In this paper, we prove that all finitely generated $3$-manifold groups are Grothendieck rigid. More precisely, for any group $G$ and proper subgroup $H\<G$, show the inclusion induced homomorphism $\widehat{i}\colon \widehat{H}\to \widehat{G}$ on profinite completions is not an isomorphism.

We fix a finitely presented group Q and consider short exact sequences 1 → N → Γ → Q → 1 with Γ finitely generated. The inclusion N ↪→ Γ induces a morphism of profinite completions N̂ → Γ̂. We prove that this is an isomorphism for all N and Γ if and only if Q is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residua...

We consider pairs of finitely presented, residually finite groups P ↪→ Γ for which the induced map of profinite completions P̂ → Γ̂ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not P is isomorphic to Γ. We construct pairs for which the conjugacy problem in Γ can be solved in quadratic time but the conjugacy problem in P is uns...

We consider finitely presented, residually finite groups G and finitely generated normal subgroups A such that the inclusion A ↪→ G induces an isomorphism from the profinite completion of A to a direct factor of the profinite completion of G. We explain why A need not be a direct factor of a subgroup of finite index in G; indeed G need not have a subgroup of finite index that splits as a non-tr...

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, it remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperbolic groups, there cannot exist algorithms to determine largeness, the existenc...

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...