# Overlap properties of geometric expanders

# Overlap properties of geometric expanders

If $G=(V,E)$ is an expander graph then for every embedding $f:V\to R$ that is extended linearly from the vertices $G$ to its edges, there must be a point $p\in R$ such that the pre-image $f^{-1}(p)$ has cardinality at least a constant multiple of the total number of edges $|E|$ (just take $p$ to be the median of $f(V)$). This property of expanders motivated Gromov to define a higher dimensional version of expansion: say that a d-dimensional simplicial complex $S$ is "expander like" if for every embedding $f:S\to R^d$ that is extended affinely from the vertices of $S$ to its d-faces, there must be a point $p\in R^d$ such that the cardinality of $f^{-1}(p)$ is at least a constant multiple of the total number of d-faces of $S$. The fact that the complete d-dimensional simplicial complex is "expander like" is a classical result of Boros-Furedi ($d=2$) and Barany ($d>2$). Gromov asked the natural question whether in every dimension there exist bounded degree simplicial complexes that are "expander like". In this talk we will answer this question positively, and describe various related results and open questions.Joint work with Jacob Fox, Misha Gromov, Vincent Lafforgue, and Janos Pach.