A Coxeter system is an ordered pair (W,S) where S is the generating set in a particular type of presentation for the Coxeter group W. A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions of a Coxeter group having special factors and special amalgamated subgroup are easily recognized from the presentation of the Coxeter group. If a Coxeter group is a subgroup of the fundamental group of a given graph of groups, then the Coxeter group is also the fundamental group of a graph of special subgroups, where each vertex and edge group is a subgroup of a conjugate of a vertex or edge group of the given graph of groups. A vertex group of an arbitrary graph of groups decomposition of a Coxeter group is shown to split into parts conjugate to special groups and parts that are subgroups of edge groups of the given decomposition. Several applications of the main theorem are produced, including the classification of maximal FA-subgroups of a finitely generated Coxeter group as all conjugates of certain special subgroups.
In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups.
The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, yielding a generalization of it to the family of so-called parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group every element is a parabolic quasi-Coxeter element). We show that such an element admits an analogue of the cycle decomposition. Elements which are not in this family still admit a generalized cycle decomposition, but it is not unique in general.
A group G has property FA if G fixes a point of every tree on which G acts without inversions. We prove that every Coxeter system of finite rank has a visual JSJ decomposition over subgroups with property FA. As an application, we reduce the twist conjecture to Coxeter systems that are indecomposable with respect to amalgamated products over visual subgroups with property FA.
We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that a non-spherical and non-affine Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. We prove that a Coxeter group has a decomposition as a direct product of indecomposable groups, and that such a decomposition is unique up to a central automorphism and a permutation of the factors. We prove that a Coxeter group has a virtual decomposition as a direct product of strongly indecomposable groups, and that such a decomposition is unique up…
Let $(W,S)$ be a Coxeter system, let $S=I \dot{\cup} J$ be a partition of $S$ such that no element of $I$ is conjugate to an element of $J$, let $\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\widetilde{W}$ be the subgroup of $W$ generated by $\widetilde{J}$. We show that $W=\widetilde{W} \rtimes W_I$ and that $(\widetilde{W},\widetilde{J})$ is a Coxeter system.
We provide examples of groups which are indecomposable by direct product, and more generally which are uniquely decomposable in direct products of indecomposable groups. Examples include Coxeter groups, for which we give an alternative approach to recent results of L. Paris. For a finitely generated linear group Gamma, we establish an upper bound on the number of factors of which Gamma can be the direct product. If Gamma is moreover torsion-free, it follows that Gamma is uniquely decomposable in direct product of indecomposable groups.
Coxeter decompositions of hyperbolic simplices where studied in math.MG/0212010 and math.MG/0210067. In this paper we use the methods of these works to classify Coxeter decompositions of bounded convex pyramids and triangular prisms in the hyperbolic space H^3.
We classify Coxeter decompositions of hyperbolic tetrahedra, i.e. simplices in the hyperbolic space H^3. The paper completes the classification of Coxeter decompositions of hyperbolic simplices.
It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. Croke and Kleiner present a right-angled Artin group with more than one visual boundary. In this paper we present a right-angled Coxeter group with non-unique equivariant visual boundary. The main theorem is that if right-angled Coxeter groups act geometrically on a Croke-Kleiner spaces, then the local angles in those spaces all have to be right angles. We present a specific right-angled Coxeter group with non-unique equivariant visual boundary. However, we conjecture that the right an- …
Let X be a space of constant curvature and P be a convex polyhedron in X. A Coxeter decomposition of the polyhedron P is a decomposition of P into finitely many Coxeter polyhedra, such that any two polyhedra having a common facet are symmetric with respect to this facet. In this paper we classify Coxeter decompositions of simplices in hyperbolic space of dimension greater than 3. The problem is close to the classification of the finite index subgroups in the discrete hyperbolic reflection groups.
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.
The idea of "JSJ-decompositions" for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of 2-sided, incompressible tori that separate a given closed irreducible 3-manifold into pieces with strong topological structure. Sela introduced the idea of JSJ-decompositions for groups, an idea that has flourished in a variety of directions. The general idea is to consider a certain class X of groups and splittings of groups in X by groups in another class Y. E.g. Rips and Sela considered splittings …
Given a class of groups C, a group G is strongly accessible over C if there is a bound on the number of terms in a sequence L(1), L(2), ..., L(n) of graph of groups decompositions of G with edge groups in C such that L(1) is the trivial decomposition (with 1-vertex) and for i>1, L(i) is obtained from L(i-1) by non-trivially and compatibly splitting a vertex group of L(i-1) over a group in C, replacing this vertex group by the splitting and then reducing. If H and K are subgroups of a group G then H is smaller than K if H intersect K has finite index in H and infinite index in K. The minimal sp…
We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.
