Charles C. Edmunds

PRODUCTS OF COMMUTATORS AND PRODUCTS OF SQUARES IN A FREE GROUP

A classification of the ways in which an element of a free group can be expressed as a product of commutators or as a product of squares is given. This is then applied to some particular classes of elements. Finally, a question about expressing a commutator as a product of squares is addressed.

Small Semigroups Generating Varieties with Continuum Many Subvarieties

The smallest finitely based semigroup currently known to generate a variety with continuum many subvarieties is of order seven. The present article introduces a new example of order six and comments on the possibility of the existence of a smaller example. It is shown that if such an example exists, then up to isomorphism and anti-isomorphism, it must be a unique monoid of order five.

The two variable substitution problem for free products of groups

Abstract We consider equations of the form W(x, y) = U with U an element of a free product G of groups. We show that with suitable algorithmic conditions on the free factors of G, one can effectively determine whether or not the equations have solutions in G. We also show that under certain hypotheses on the free factors of G and the equation itself, the equation W(x, y) = U has only finitely many solutions, up to the action of the stabilizer of W(x, y) in Aut(〈x, y; 〉).

On the rank of quadratic equations in free groups

Let Fd and H be free groups freely generated by sets X = {x1,…,xd} (variables) and A (constants). An equation over H is an expression W = 1 where W is a reduced word on X±1 ∪ A±1; this equation is called quadratic if for each i, 1≤i≤d, the total number of occurrences of xi and xi−1 in W is zero or two. A solution to an equation W = 1 over H is an endomorphism o of Fd∗H whose restriction to H is the identity on H and such that Wo = 1. The rank of a solution o to W = 1 is the rank of the free group Foπ, where π is the projection of Fd∗H onto Fd, and the rank of an equation W=1 is the maximum of the ranks of its solutions. p]A formula for ranks of quadratic equations over free groups is given, and is shown to include earlier rank formulas for quadratic equations without constants.