It is well-known that there are 19 classes of geometries for 4-dimensional manifolds in the sense of Thurston. We could ask that to what extent the geometric information is revealed by the profinite completion of the fundamental group of a closed smooth geometric 4-manifold. In this paper, we show that the geometry of a closed orientable 4-manifold in the sense of Thurston could be detected by the profinite completion of its fundamental group except when the geometry is $ \mathbb{H}^{4}$, $\mathbb{H}^{2}_{\mathbb{C}}$ or $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, despite the fact that not every smooth 4-manifold could admit one geometry in the sense of Thurston, some 4-dimensional manifolds with Seifert fibred structures are indeed geometric. For a closed orientable Seifert fibred 4-manifold $M$, we show that whether $M$ is geometric could be detected by the profinite completion of its fundamental group.
In this paper we extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary. We also illustrate an alternative and perhaps more natural approach to part of the original theorem, using relative cohomology to analyse the actions of an-annular atoroidal groups on profinite trees.
Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.
We study several properties of the completed group ring $\widehat{\mathbb{Z}}[[t^{\widehat{\mathbb{Z}}}]]$ and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots $J$ and $K$ are isomorphic, then their Alexander polynomials $Δ_J(t)$ and $Δ_K(t)$ coincide.
The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.
Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in $\mathrm{PU}(n,1)$ with isomorphic profinite completions for all $n \ge 2$. This disproves a conjecture of D. Kazhdan and gives the first examples nonisomorphic lattices in a semisimple Lie group of real rank one with isomorphic profinite completions, answering two questions of A. Reid.
Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.
It is proved that the profinite completion of the mapping class group Mod (g,n) of a surface of genus g with n boundary components is isomorphic to such of the arithmetic group GL(6g-6+2n, Z). We establish a relation between the normal subgroups of Mod (g,n) and the absolute Galois group G(K) of a number field K. Using the Tits alternative, we prove the Shafarevich Conjecture saying that the group G(Q^ab) of the maximal abelian extension of the field of rationals is isomorphic to a free profinite group.
