Study on Seifert 4-manifolds to classify profinite rigidity.
problem Understanding profinite completions of Seifert 4-manifolds' fundamental groups.
method Classification of monodromies and conditions for profinite rigidity.
result Classification of Seifert 4-manifolds based on their profinite completions.
Surveying recent progress on hyperbolic 3-manifold rigidity.
problem Profinite rigidity of hyperbolic 3-manifolds.
method Review of profinite completion and rigidity of groups, evidence from other types of 3-manifolds, and existing ideas.
result Positive evidence for profinite rigidity in hyperbolic 3-manifolds.
Profinite rigidity of certain 3-manifolds detected through Dehn fillings.
problem Detecting profinite rigidity in 3-manifolds.
method Observation of profinite isomorphisms between cusped hyperbolic 3-manifolds and their Dehn fillings.
result Some cusped hyperbolic 3-manifolds are profinitely rigid.
Profinite rigidity proven for many hyperbolic manifolds.
problem Profinite rigidity of hyperbolic manifolds.
method Geometric topology and bubble-drilling construction.
result Profinite rigidity of many cusped hyperbolic manifolds.
New insights into profinite rigidity of Kleinian groups and their subgroups.
problem Characterizing profinite completions of Kleinian groups and their subgroups.
method Analyzing profinite completions and using finite index subgroups to distinguish completions.
result Profinite completions of certain subgroups of finite index in Kleinian groups are not isomorphic.
Profinite rigidity studied for algebraic fibring of groups.
problem Profinite rigidity of groups through algebraic fibring.
method Introducing TAP groups and proving algebraic fibring is a profinite property.
result Algebraic fibring is a profinite property for LERF groups.
New examples of hyperbolic 3-manifolds with unique profinite structure.
problem Finding hyperbolic 3-manifolds with unique profinite structure.
method Examining fundamental groups of closed fibered hyperbolic 3-manifolds.
result First examples of closed fibered hyperbolic 3-manifolds with unique profinite structure.
Triangle groups uniquely identified by their finite quotients.
problem Identifying triangle groups among finitely generated residually finite groups.
method Character varieties method to distinguish profinite completions.
result Certain Fuchsian triangle groups are profinitely rigid.
Proves almost profinite rigidity for certain free-by-cyclic groups.
problem Profinite rigidity of free-by-cyclic groups.
method Analyzes ranks of fibres, characteristic polynomials, and stretch factors of monodromies.
result Generic free-by-cyclic groups are almost profinitely rigid.
3-manifolds with toral boundary are uniquely determined by their profinite completions.
problem Determining the homeomorphism type of 3-manifolds based on their profinite completions.
method JSJ-decomposition and profinite completions of fundamental groups.
result Profinitely almost rigid 3-manifolds have finitely many homeomorphism types.
The paper shows how certain Kähler groups are uniquely determined by their profinite completions.
problem Understanding the uniqueness of Kähler groups within residually finite groups.
method Holomorphic fibrations and profinite completions of fundamental groups.
result Aspherical smooth projective varieties are determined by their algebraic fundamental groups.
Proves a property of certain 3D knots and links.
problem Understanding the fundamental groups of 3D knots and links.
method Uses profinite rigidity to analyze two-bridge links and 3-tangle Montesinos links.
result Profinite rigidity for two-bridge links and 3-tangle Montesinos links is established.
3-manifold groups are uniquely identifiable via their profinite completions.
problem Identifying uniquely 3-manifold groups among finitely generated groups.
method Proving that the inclusion of profinite completions is not an isomorphism for proper subgroups.
result All finitely generated 3-manifold groups are Grothendieck rigid.
Surface groups are uniquely identified by their profinite completions.
problem Identifying surface groups among limit groups.
method Using profinite completions and topology.
result The set of surface words is closed in the profinite topology.
The study explores rigidity in groups and their products, finding uncountable families of non-isomorphic subgroups.
problem Rigidity in groups and their direct products.
method Constructing Grothendieck pairs and exploiting relative hyperbolicity.
result Uncountable families of non-isomorphic subgroups Pλ in GimesG. Study on profinite rigidity of direct products of free and surface groups.
problem Profinite rigidity of direct products in residually free groups.
method Rank gradients of pro-p groups, virtual homology of limit groups. result Direct products of free and surface groups are profinitely rigid.
Proves 3-manifold groups uniquely identify hyperbolic bundles.
problem Identifying hyperbolic 3-manifold groups from their finite quotients.
method Upgraded Liu's result to detect fiber type via profinite completion.
result Proves hyperbolic bundles are distinguished by their profinite completions.
Groups with similar pro-p completions have two-generated subgroups that are free.
problem Understanding the profinite rigidity of groups with specific pro-p completions. method Utilizing new results on residually-(torsion-free nilpotent) groups and their subgroups, and applying these to prove profinite rigidity.
result Groups with the same pro-p completion have two-generated subgroups that are free. The paper characterizes simple closed curves on surfaces using profinite rigidity.
problem Characterizing simple closed curves on surfaces using profinite rigidity.
method Proving that elements with the same images under all finite groups are simple closed curves.
result The set of simple closed curves is closed in the profinite topology of the surface group.
