Título: Prosoluble subgroups of the profinite completion of 3-manifolds groups

Palestrante: Lucas C. Lopes (UnB)

 

Resumo:

 

In recent years there has been a great deal of interest in detecting properties of the fundamental group π1(M) of a 3-manifold via its finite quotients, or more  conceptually by its profinite completion. This motivates the study of the profinite completion π1(M) of the fundamental group of a 3-manifold. A recent work by H. Wilton and P. Zalesskii shows that the typical splittings of 3-manifold groups such as free products with amalgamation, HNN-extensions, and graphs of groups are preserved under profinite completion. So one can use the profinite analogue of Bass-Serre theory for groups acting on trees. However, this theory does not have the full strength of its classical original. The main theorem of Bass-Serre theory does not hold in the profinite case. As a consequence, subgroup theorems do not hold, in general, for profinite groups and even for free profinite products, i.e., the profinite version of the Kurosh subgroup theorem is not valid. This implies that it is reasonable to study subgroup structure of free constructions for important subclasses of profinite groups. The most important subclasses of profinite groups are prosoluble and pro-p as they play the same role as finite p-groups and soluble groups in finite group theory. In particular, this applies to the profinite completion of 3-manifold groups. Such a study for pro-p subgroups was performed in [WZ18], where the structure of finitely generated pro-p subgroups of profinite completions of the fundamental groups of a 3-manifold was described. The objective of this work is to study prosoluble subgroups of π1(M).