Constructing uncountably many groups with the same profinite completion

Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.

THE PROFINITE COMPLETION OF A PROFINITE PROJECTIVE GROUP

We prove that the profinite completion of a profinite projective group is projective.

The weight of nonstrongly complete profinite groups

We compute the local weight of the completion of a nonstrongly complete profinite group and conclude that, if a profinite group is abstractly isomorphic to its own profinite completion, then they are equal. The local weights of all the groups in the tower of completions are computed as well.

Profinite genus of the fundamental groups of compact flat manifolds with holonomy group of prime order

In this article, we calculate the profinite genus of the fundamental group of an 𝑛-dimensional compact flat manifold 𝑋 with holonomy group of prime order. As consequence, we prove that if n ⩽ 21 n\leqslant 21 , then 𝑋 is determined among all 𝑛-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of the profinite completion of the fundamental group of 𝑋 in terms of the representation genus of its holonomy group.

The profinite completion of multi-EGS groups

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

Profinite invariants of arithmetic groups

We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$ -torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.