The following results are proved: The center of any finite index subgroup of an irreducible, infinite, non-affine Coxeter group is trivial; Any finite index subgroup of an irreducible, infinite, non-affine Coxeter group cannot be expressed as a product of two nontrivial subgroups. These two theorems imply a unique decomposition theorem for a class of Coxeter groups. We also obtain that the orbit of each element other than the identity under the conjugation action in an irreducible, infinite, non-affine Coxeter group is an infinite set. This implies that an irreducible, infinite Coxeter group i…
Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\si_1\si_2 cdots\si_d$ of a Coxeter element $c$ of $W$, such that $\si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$, and such that the factorisation is "minimal" in the sense that the sum of the ranks of the $T_i$'s, $i=1,2,...,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author. The type $A_n$ d…
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.
In this work we introduce a new combinatorial notion of boundary $\Re C$ of an $ω$-dimensional cubing $C$. $\Re C$ is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of $C$, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When $C$ arises as the dual of a cubulation -- or discrete system of halfspaces -- $\HH$ of a CAT(0) space $X$ (for example, the Niblo-Reeves cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we show how $\HH$ induces a function $ρ:\bd X\to\Re…
To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\cal X}^r(X)$ which we call {\it the reflection tree of graphs $X$}. This space is of topological dimension $\le1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $Γ$ for which $X$ is the geometric realization) then the visual boundary $\partial_\infty(W,S)$ of the right angled Coxeter system $(W,S)$ with the nerve isomorphic to $Γ$ is homeomorphic to ${\cal X}^r(X)$. For each $X$, this yields in particular many w…
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group $W$ afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of $W$ related to the descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of the Orlik-Solomon algebra have parallel, graded de…
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about the factorization of the Alexander-Conway polynomial permit to connect Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. One uses reconstractions of A'Campo links and Coxeter links. There is formulated a conjecture connect…
We study global fixed points for actions of Coxeter groups on nonpositively curved singular spaces. In particular, we consider property FA_n, an analogue of Serre's property FA for actions on CAT(0) complexes. Property FA_n has implications for irreducible representations and complex of groups decompositions. In this paper, we give a specific condition on Coxeter presentations that implies FA_n and show that this condition is in fact equivalent to FA_n for n=1 and 2. As part of the proof, we compute the Gersten-Stallings angles between special subgroups of Coxeter groups.
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in…
An element of a Coxeter group $W$ is called fully commutative if any two of its reduced decompositions can be related by a series of transpositions of adjacent commuting generators. In the preprint "Fully commutative elements in finite and affine Coxeter groups" (arXiv:1402.2166), R. Biagioli and the authors proved among other things that, for each irreducible affine Coxeter group, the sequence counting fully commutative elements with respect to length is ultimately periodic. In the present work, we study this sequence in its periodic part for each of these groups, and in particular we determi…
An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group W is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied in Boothby et al.. In particular the authors enumerated cyclically fully commutative elements in all Coxeter groups having a finite number of them. In this work we characterize and enumerate cyclically fully commutative elements according to their Coxeter length in all finite or aff…
Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our construction we identify a large class of such groups for which the JSJ tree, and hence the visual boundary, is a complete quasi-isometry invariant, and thus the quasi-isometry problem is decidable. We also give a direct proof of the fact that among the Coxeter groups we consider, the coco…
This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition.
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the "topograph," Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell's equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({\mathbb Z})$ and the Coxeter group of type $(3,\infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway's topograph, and generalizations to …
Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $Σ$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(Σ)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(Σ)$ are t…
The numbers game is a one-player game played on a finite simple graph with certain ``amplitudes'' assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. In particular, Eriksson connects certain geometric representations of Coxeter groups with games on graphs with certain real number amplitudes. Games played on such graphs are ``E-games.'' Here we in…
A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve is a complete graph on $n$ vertices for $n\geq 5$. The construction depends on a slight extension of Sierpiński's theorem on embedding $1$--dimensional planar compacta into the Sierpiński carpet. See the appendix for a brief erratum.
We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well-known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton's formula for the number of chains of given length in the non-crossing partitions lattice NCP_W. Our proof uses the properties of the…
In this paper we consider a function $L(\vec{x})=L(a,b,c,d;e;f,g)$, which can be written as a linear combination of two Saalschützian ${}_4F_3(1)$ hypergeometric series or as a very-well-poised ${}_7F_6(1)$ hypergeometric series. We explore two-term and three-term relations satisfied by the $L$ function and put them in the framework of group theory. We prove a fundamental two-term relation satisfied by the $L$ function and show that this relation implies that the Coxeter group $W(D_5)$, which has 1920 elements, is an invariance group for $L(\vec{x})$. The invariance relations for $L(\vec{x})$ …
For every finite Coxeter group $Γ$, each positive braids in the corresponding braid group admits a unique decomposition as a finite sequence of elements of $Γ$, the so-called Garside-normal form.The study of the associated adjacency matrix $Adj(Γ)$ allows to count the number of Garside-normal form of a given length.In this paper we prove that the characteristic polynomial of $Adj(B_n)$ divides the one of $Adj(B_{n+1})$. The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type $A$. We observe that this does not hold for type $D$. T…
