Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering of G.
We give a criterion for H to be left relatively convex in G that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a result of Duncan and Howie.
We show that if G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell.
We show that all maximal cyclic subgroups in locally residually torsion-free nilpotent groups are left relatively convex.
We define relatively quasiconvex subgroups of relatively hyperbolic groups in the sense of Osin and show that such subgroups have expected properties. Also we state several definitions equivalent to the definition of relatively hyperbolic groups in the sense of Osin.
A conjecture of Dicks and the author on rank of the intersection of factor-free subgroups in free products of groups is proved for the case of left ordered groups.
This paper proves under certain conditions the existence of an algorithm, which detects relatively quasiconvex subgroups $H$ of relatively hyperbolic groups $(G,\mathbb{P})$. Additionally, this algorithm outputs an induced peripheral structure of $(G,\mathbb{P})$ on $H$.
We prove that for any finitely generated relatively hyperbolic group G and any symmetric endomorphism f of G with relatively quasiconvex image, Fixf is relatively quasiconvex subgroup of G.
We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a group and relative quasiconvexity for subgroups of the group.
Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follows that the fixed subgroup is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. If all peripheral subgroups of G are slender (respectively, slender and coherent), our result implies that the fixed subgroup of every automorphism of G is finitely generated (respectively, finitely presented). In particular, this hap…
There has been interest recently concerning when a left ordered group is locally indicable. Bergman and Tararin have shown that not all left ordered groups are locally indicable, but all known examples contain a nonabelian free subgroup. We shall show for a large class of groups not containing a nonabelian free subgroup, that any left ordered group in this class is locally indicable. Specifically this class is the smallest class of groups containing NS and Thompson's group of piecewise linear homeomorphisms of the unit interval, and is closed under taking subgroups, quotient groups, extensions…
We indicate a natural generalization of the concept of subgroup commutativity degree of a finite group and a list of open problems on these new concepts.
For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two quasiconvex subgroups $Q$ and $R$ is quasiconvex and isomorphic to $Q \ast_{Q\cap R} R$. Our results generalized known combination theorems for quasiconvex subgroups of word-hyperbolic groups. Some applications are presented.
We introduce the bounded packing property for a subgroup of a countable discrete group G. This property gives a finite upper bound on the number of left cosets of the subgroup that are pairwise close in G. We establish basic properties of bounded packing, and give many examples; for instance, every subgroup of a countable, virtually nilpotent group has bounded packing. We explain several natural connections between bounded packing and group actions on CAT(0) cube complexes. Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, und…
Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\mathbb{P}=\{P_1,\dots,P_m\}$. Let $H_1,H_2$ be subgroups of $G$ such that $H_1$ is relatively quasiconvex with respect to $\mathbb{P}$ and $H_2$ is not parabolic. Suppose that $H_2$ is elementwise conjugate into $H_1$. Then there exists a finite index subgroup of $H_2$ which is conjugate into $H_1$. The minimal length of the conjugator can be estimated. In the case where $G$ is a limit group, it is sufficient to assume only that $H_1$ is a finitely generated and $H_2$ is an arbitrary subgroup of $…
For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $γ_{\text{cp}}\left( G\right) $ to be the smallest number $k$ such that $G$ is a product, in some order, of $k$ pairwise conjugated proper subgroups of $G$. We prove that if $G$ is non-solvable then $γ_{\text{cp}}\left( G\right) \leq36$ while if $G$ is solvable then $γ_{\text{cp}}\left( G\right) $ can attain any integer value bigger than $2$, while, on the other hand, $γ_{\text{cp}}\left( G\right) \leq4\log_{2}\left\vert…
We consider the general question of how the homological finiteness property left-FPn holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. In particular we show that left-FPn is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups we prove the converse, and as a corollary show that a completely simple semigroup is of type left- and right-FPn if and only if it has finitely many left and right ideals and all of its m…
Let $G$ be a group hyperbolic relative to a collection of subgroups $\{H_λ,λ\in Λ\} $. We say that a subgroup $Q\le G$ is hyperbolically embedded into $G$, if $G$ is hyperbolic relative to $\{H_λ,λ\in Λ\} \cup \{Q\} $. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element $g\in G$ has infinite order and is not conjugate to an element of $H_λ$, $λ\in Λ$, then the (unique) maximal elementary subgroup contained $g$ is hyperbolically embedded into $G$. This allows to prove that if $G$ is boundedly generated, then $G$ is elementar…
For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces.
