On the Burnside problem for periodic groups

Narain Gupta 1,. and Said Sidki 2 1 University of Manitoba, Department of Mathematics, Winnipeg, Manitoba R3T 2N2, Canada 2 University of Brasilia, Department of Mathematics, Brasilia, D.F., Brazil Introduction The generalized Burnside problem refers to the question: Are finitely generated periodic groups finite? This was answered in the negative by Golod [1] who proved that, for each prime p, there exists a finitely generated infinite p-group. Golod's construction in not, however, direct and is based on his celebrated work with Safarevi~. Recently, Grigor~uk [2] has given a direct and elegant construction of an infinite 2-group which is generated by three elements of order 2. In this paper we give, for each odd prime p, a direct construction of an infinite p-group on two generators, each of order p. Our group is a subgroup of the automorphism group of a regular tree of degree p; and as might be expected, it is residually finite and has infinite exponent. Preliminaries. Let p be an odd prime and let T(0) be the infinite regular tree of degree p with vertex 0, so that through each vertex u of T(0) there are p regular subtrees