arXiv research

Surface groups are uniquely identified by their profinite completions.

3-manifolds with toral boundary are uniquely determined by their profinite completions.

Study on Seifert 4-manifolds to classify profinite rigidity.

Non-isomorphic groups with similar profinite completions found.

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 $M$.

Study describes prosoluble subgroups in 3-manifold groups.

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all…

The paper shows how certain Kähler groups are uniquely determined by their profinite completions.

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on prof…

Two groups with same profinite completion have different co-Hopfian properties.

Study of profinite quandles with constructions and characterizations.

The paper shows how profinite completions can reveal 4D geometries except for specific cases.

Surveying recent progress on hyperbolic 3-manifold rigidity.

Proves regularity of isomorphisms between hyperbolic 3-manifolds.

3-manifold groups are uniquely identifiable via their profinite completions.

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, and furthermore of a very particular form. In particular, the profinite completion of th…

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 …

Turaev-Viro invariants match for certain surface bundles.

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 coh…

There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish …

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 finitely generated pro-$p$ groups.

Study automorphisms of profinite mapping class groups for surfaces with negative Euler characteristic.

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.

Proves almost profinite rigidity for certain free-by-cyclic 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.

New insights into profinite rigidity of Kleinian groups and their subgroups.

Study of hyperelliptic mapping class groups with applications and profinite completions.

Knot groups of hyperbolic 2-bridge knots are uniquely identified by their finite quotients.

Triangle groups uniquely identified by their finite quotients.

Finite subgroups of good groups correspond to their profinite completions.

Study ramification in knot groups through finite covers and their quotients.

We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distingui…

Groups with similar pro-$p$ completions have two-generated subgroups that are free.

In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give…

The study explores rigidity in groups and their products, finding uncountable families of non-isomorphic subgroups.

We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow Γ$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}: \widehat{P} \to \widehatΓ$ is an isomorphism. Nor do there exist algorithms…

Automorphisms of pants complex are shown to be inner.

The paper connects arithmetic invariants of hyperbolic 3-manifolds.

Proves 3-manifold groups uniquely identify hyperbolic bundles.

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 remains undecidable among the fundamental groups of compact, non-positively curved square complexes. We deduce that many other properties of groups are undecidable. For hyperb…

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…

Study on contact forms with constant curvature on CR manifolds.

Galois action on manifold structures of complex varieties is abelian.

3-manifolds with similar completions have matching slopes and polynomials.

Study on endomorphism and automorphism groups of specific quandles.

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, $C'(1/6)$ one-relator groups are conjugacy separable; we provide a…

Extends Lannes-Quillen theorem to all profinite groups.

Bi-orderable groups from left-orderable ones, showing non-profinite properties.