Conjugacy Separability Research Articles

Quantifying conjugacy separability in wreath products of groups

Abstract We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of A and B, the growth of $\mathcal{C}$-cyclic subgroup separability of B and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterization of when $A \wr B$ is p-conjugacy separable. We use this characterization to provide for each prime p an example of a wreath product with infinite base group that is p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups, which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups.

Infinitely many virtual geometric triangulations

We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal triangulations. This cover is constructed in several stages, using results about separability of peripheral subgroups and their double cosets, in addition to a new conjugacy separability theorem that may be of independent interest. The infinite sequence of geometric triangulations is supported in a geometric submanifold associated to one cusp, and can be organized into an infinite trivalent tree of Pachner moves.

The twisted commutator subgroups and questions of Fel’shtyn and Troitsky

In this paper, we study the notion of the twisted conjugacy separability developed in Fel’shtyn and Troitsky (J K-Theory 1:1–40, 2008). We give an example of group with twisted conjugacy separability, but without residual finiteness. Thus, we find the negative answers on the several open questions formulated by Fel’shtyn and Troitsky (Topol Appl 157:1724–1735, 2010). In addition, we discuss the generalization of usual the commutator subgroup, especially for some relations among Nielsen fixed point theory and the twisted commutator subgroup.

Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

A group G is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of G, there exists a finite quotient of G where the images of these subgroups are not conjugate. It is proved that the fundamental group of a hyperbolic 3-manifold (closed or with cusps) is subgroup conjugacy separable

Profinite rigidity of graph manifolds, II: knots and mapping classes

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 their profinite fundamental groups. We also prove a strong conjugacy separability result for certain mapping classes of surfaces.

Effective twisted conjugacy separability of nilpotent groups

This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a possibly sharp upper bound for the conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.

On conjugacy separability of fibre products

In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group $G_2$ will always posses an index $3$ overgroup that is not conjugacy separable. Finally, we characterize $p$-conjugacy separable subdirect products of two free groups, where $p$ is a prime. We show that fibre products provide a natural correspondence between residually finite $p$-groups and $p$-conjugacy separable subdirect products of two non-abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite $p$-group is equivalent to the question about the existence of a finitely generated $p$-conjugacy separable full subdirect product of infinite index in the direct product of two free groups.

On conjugacy separability of graph products of groups

We show that the class of C-hereditarily conjugacy separable groups is closed under taking arbitrary graph products whenever the class C is an extension closed variety of finite groups. As a consequence we show that the class of C-conjugacy separable groups is closed under taking arbitrary graph products. In particular, we show that right angled Coxeter groups are hereditarily conjugacy separable and 2-hereditarily conjugacy separable, and we show that infinitely generated right angled Artin groups are hereditarily conjugacy separable and p-hereditarily conjugacy separable for every prime number p.

On the conjugacy separability of some free constructions of groups by root classes of finite groups

Let C be an arbitrary class of groups which has the root property, consists of finite groups only, and contains at least one nonidentity group. It is proved that every extension of a free group by a C-group is conjugacy C-separable. It is also proved that, if G is a free product of two conjugacy C-separable groups with finite amalgamated subgroup or an HNN-extension of a conjugacy C-separable group with finite associated subgroups, then the group G is residually C if and only if it is conjugacy C-separable.

Subgroup conjugacy separability of free-by-finite groups

A group G is said to be conjugacy subgroup separable if for every pair of non-conjugate finitely generated subgroups H and K of G, there exists a finite quotient of G where the images of these subgroups are not conjugate. The notion was introduced recently by O. Bogopolski and F. Grunewald. We prove here that finitely generated free-by-finite groups are subgroup conjugacy separable. This generalizes Theorem 1.5 in Bogopolski and Grunewald (On subgroup conjugacy separability in the class of virtually free groups, vol 110, 18 pages, 2010). We also show that free products preserve subgroup conjugacy separability. The methods are based on the profinite version of Bass–Serre’s theory of groups acting on trees. In particular, we use essentially the results of Ribes and Zalesskii (Rev Mat Iberoam 30:165–190, 2014).

