Bipolar Coxeter groups

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.

In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.

For an element 𝑤 of a Coxeter group 𝑊, there are two important attributes, namely its length, and its expression as a product of disjoint cycles in its action on Φ, the root system of 𝑊. This paper investigates the interaction between these two features of 𝑤, introducing the notion of the crossing number of 𝑤, κ ⁢ ( w ) \kappa(w) . Writing w = c 1 ⁢ ⋯ ⁢ c r w=c_{1}\cdots c_{r} as a product of disjoint cycles, we associate to each cycle c i c_{i} a “crossing number” κ ⁢ ( c i ) \kappa(c_{i}) , which is the number of positive roots 𝛼 in c i c_{i} for which w ⋅ α w\cdot\alpha is negative. Let Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) be the sequence of κ ⁢ ( c i ) \kappa(c_{i}) written in increasing order, and let κ ⁢ ( w ) = max ⁡ Seq κ ⁢ ( w ) \kappa(w)=\max{\mathrm{Seq}}_{\kappa}({w}) . The length of 𝑤 can be retrieved from this sequence, but Seq κ ⁢ ( w ) {\mathrm{Seq}}_{\kappa}({w}) provides much more information. For a conjugacy class 𝑋 of 𝑊, let κ min ⁢ ( X ) = min ⁡ { κ ⁢ ( w ) ∣ w ∈ X } \kappa_{\min}(X)=\min\{\kappa(w)\mid w\in X\} and let κ ⁢ ( W ) \kappa(W) be the maximum value of κ min \kappa_{\min} across all conjugacy classes of 𝑊. We call κ ⁢ ( w ) \kappa(w) and κ ⁢ ( W ) \kappa(W) , respectively, the crossing numbers of 𝑤 and 𝑊. Here we determine the crossing numbers of all finite Coxeter groups and of all universal Coxeter groups. We also show, among other things, that for finite irreducible Coxeter groups, if 𝑢 and 𝑣 are two elements of minimal length in the same conjugacy class 𝑋, then Seq κ ⁢ ( u ) = Seq κ ⁢ ( v ) {\mathrm{Seq}}_{\kappa}({u})={\mathrm{Seq}}_{\kappa}({v}) and κ min ⁢ ( X ) = κ ⁢ ( u ) = κ ⁢ ( v ) \kappa_{\min}(X)=\kappa(u)=\kappa(v) .

In 1986, G.X. Viennot introduced the theory of heaps of pieces as a visualization of Cartier and Foata’s “partially commutative monoids”. These are essentially labeled posets satisfying a few additional properties, and one natural setting where they arise is as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version of a heap, which loosely speaking, can be thought of as taking a heap and wrapping it into a cylinder. We call this object a toric heap, because we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. Defining the category of toric heaps leads to the notion of certain morphisms such as toric extensions. We study toric heaps in Coxeter theory, because a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. As such, we formalize and study a framework that we call cyclic reducibility in Coxeter theory, which is closely related to conjugacy. We introduce what it means for elements to be torically reduced, which is a stronger condition than simply being cyclically reduced. Along the way, we encounter a new class of elements that we call torically fully commutative (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the cyclically fully commutative (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-convex in X, and in Cayley graphs for Coxeter groups, elements of infinite order are tracked by geodesics.

The Coxeter groups that act geometrically on eu- clidean space have long been classied and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors and I have resolved several long- standing conjectures about these groups, proving for the rst time that every irreducible euclidean Artin group is a torsion-free center- less group with a decidable word problem and a nite-dimensional classifying space. This article surveys our results and the tech- niques we use to prove them.

A Coxeter system (W, S ) is called affine-free if its Coxeter diagram contains no affine subdiagram of rank ≥ 3. Let (W, S ) be a Coxeter system of finite rank (i.e. |S | is finite). The main result is that W is affine-free if and only if W has finitely many conjugacy classes of reflection triangles. This implies that the action of W on its Coxeter cubing (defined by Niblo and Reeves [G. Niblo and L. Reeves. Coxeter groups act on CAT(0) cube complexes. J. Group Theory 6 (2003), 399–413]) is cocompact if and only if (W, S ) is affine-free. This result was conjectured in loc. cit. As a corollary, we obtain that affine-free Coxeter groups are biautomatic.

In this article, we investigate the set of $\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_{\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_{\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\gamma\in W$ the lattice $\mathcal{B}_{\gamma}$ is in fact distributive. It turns out that this is the case for the "coincidental" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\gamma$ of said groups for which $\mathcal{B}_{\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.

Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of $W$, thereby proving a conjecture of A. Cohen from 1994.

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups and to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group so they are not classified in the class of chiral polyhedra. It is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsand respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by-product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.

We compute Aut(W) for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in terms of the given presentation for the Coxeter group and admits an easy characterization of those groups W for which Out(W) is finite.

We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.

Huang proved that every set of more than half the vertices of the d-dimensional hypercube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs.First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encountered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d-regular containing an induced matching on a d2d−1-fraction of the vertices. This is largest possible and answers a question of Lehner and Verret.Second, we consider Huang's lower bound for graphs with subcubes and show that the corresponding lower bound is tight for products of Coxeter groups of type An, I2(2k+1), and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang's question.Finally, we show that induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This gives classes of Cayley graphs with properties similar to the ones provided by Huang's results. However, in contrast to Coxeter groups these graphs have no subcubes.

Conjugacy classes and straight elements in Coxeter groups

It is possible for a group W that is abstractly isomorphic to a Coxeter group to have more than one conjugacy class of Coxeter generating sets, and if S and R are two non-conjugate Coxeter generating sets then it may or may not be the case that some element s ∈ S is conjugate to an element r ∈ R . In this paper we classify the so-called intrinsic reflections: those elements of W whose conjugacy class intersects non-trivially every Coxeter generating set. In combination with previously known results, this leads us to a classification of Coxeter groups for which all Coxeter generating sets are conjugate.