Martin Bridson
Subgroups of direct products :
I shall discuss the structure of subgroups of direct products
of groups, paying particular attention to finite conditions such as
finite presentability and its higher-dimensional refinements. A special
role will be played by direct products of surface groups. The techniques
that appear will be both geometric/topological and algebraic. I shall
attempt to give a fairly complete proof of the following theorem of
Bridson, Howie, Miller and Short: there is an algorithm which, given
finite presentations of a finite number of groups G_1,...,G_n and a
finite subset S of their direct product, will construct a finite
presentation of the subgroup H generated by S provided that the projection of H to
each of the subproducts G_i x G_j is of finite index. If time allows, I
will discuss applications to profinite completions of groups.