs) into an \(n\)-dimensional space with \(n\geq 2k+1\). In this setting, they prove, as in the case of Kolmogorov and Barzdin, a result relating the “thickness” and the number of simplices in the simplicial complex, generalizing an estimate Kolmogorov and Barzdin obtained for graphs. The second result deals with closed arithmetic hyperbolic 3-manifolds. The key property used is that this class of manifolds satisfies an expander-like isoperimetric inequality. The third result concerns distortion of knots. The authors give an alternate proof of a result obtained recently by J. Pardon saying that there exist isotopy classes of knots requiring arbitrarily large distortion. Here, a knot \(K\) is said to have distortion at least \(D\) if there exist two points \(x,y\in K\) with \(d_K(x,y)\geq D \mathrm{dist}_{{\mathbb R^3}}(x,y)\), where \(d_K(x,y)\) denotes the distance along \(K\), that is, the shortest distance from \(x\) to \(y\) measured on \(K\). The knots they use are built using arithmetic hyperbolic 3-manifolds.