Abstract

The free Burnside group B(r,n) is the quotient of the free group of rank r by the normal subgroup generated by the n-th power of all its elements. It was introduced in 1902 by Burnside who asked whether B(r,n) is necessarily a finite group or not. In 1968 Novikov and Adian proved that if r > 1 and n is a sufficiently large odd exponent, then B(r,n) is actually infinite. It turns out that B(r,n) has a very rich structure. In this talk we are interested in understanding equations in B(r,n). In particular we want to investigate the following problem. Given a set of equations S, under which conditions, every solution to S in B(r,n) already comes from a solution in the free group of rank r.

Along the way we will explore other aspects of certain periodic groups (i.e. quotients of a free Burnside groups) such that the Hopf / co-Hopf property, the isomorphism problem, their automorphism groups, etc.

Joint work with Z. Sela