arXiv research

Surface groups are uniquely identified by their profinite completions.

Study of profinite quandles with constructions and characterizations.

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…

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

In this paper we determine the automorphism groups of the profinite braid groups with four or more strings in terms of the profinite Grothendieck-Teichmüller group.

Study on endomorphism and automorphism groups of specific quandles.

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

Extends Lannes-Quillen theorem to all profinite groups.

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

Profinite rigidity studied for algebraic fibring of groups.

Study on Seifert 4-manifolds to classify profinite rigidity.

Non-isomorphic groups with similar profinite completions found.

We give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of the direct product of two free groups. We show that the profinite topology of the above group is strongly connected wit…

Triangle groups uniquely identified by their finite quotients.

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

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

Study describes prosoluble subgroups in 3-manifold groups.

New examples of hyperbolic 3-manifolds with unique profinite structure.

Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on prof…

Surveying recent progress on hyperbolic 3-manifold rigidity.

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

Proves regularity of isomorphisms between hyperbolic 3-manifolds.

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

The study shows that several properties are not profinite invariants.

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 $M$.

Turaev-Viro invariants match for certain surface bundles.

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

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…

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…

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.

In this paper we study some consequences of the author's classification of graph manifolds by their profinite fundamental groups. In particular we study commensurability, the behaviour of knots, and relation to mapping classes. We prove that the exteriors of graph knots are distinguished among all 3-manifold groups by …

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

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.

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…

Proves a property of certain 3D knots and links.

Proves 3-manifold groups uniquely identify hyperbolic bundles.

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

Study on profinite rigidity of direct products of free and surface groups.

Finite subgroups of good groups correspond to their profinite completions.

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 …

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

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

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.

Defines tensor product of profinitely many vector spaces over F2.

The paper characterizes simple closed curves on surfaces using profinite rigidity.

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 …

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.