Local spectral expansion approach to high dimensional expanders part I: Descent of spectral gaps

Abstract

This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex.

As advised by the referees, the arxiv paper "Local spectral expansion approach to high dimensional expanders" arXiv:1407.8517 was split into two parts since it was too long to publish as a whole. This is part I of the split article