On the Conjugacy Separability of Some Free Constructions of Groups by Root Classes of Finite Groups

Пусть $\mathcal{C}$ - произвольный корневой класс групп, состоящий лишь из конечных групп и содержащий хотя бы одну неединичную группу. Доказано, что любое расширение свободной группы при помощи $\mathcal{C}$-группы аппроксимируется классом $\mathcal{C}$ относительно сопряженности. Установлено также, что если $G$ - свободное произведение двух $\mathcal{C}$-аппроксимируемых относительно сопряженности групп с конечной объединенной подгруппой или HNN-расширение $\mathcal{C}$-аппроксимируемой относительно сопряженности группы с конечными связанными подгруппами, то $\mathcal{C}$-аппроксимируемость группы $G$ равносильна ее $\mathcal{C}$-аппроксимируемости относительно сопряженности. Библиография: 14 названий.

An effective algebraic detection of the Nielsen–Thurston classification of mapping classes

In this paper, we propose two algorithms for determining the Nielsen–Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in 〈X〉. We show that if ψ represents a reducible mapping class in Mod (S), then ψ admits a canonical reduction system whose total length is exponential in the word length of ψ. We use this fact to find the canonical reduction system of ψ. We also prove an effective conjugacy separability result for π1(S) which allows us to lift the action of ψ to a finite cover [Formula: see text] of S whose degree depends computably on the word length of ψ, and to use the homology action of ψ on [Formula: see text] to determine the Nielsen–Thurston classification of ψ.

On conjugacy separability of some Coxeter groups and parabolic-preserving automorphisms

We prove that even Coxeter groups, whose Coxeter diagrams contain no (4, 4, 2) triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application, we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups. © 2014 University of Illinois.

Residual finiteness, subgroup separability and conjugacy separability of certain HNN extensions

Abstract In this note we shall give characterisations for HNN extensions of non-cyclic polycyclic-by-finite groups with normal infinite cyclic associated subgroups to be residually finite, subgroup separable and conjugacy separable.

Hereditary conjugacy separability of free products with amalgamation

A group G is called hereditarily conjugacy separable if every finite index subgroup of G is conjugacy separable. The property is not preserved in general by the formation of free products with amalgamation. Here we find conditions (a)−(fNH) for a free product with cyclic amalgamation to preserve this property. We define a class XH to consist of all groups satisfying conditions (a)−(fNH) and prove that polycyclic-by-finite and free-by-finite groups belong to this class. We also prove that XH contains right angled Artin groups and virtual retracts of them (i.e. semidirect factors of finite index subgroups). Combining this with recent results of Wise and Agol we deduce that the fundamental group of a compact 3-manifold contains a finite index subgroup that belongs to the class XH.

Hereditary conjugacy separability of right-angled Artin groups and its applications

We prove that finite-index subgroups of right-angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina–Brady groups are conjugacy separable and have solvable conjugacy problem.

Conjugacy separability and outer automorphism groups of certain HNN extensions

In this note we first establish a criterion for the conjugacy separability of HNN extensions of polycyclic-by-finite groups with central associated subgroups. Using this result, we prove the equivalence of residually finiteness and conjugacy separability in these HNN extensions. Then we prove that the outer automorphism groups of these HNN extensions are residually finite if the HNN extensions themselves are residually finite.

CONJUGACY SEPARABILITY OF GENERALIZED FREE PRODUCTS OF FINITELY GENERATED NILPOTENT GROUPS

In this paper, we prove a criterion of conjugacy separability of generalized free products of polycyclic-by-finite groups with a non cyclic amalgamated subgroup. Applying this criterion, we prove that certain generalized free products of polycyclic-by-finite groups are conjugacy separable.

Bianchi groups are conjugacy separable

We prove that non-uniform arithmetic lattices of S L 2 ( C ) and consequently the Bianchi groups are conjugacy separable. The proof is based on recent deep results of Agol, Long, Reid and Minasyan. The conjugacy separability of groups commensurable with limit groups is also established.

Conjugacy Separability of Some One‐Relator Groups

Conjugacy separability of any group of the class of one‐relator groups given by the presentation 〈a, b; [am, bn] = 1〉 (m, n > 1) is proven. The proof made used of theoretical combinatorial group methods, namely the structure of amalgamated free products and some properties of the subgroups and quotients of any group of the class of one‐relator groups given above.