# 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.

## Key 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.