The congruence subgroup problem for a finitely generated group $Γ$ asks whether the map $\hat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C\left(\mathbb{Z}^{n}\right)=\left\{ 1\right\}$ for every $n\geq3$, but $C\left(\mathbb{Z}^{2}\right)=\hat{F}_ω$, where $\hat{F}_ω$ is the free profinite group on countably many generators. Considering $Φ_{n}$, the free metabelian group on $n$ generators, it was also proven that…
The congruence subgroup problem for a finitely generated group $Γ$ asks whether $\widehat{Aut\left(Γ\right)}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In this paper we first give two new short proofs of two known results (for $Γ=F_{2}$ and $Φ_{2}$) and a new result for $Γ=Φ_{3}$: 1. $C\left(F_{2}\right)=\left\{ e\right\}$ when $F_{2}$ is the free group on two generators. 2. $C\left(Φ_{2}\right)=\hat{F}_ω$ when $Φ_{n}$ is the free metabelian group on $n$ generators, and $\hat{F}_ω$ is the free pr…
Let $\mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $\mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $\mathfrak{X}$-maximal in $G/A$. We say that the reduction $\mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $\mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $\…
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
This paper proves that in a non-elementary relatively hyperbolic group, the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set. As a consequence, any non-elementary subgroup has uniform exponential growth.
In this paper, we state two combination theorems for relatively quasiconvex subgroups in a relatively hyperbolic group. Applications are given to the separability of double cosets of certain relatively quasiconvex subgroups and the existence of closed surface subgroups in relatively hyperbolic groups.
We show that the restriction of the Dehornoy ordering to an appropriate free subgroup of the three-strand braid group defines a left ordering of the free group on k generators, k>1, that has no convex subgroups.
Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. The method is to reduce, via combination and filling theorems, the separability of a quasiconvex s…
Let $(W, S)$ be a Coxeter system. We give necessary and sufficient conditions on the Coxeter diagram of $(W, S)$ for $W$ to be relatively hyperbolic with respect to a collection of finitely generated subgroups. The peripheral subgroups are necessarily parabolic subgroups (in the sense of Coxeter group theory). As an application, we present a criterion for the maximal flats of the Davis complex of $(W,S)$ to be isolated. If this is the case, then the maximal affine sub-buildings of any building of type $(W,S)$ are isolated.
Left braces, introduced by Rump, have turned out to provide an important tool in the study of set theoretic solutions of the quantum Yang-Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace $(B,+,\cdot )$ is a structure determined by two group structures on a set $B$: an abelian group $(B,+)$ and a group $(B,\cdot)$, satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group $A$ is a subgroup of the additive group of a finite simple left brace $B$ with metabelian multiplicative grou…
The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include…
Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{H_1, ..., H_m\} $. We prove that if each of the subgroups $H_1, ..., H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$ is also finite.
Let ${\mathfrak{X}}$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Denote by ${\mathrm{k}}_{\mathfrak{X}}(G)$ the number of conjugacy classes ${\mathfrak{X}}$-maximal subgroups of a finite group $G$. The natural problem to describe up to conjugacy ${\mathfrak{X}}$-maximal subgroups of a given finite group is complicated by the fact that it is not inductive. In particular, generally speaking, the image of an ${\mathfrak{X}}$-maximal subgroup is not ${\mathfrak{X}}$-maximal in the image of a homomorphism. Nevertheless, there are group homomorphism…
We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $ϕ$ into a relatively hyperbolic group $G$ is relatively hyperbolic with respect to a system of subgroups whose image under $ϕ$ is situated in a uniformly bounded distance from the parabolic subgroups of $G$.
