  We study the growth of product sets in some finite three-dimensional matrix
groups. In particular, we prove two results about the group of $2\times 2$
upper triangular matrices over arbitrary finite fields: a product set estimate
using techniques from multiplicative combinatorics, and an energy estimate
using incidence geometry. The energy method gives better quantitative results,
but only applies to small sets. We also prove an energy result for the
Heisenberg group.
