Growth and generation in SL_2(Z/pZ)

TL;DR

This paper proves that subsets of SL_2(Z/pZ) expand quickly under group action, leading to efficient representations of elements as products of a logarithmic number of generators.

Contribution

It establishes rapid growth properties of subsets in SL_2(Z/pZ) and bounds the length of products needed to express any element using generators.

Findings

Subsets of SL_2(Z/pZ) grow rapidly under group action.

Any element can be expressed as a product of O((log p)^c) generators.

Results imply efficient generation of the entire group from small sets.

Abstract

We show that every subset of SL_2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL_2(Z/pZ), every element of SL_2(Z/pZ) can be expressed as a product of at most O((log p)^c) elements of the union of A and A^{-1}, where c and the implied constant are absolute.