Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_λ\}_{λ\in Λ}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We give an upper bound on the orders of finite non-parabolic subgroups of $G$ in terms of some fundamental constants associated with $\mathcal{P}$. This upper bound is computable if $G$ is finitely generated and the word problem in each $H_λ$, $λ\in Λ$, is decidable.
We prove that a parabolic subgroup $P$ contained in another parabolic subgroup $P'$ of an Artin group $A$ is a parabolic subgroup of $P'$. This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction $A \to P$ of the inclusion map from a parabolic subgroup $P$ into $A$. This retraction was implicitly constructed in a previous paper by Charney and the second author.
Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed using parabolic subgroups. We extend the construction of the complex of parabolic subgroups to FC type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is $δ-$…
Given an Artin group $A_Γ$, a common strategy in the study of $A_Γ$ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that $A_Γ$ has a specific property if and only if all "small" parabolic subgroups of $A_Γ$ have this property. Since "small" parabolic subgroups are the puzzle pieces of $A_Γ$ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of $A_Γ$ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Ar…
The aim of this note is to prove that the parabolic closure of any subset of a Coxeter group is a parabolic subgroup. To obtain that, several technical lemmas on the root system of a parabolic subgroup are established.
Despite the significance of the notion of parabolic closures in Coxeter groups of finite ranks, the parabolic closure is not guaranteed to exist as a parabolic subgroup in a general case. In this paper, first we give a concrete example to clarify that the parabolic closure of even an irreducible reflection subgroup of countable rank does not necessarily exist as a parabolic subgroup. Then we propose a generalized notion of "locally parabolic closure" by introducing a notion of "locally parabolic subgroups", which involves parabolic ones as a special case, and prove that the locally parabolic c…
We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable. We also use this geometric perspective to recover and unify results describing the normalisers of para…
Abstract. We address the conjecture which states that an intersection of parabolic subgroups of an Artin-Tits group is a parabolic subgroup. We prove that the conjecture is equivalent to a, a priori, weaker conjecture. We also prove the conjecture in a specific case. Along the way, we provide short and almost self-contain algebraical proofs of several classical results on Artin-Tits groups, such as those of Van der Lek on intersection of standard parabolic subgroups.
We introduce the notion of two-dimensional Coxeter system and show that parabolic subgroups of GL_n(F_2) can be described by an appropriate two-dimensional Coxeter system.
A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion of a Garside subgroup of a Garside group, which is related to the notion of LCM-homomorphisms between Artin-Tits groups. We prove that most of the properties of parabolic subgroups extend to this subgro…
We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small …
We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots.
It is well known that the normaliser of a parabolic subgroup of a finite Coxeter group is the semidirect product of the parabolic subgroup by the stabiliser of a set of simple roots. We show that a similar result holds for all finite unitary reflection groups: namely, the normaliser of a parabolic subgroup is a semidirect product and in many, but not all, cases the complement can be obtained as the stabiliser of a set of roots. In addition we record when the complement acts as a reflection group on the space of fixed points of the parabolic subgroup.
We determine the detailed structure of parabolic subgroups of orthogonal groups over $\mathbb{Z}$, and deduce the precise form of canonical boundary components in toroidal compactifications of orthogonal Shimura varieties.
Oshima's Lemma describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras.
We give a complete classification of conjugacy stable parabolic subgroups of Artin-Tits groups of spherical type. This answers a question posed by Ivan Marin and generalizes a theorem obtained by Juan González-Meneses in the specific case of Artin braid groups.
The conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.
Let S be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph S, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions.
We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink's result that the non-reflection part of a reflection centralizer is free. Namely, the non-reflection part of the normalizer of parabolic subgroup of type D5 or Aodd is either free or has a free subgroup of index 2.
Let G be a complete Kac-Moody group over a finite field. It is known that G possesses a BN-pair structure, all of whose parabolic subgroups are open in G. We show that, conversely, every open subgroup of G is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.
By constructing, in the relative case, objects analoguous to Rips and Sela's canonical representatives, we prove that the set of images by morphisms without accidental parabolic, of a finitely presented group in a relatively hyperbolic group, is finite, up to conjugacy.
One can observe that Coxeter groups and right-angled Artin groups share the same solution to the word problem. On the other hand, in his study of reflection subgroups of Coxeter groups Dyer introduces a family of groups, which we call Dyer groups, which contains both, Coxeter groups and right-angled Artin groups. We show that all Dyer groups have this solution to the word problem, we show that a group which admits such a solution belongs to a little more general family of groups that we call quasi-Dyer groups, and we show that this inclusion is strict. Then we show several results on parabolic…
In this paper, we investigate boundaries of parabolic subgroups of Coxeter groups. Let $(W,S)$ be a Coxeter system and let $T$ be a subset of $S$ such that the parabolic subgroup $W_T$ is infinite. Then we show that if a certain set is quasi-dense in $W$, then $W \partialΣ(W_T,T)$ is dense in the boundary $\partialΣ(W,S)$ of the Coxeter system $(W,S)$, where $\partialΣ(W_T,T)$ is the boundary of $(W_T,T)$.
We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to established the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin-Tits groups of spherical type.
