This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In §2 we collect some questions that motivated the lectures and this article, and in §3 discu...

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projecti...

The theory of special groups is an axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia (see [4]). The class of special groups, together with its morphisms, forms a category. As for other such axiomatisations, the main examples of special groups are provided by fields, in this case by applying the special group functor, which associates to each field F ...

We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a finite p-group. We also prove a version of Alperin’s Fusion Theorem for pro-fusion systems.

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the class...

In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group G is a group Comm(G) which depends only on the commensurability class of G. We study various properties of Comm(G); in particular, we find two natural ways to turn it into a topological group. We also use Comm(G) to study topological groups which conta...

Let G be a profinite group and let α be an automorphism of G. Then α is topologically intense if, for every closed subgroup H of G, there exists x ∈ G such that α(H) = xHx. Topologically intense automorphisms are automatically continuous, because they stabilize each open normal subgroup of the group on which they are defined. We denote by Intc(G) the group of topologically intense automorphisms...

We investigate whether a finitely generated profinite group G could have a finitely generated infinite image. A result of Dan Segal shows that this is impossible if G is prosoluble. We prove that such an image does not exist if G is semisimple or nonuniversal.

1 2 3 Let T be a rooted tree and Iso(T ) be the group of its isometries. We study closed subgroups G of Iso(T ) with respect to the number of conjugacy classes of Iso(T ) having representatives in G.