  All groups are 2-generator. For any prime-power q, Theorem 1 constructs a
solvable matrix group over a quotient of a Laurent polynomial ring. This group
is closely related to a group of exponent q as shown in Theorems 2 & 3 .
Theorem 4 in section 5 shows that a group of prime-power exponent contains the
relations of a solvable group. It follows that the Burnside groups of exponent
q are solvable, and it is easy to deduce that the solvability class of these
groups tends to infinity with q. Crucial to this work, especially for precise
bounds on the solvability class, is an earlier paper with Heilbronn and
Mochizuki.
