Symmetric Ranks of Matrix Groups

Roughly speaking, an n-dimensional integral matrix group is a set of n-by-n matrices with integer entries which is closed under matrix inversion and matrix multiplication. The symmetric rank of an integral matrix group is a measure of how much the group ``shuffles around" vectors of integers. Before formally defining these notions and looking at examples, we will begin by motivating their study with a geometric question about algebraic tori which is equivalent to finding an upper bound on the symmetric ranks of all finite integral matrix groups of a given dimension. To conclude, I will present my current progress toward finding this upper bound, including partial results and a general conjecture