Isoperimetric Inequalities in Simplicial Complexes

Understand

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.