Expansion for the product of matrices in groups

TL;DR

This paper establishes lower bounds on the size of product sets of matrices in specific groups, extending known results to matrix groups like SL_2 over finite fields and the Heisenberg group.

Contribution

It provides new lower bounds for matrix product sets in groups, generalizing previous scalar and cube expansion results to matrix groups.

Findings

Proved lower bounds for matrix product sets in SL_2(𝔽_p).

Extended expansion results to the Heisenberg group.

Generalized scalar expansion results to matrix groups.

Abstract

In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in $SL_2(\mathbb{F}_p)$ with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyv\'{a}ri and Hennecart.