A random group contains many subgroups which are isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. These subgroups can be taken to be quasi-isometrically embedded. This ...

We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, ...

We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary 3-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass–Serre theory for ...

We prove Wise’s W-cycles conjecture: Consider a compact graph Γ′ immersing into another graph Γ. For any immersed cycle Λ : S¹ → Γ, we consider the map Λ′ from the circular components S of the pullback to Γ′. Unless Λ′ is ...

We prove that there is no algorithm that can determine whether or not a finitely presented group has a non-trivial finite quotient; indeed, this property remains undecidable among the fundamental groups of compact, ...

A $\textit{permutoid}$ is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there ...