Let $(W,S)$ be a Coxeter system, and let $X$ be a subset of $S$. The subgroup of $W$ generated by $X$ is denoted by $W_X$ and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of $W_X$ in $W$ is the subgroup of $w$ in $W$ such that $wW_Xw^{-1}\cap W_X$ has finite index in both $W_X$ and $wW_Xw^{-1}$. The subgroup $W_X$ can be decomposed in the form $W_X = W_{X^0} \cdot W_{X^\infty} \simeq W_{X^0} \times W_{X^\infty}$ where $W_{X^0}$ is finite and all the irreducible components of $W_{X^\infty}$" > are …
We extend the notion of induced conjugacy classes in reductive groups, introduced by Lusztig and Spaltenstein for unipotent classes, to arbitrary classes. We study properties of equivariant fibrations of prehomogeneous affine spaces, especially the existence of relative invariants. We also detect prehomogeneous affine spaces as subquotients of canonical parabolic subgroups attached to elements of reductive groups in the sense of Jacobson-Morozov. These results are prerequisites for making the geometric expansion of the Arthur-Selberg trace formula more explicit.
Let $W$ be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer $Z_W(W_I)$ of an arbitrary parabolic subgroup $W_I$ into the center of $W_I$, a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. Moreover, by using our description of $Z_W(W_I)$, we reveal a further strong property of the action of the third factor on the second factor, in particular on the finite irreducible components of the second factor.
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.
We define the notion of preGarside group slightly lightening the definition of Garside group so that all Artin-Tits groups are preGarside groups. This paper intends to give a first basic study on these groups. Firstly, we introduce the notion of parabolic subgroup, we prove that any preGarside group has a (partial) complemented presentation, and we characterize the parbolic subgroups in terms of these presentations. Afterwards we prove that the amalgamated product of two preGarside groups along a common parabolic subgroup is again a preGarside group. This enables us to define the family of pre…
We study density of parabolic elements in a finitely generated relatively hyperbolic group $G$ with respect to a word metric. We prove this density to be zero (apart from degenerate cases) and the limit defining the density to converge exponentially fast; this has recently been proven independently by W. Yang. As a corollary, we obtain the analogous result for the set of commuting pairs of elements in $G^2$, showing that the degree of commutativity of $G$ is equal to zero.
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.
Given an Artin group $A$ and a parabolic subgroup $P$, we study if every two elements of $P$ that are conjugate in $A$, are also conjugate in $P$. We provide an algorithm to solve this decision problem if $A$ satisfies three properties that are conjectured to be true for every Artin group. We partially solve the problem if $A$ has $FC$-type, and we totally solve it if $A$ is isomorphic to a free product of spherical Artin groups. In particular, we show that in this latter case, every element of $A$ is contained in a unique minimal (by inclusion) parabolic subgroup.
We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets …
A subgroup of a group $G$ is called algebraic if it can be expressed as a finite union of solution sets to systems of equations. We prove that a non-elementary subgroup $H$ of an acylindrically hyperbolic group $G$ is algebraic if and only if there exists a finite subgroup $K$ of $G$ such that $C_G(K) \leq H \leq N_G(K)$. We provide some applications of this result to free products, torsion-free relatively hyperbolic groups, and ascending chains of algebraic subgroups in acylindrically hyperbolic groups.
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.
The principal objects studied in this note are Coxeter groups $W$ that are neither finite nor affine. A well known result of de la Harpe asserts that such groups have exponential growth. We consider quotients of $W$ by its parabolic subgroups and by a certain class of reflection subgroups. We show that these quotients have exponential growth as well. To achieve this, we use a theorem of Dyer to construct a reflection subgroup of $W$ that is isomorphic to the universal Coxeter group on three generators. The results are all proved under the restriction that the Coxeter diagram of $W$ is simply l…
There is a family of hyperbolic groups known as hyperbolic hydra which contain heavily distorted free subgroups. We prove the existence of Cannon--Thurston maps (that is, maps of the boundaries induced by subgroup inclusion) for these free subgroups. It is known that Cannon--Thurston maps between hyperbolic space boundaries can exist even in the presence of arbitrarily heavy (even non-recursive) distortion. The hyperbolic hydra examples show that Cannon--Thurston maps can exist even between hyperbolic group boundaries in the presence of arbitrarily heavy primitive recursive distortion.
We extend previous results by Cumplido, Martin and Vaskou on parabolic subgroups of large-type Artin groups to a broader family of two-dimensional Artin groups. In particular, we prove that an arbitrary intersection of parabolic subgroups of a $(2,2)$-free two-dimensional Artin group is itself a parabolic subgroup. An Artin group is $(2,2)$-free if its defining graph does not have two consecutive edges labeled by $2$. As a consequence of this result, we solve the conjugacy stability problem for this family by applying an algorithm introduced by Cumplido. All of this is accomplished by consider…
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 classify up to conjugation by $\operatorname{GL}(2,\mathbb{R})$ (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of $\operatorname{Sp}(2,\mathbb{R})$ by means of an essentially geometric argument. This classification has already been established without geometry, under a stricter notion of equivalence, namely conjugation by arbitrary symplectic matrices. The present approach might be useful in higher dimensions and provides some insight.
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…
Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its word problem, and generating sets of the parabolic subgroups. We also give an algorithm that finds parabolic subgroups in a given recursively enumerable class of groups.
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.
Brady proved that there are hyperbolic groups with finitely presented subgroups that are not of type $FP_3$ (and hence not hyperbolic). We reprove Brady's theorem by presenting a new construction. Our construction uses Bestvina-Brady Morse theory, but does not involve branched coverings.
We characterize the double centralizer of all parabolic subgroups of the braid groups. We apply this result to provide a new and potentially more efficient solution to the subgroup conjugacy problem for parabolic subgroups. In the course of the proof we also characterize the centralizer for all parabolic subgroups.
We consider two families of subgroups of a group. Each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is supposed to be hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.