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group PSL(2,Z[ω]) with ω2+ω+1=0 is rigid in this sense. Other examples include th…
We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a Z-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley…
Using conjugation of Shimura varieties, we produce nonisomorphic, cocompact, torsion-free lattices in PU(n,1) with isomorphic profinite completions for all n≥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 …
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…
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 …
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 …
Computes second homology groups of orbifold groups, proving profinite rigidity and Grothendieck pairs.
problem Computing the second homology group of orbifold groups.
method Effective method for computing H2 in specific cases of orbifold groups. result Explicit computations of second homology groups, including H2(ΓO,Z)≅Z/2Z. Study automorphisms of profinite mapping class groups for surfaces with negative Euler characteristic.
problem Determine automorphism groups of profinite completions of mapping class groups.
method Analyzes the structure of automorphism groups and embeddings of profinite Grothendieck-Teichmüller groups.
result Provides natural isomorphisms and embeddings for specific cases of surfaces.
The study shows that certain groups can be uniquely identified by their finite abelian summands.
problem Identifying groups based on their finite abelian summands.
method Analyzing hyperbolic groups as graphs of free groups with cyclic edge groups.
result Free products of free and surface groups are profinitely rigid.
Finite quotients of fibered hyperbolic 3-manifold groups detect taut polynomials.
problem Detecting taut polynomials of fibered faces of Thurston norm balls
method Developing a framework for profinite invariance of twisted multivariable Alexander polynomials
result Proving finite quotients detect taut polynomials
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⋊Z are distinguished from one another by their profinite…
We establish results concerning the profinite completions of 3-manifold groups. In particular, we prove that the complement of the figure-eight knot S3−K 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-manifo…
Study of profinite quandles with constructions and characterizations.
problem Characterizing and constructing profinite quandles.
method Several constructions and characterizations of profinite quandles from profinite groups and other quandles.
result Characterization of algebraically connected profinite quandles in terms of $\widehat{\Inn(Q)}$.
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…
Study on endomorphism and automorphism groups of specific quandles.
problem Characterizing endomorphism and automorphism groups of residually finite and profinite quandles.
method Proved properties of endomorphism monoids and automorphism groups for residually finite and profinite quandles.
result Endomorphism and automorphism groups of residually finite quandles are residually finite.
Extends Lannes-Quillen theorem to all profinite groups.
problem Conjugacy separability of p-torsion elements and finite p-subgroups. method Developed a theory of products for families of discrete torsion modules.
result Proves a full version of the Lannes-Quillen theorem for all profinite groups.
Bi-orderable groups from left-orderable ones, showing non-profinite properties.
problem Non-profinite properties in bi-orderability and generalized torsion elements.
method Using fiber products to construct bi-orderable groups, showing non-profinite properties.
result Bi-orderability and generalized torsion elements are not profinite properties.
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.
Non-isomorphic groups with similar profinite completions found.
problem Finding non-isomorphic groups with similar profinite completions.
method Exhibited infinitely many pairs of non-isomorphic groups with specific properties.
result Groups with Property FA and non-trivial actions on trees have isomorphic profinite completions.
A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial compu…
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.
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…
The study shows that several properties are not profinite invariants.
problem Determining which properties are profinite invariants.
method Combining Rips constructions and iterated group-theoretic Dehn filling on hyperbolic virtually special groups.
result Several properties (stable commutator length, quasimorphisms, property NL, property FW∞, property FA, and non-abelian free subgroups) are not profinite invariants. Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Regularity of isomorphisms between cusped hyperbolic 3-manifolds.
method Proves regularity of profinite completions of fundamental groups.
result Proves A-polynomial is a profinite invariant. 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…
Study describes prosoluble subgroups in 3-manifold groups.
problem Characterizing prosoluble subgroups in 3-manifold groups.
method Analyzes finitely generated prosoluble subgroups of profinite completions.
result Provides a description of these subgroups.
Two groups with same profinite completion have different co-Hopfian properties.
problem Understanding co-Hopfian properties in residually finite groups.
method Using a specific construction involving a finitely presented acyclic group with trivial profinite completion.
result Found two groups with same profinite completion but different co-Hopfian properties.
Proves regularity of isomorphisms between hyperbolic 3-manifolds.
problem Understanding the regularity of isomorphisms between hyperbolic 3-manifolds.
method Strengthened a previous result by proving regularity in the sense of Boileau and Friedl.
result Profinite isomorphisms of hyperbolic 3-manifolds are regular.
Defines tensor product of profinitely many vector spaces over F2.
problem Defining tensor product of profinitely many copies of a vector space.
method Proposes a definition for finite-dimensional vector spaces over F2 with specific group actions.
result Organizes computations in Heegaard Floer homology.