We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite by its set of quotients. also develop a method based on character varieties can be used to distinguish between profinite completions groups.

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Also, we show that right-angled Artin groups are conjugacy separable. Moreover, we investigate the profinite topology of F 2 × F 2 and of the group L in [22], which are the only obstructions for the subgroup separability of th...

We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of...

We show that the quasiconvex subgroups in doubles of certain negatively curved groups are closed in the profinite topology. This allows us to construct the first known large family of hyperbolic 3-manifolds such that any finitely generated subgroup of the fundamental group of any member of the family is closed in the profinite topology.

This work is a development of some model-theoretic aspects of the theory of Special Groups ([9]) − a first order axiomatization of the algebraic theory of quadratic forms. In [33] we constructed a functor, the profinite hull functor, P, from the category RSG of reduced special groups with SG-morphisms to the category RSGpf of profinite reduced special groups with continuous SG-morphisms. From r...

We extend a finite group solvability criterion of J.G. Thompson, based on his classification of finite minimal simple groups, to a prosolvability criterion. Moreover, we generalize to the profinite setting subsequent developments of Thompson’s criterion by G. Kaplan and the second author, which recast it in terms of properties of sequences of Sylow subgroups and their products. This generalizat...