Delram Kahrobaei, City University of New York

Abstract:
Residual Properties of groups is a term introduced by Philip Hall in 1954. Let X be a class of groups: a group G is residually-X if and only if for every non-trivial element g in G there is an epimorph of G to a group in X such that the element corresponding to g is not the identity. In the literature, studying the residual solubility of groups was pioneered by Gilbert Baumslag in his celebrated paper in 1971, where he showed that positive one-relator groups are residually soluble. I have studied the notion of residual solubility and verified this property for several structures of groups. In this talk, I will give an overview of some of these results, from generalized free products to one-relator groups.

The true prosoluble completion P\Cal S (G) of a group G is the inverse limit of the projective system of soluble quotients of G. The definition proposed by G. Arzhantseva, P. de la Harpe and myself. In this talk I will describe examples including free groups, free soluble groups, wreath products, SL_d(Z) and its congruent subgroups, the Grigorchuk group, and non-free parafree groups(discovered by Baumslag). I will point out some natural open problems, particularly I will discuss a question of Grothendieck for profinite completions and its analogue for true prosoluble and true pronilpotent completions.