We prove that there exist profinite Heyting algebras are not isomorphic to the completion of any algebra. This resolves an open problem from 2009. More generally, we characterize those varieties in which completions. It turns out exists largest such. give different characterizations this variety and show it is finitely axiomatizable locally finite. From follows decidable whether a all members I...

We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot SrK is distinguished from all other compact 3-manifolds by the set of finite quotients of its fundamental group. In addition, we show that if M is a compact 3-manifold with b1(M) = 1, and π1(M) has the same finite quotients as a free-by-cyclic grou...

We establish some sufficient conditions for the profinite and prop completions of an abstract group G of type FPm (resp of finite cohomological dimension, of finite Euler characteristics) to be of type FPm over the field Fp for a fixed natural prime p (resp. of finite cohomological p-dimension, of finite Euler p-characteristics). We apply our methods for orientable Poincaré duality groups G of ...

If M is a compact 3-manifold whose first betti number is 1, and N is a compact 3-manifold such that π1N and π1M have the same finite quotients, then M fibres over the circle if and only if N does. We prove that groups of the form F2 o Z are distinguished from one another by their profinite completions. Thus, regardless of betti number, if M and N are punctured torus bundles over the circle and ...