It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense Thurston. We could ask to what extent geometric information revealed by profinite completion fundamental group a closed smooth 4-manifold. In this paper, we show if two manifolds, neither whose $$\mathbb {H}^{4}$$ or {H}^{2}_{\mathbb {C}}$$ , share same then they have geometry. Moreover, despite fa...

We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V . The converse holds for varieties of finite type. It is a matter of folklore that the profinite completion of a Boolean algebra B is given by the power set of the Stone space of B, or in the terminology of Jónsson and Tarski [5], by the canonical ext...

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ−module, whose restriction to RH is projective. Moore’s conjecture [5]: Assume for every nontrivial element x in Γ, at least one of the following two conditions holds: M1) 〈x〉 ∩ H 6= {e} (in particular this holds if Γ is torsion free) M2) ord(x) is finite and invert...

We define, for any group G, finite approximations ; with this tool, we give a new presentation of the profinite completion b π : G → b G of an abtract group G. We then prove the following theorem : if k is a finite prime field and if V is a k-vector space, then, there is a natural isomorphism between b V (for the underlying additive group structure) and the additive group of the double-dual V ∗...

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.

We study finitely generated Heyting algebras from algebraic and model theoretic points of view. We prove amon others that finitely generated free Heyting algebras embed in their profinite completions, which are projective limits of finitely generated free Heyting algebras of finite dimension.