Geometric complexity of embeddings in ${\mathbb R}^d$

Given a simplicial complex $K$, we consider several notions of geometric complexity of embeddings of $K$ in a Euclidean space ${\mathbb R}^d$: thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any $n$-complex with $N$ simplices which topologically embeds in ${\mathbb R}^{2n}$, $n>2$, can be PL embedded in ${\mathbb R}^{2n}$ with refinement complexity $O(e^{N^{4+{\epsilon}}})$. Families of simplicial $n$-complexes $K$ are constructed such that any embedding of $K$ into ${\mathbb R}^{2n}$ has an exponential lower bound on thickness and refinement complexity as a function of the number of simplices of $K$. This contrasts embeddings in the stable range, $K\subset {\mathbb R}^{2n+k}$, $k>0$, where all known bounds on geometric complexity functions are polynomial. In addition, we give a geometric argument for a bound on distortion of expander graphs in Euclidean spaces. Several related open problems are discussed, including questions about the growth rate of complexity functions of embeddings, and about the crossing number and the ropelength of classical links.