Growth of small generating sets in SL_n(Z/pZ)

TL;DR

This paper proves that small generating sets in the special linear group over finite fields expand significantly when multiplied, with the expansion rate depending on the set size and the group's dimension.

Contribution

It establishes a quantitative growth result for small generating sets in SL_n(Z/pZ), showing they expand rapidly under multiplication, depending only on the set size and dimension.

Findings

Small generating sets expand by a factor depending on their size and the group's dimension.

The expansion rate is at least |A|^{1+epsilon} for some epsilon>0.

The result depends only on n and delta, not on p or the specific set.

Abstract

Let G=SL_n. Let K=Z/pZ, p a prime. Let A\subset G(K) generate G(K). Suppose that |A|<p^{n+1-\delta}, delta>0. Then |A A A|>>|A|^{1+\epsilon}, where epsilon>0 and the implied constant depend only on n and delta.