Local-global properties of word varieties

We consider equations of the form w(x,y)=g where w
is a group word in two letters (= an element of the free
group on two generators), g is a fixed element of a fixed
group G, and solutions are sought among pairs of elements
of G. Our focus is on the case where G is a simple linear
algebraic group. In this talk we consider the case
G=SL(2,K) where K is a number field. We discuss approximation
properties of the corresponding algebraic K-variety.

For a broad class of words, we reduce the problem to a similar
problem for a certain surface, using the fibration method.
For the commutator word w=[x,y] we obtain conclusive results.

Several related open problems will also be discussed.

The talk is based on a joint work with T. Bandman.