Expansion of Integer Matrices over Various Rings

In this article, we explore the problem of constructing high-dimensional expanders through the study of relations between expansion constants over different rings. We investigate expansion constants of integer matrices regarded as morphisms between free modules over R, Z, and Z/pZ. We introduce a new condition which we call integral spanning regarding kernels of integer matrices, and prove that it ensures equality of real and integral expansions. In addition, we prove a bound on expansion constants over finite fields for a certain class of matrices in terms of the corresponding integral expansions. As an application, one may use this theorem to bound the expansion of codifferentials over Z/2Z in degrees $0$ and $1$.