On the metabelianization of Burnside groups

BARMAK, JONATHAN ARIEL

Karlsruhe (virtual)

Conferencia; 3rd International Meeting on Geometric Group Theory and Low Dimensional Topology; 2022

The Burnside group B(d,n) is presented by d generators and the relators are all the nth powers. It is known that some of these groups are finite and some others are not, but for given d,n, the question in general is open. The existence of a largest finite quotient R(d,n) of B(d,n) (the Restricted Burnside Problem) was settled in 1991 by Zelmanov, but again little is known about these groups. The free metabelian group d generators is the quotient of B(d,n) by its second derived subgroup. This is an approximation to R(d,n). It is always finite and easier to study. However, the order of these groups is not known in general. In this talk I will present invariants used to identify nontrivial elements of M(d,n), and in particular we will improve known bounds for the order of these groups. Our invariants are based on the Winding invariant and have a strong combinatorial flavor.