Let A = Z[c], where c is an irrational number whose square is rational, or let A = Z[1/r], where r > 1 is a square-free natural number. We show that no finite-index subgroup of SL(2,A) is left orderable. (Equivalently, these subgroups have no nontrivial orientation-preserving actions on the real line.) This implies that if G is an isotropic F-simple algebraic group over an algebraic number field F, then no nonarchimedean S-arithmetic subgroup of G is left orderable. Our proofs are based on the fact, proved by B.Liehl, that every element of SL(2,A) is a product of a bounded number of elementary…
We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller-Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic gro…
We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin, and Bowditch's definitions of relative hyperbolicity for countable groups. We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperb…
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…
We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly contained in the left-regular representation, but not weakly equivalent to it. We discuss applications of this result and pose the problem to construct non-discrete topologically simple groups with a cocompact amenable closed subgroup but without a Gelfand pair.
We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over cohe…
Let $G$ be a group hyperbolic relative to a finite collection of subgroups $\mathcal P$. Let $\mathcal F$ be the family of subgroups consisting of all the conjugates of subgroups in $\mathcal P$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{\mathcal F} G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.
This is the fourth and last in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of $\text{Out}(F_n)$. In this paper we develop general ping-pong techniques for the action of $\text{Out}(F_n)$ on the space of lines of $F_n$. Using these techniques we prove the main results stated in the research announcement, Theorem C and its special case Theorem I, the latter of which says that for any finitely generated subgroup $\mathcal H$ of $\text{Out}(F_n)$ that acts trivially on homology with $\mathbb{Z}/3$ coefficients, and for…
We study different notions of quasiconvexity for a subgroup $H$ of a relatively hyperbolic group $G.$ The first result establishes equivalent conditions for $H$ to be relatively quasiconvex. As a corollary we obtain that the relative quasiconvexity is equivalent to the dynamical quasiconvexity. This answers to a question posed by D. Osin \cite{Os06}. In the second part of the paper we prove that a subgroup $H$ of a finitely generated relatively hyperbolic group $G$ acts cocompactly outside its limit set if and only if it is (absolutely) quasiconvex and every its infinite intersection with a pa…
Let F be an infinitely generated free group and R a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F' of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group Aut(F/R') of the group F/R' is complete. In particular, the automorphism group of any infinitely generated free solvalbe group of derived length at least two is complete. This extends a result by Dyer and Formanek (1977) on finitely generated groups F_n/R' where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.
Consider a finitely generated group $G$ that is relatively hyperbolic with respect to a family of subgroups $H_1, ..., H_n$. We present an axiomatic approach to the problem of extending metric properties from the subgroups $H_i$ to the full group $G$. We use this to show that both (weak) finite decomposition complexity and straight finite decomposition complexity are extendable properties. We also discuss the equivalence of two notions of straight finite decomposition complexity.
Given a monoid defined by a presentation, and a homotopy base for the derivation graph associated to the presentation, and given an arbitrary subgroup of the monoid, we give a homotopy base (and presentation) for the subgroup. If the monoid has finite derivation type (FDT), and if under the action of the monoid on its subsets by right multiplication the strong orbit of the subgroup is finite, then we obtain a finite homotopy base for the subgroup, and hence the subgroup has FDT. As an application we prove that a regular monoid with finitely many left and right ideals has FDT if and only if all…
This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise's special groups. More precise, we introduce a new combinatorial formalism to study (virtually) special groups, and we prove that, given a cocompact special group $G$ and a finite collection of subgroups $\mathcal{H}$, then $G$ is hyperbolic relative to $\mathcal{H}$ if and only if (i) each subgroup of $\mathcal{H}$ is convex-cocompact, (ii) $\mathcal{H}$ is an almost malnormal collection, and (iii) every non-virtually cyclic abelian subgroup of $G$ is contained in a conjugate of some group of $…
It is known that the Kurosh Subgroup Theorem does not hold for pro-$p$ groups of big cardinality. However, a subgroup $G$ of a free pro-$p$ product is projective relative to the Kurosh family of subgroups. In this paper we prove the converse of this fact.
Let $G$ be a group and $H$ a subgroup of $G$. This note introduces an equivalent definition of hyperbolic embedded subgroup based on Bowditch's approach to relatively hyperbolic groups in terms of fine graphs.
Let k be a local field, and G a linear group over k. We prove that either G contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups.
We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically associated with the corresponding subgroups. We show that this notion of Stallings graphs allows a unified approach to many algorithmic problems: some which had already been solved like the generalized membership problem or the computation of a quasi-convexity constant (Kapovich, 1996); and others such as the computation of intersections, the con…
