Related papers: Coxeter systems with two-dimensional Davis-Vinberg…

In this paper, we show that the center of every Coxeter group is finite and isomorphic to $(\Z_2)^n$ for some $n\ge 0$. Moreover, for a Coxeter system $(W,S)$, we prove that $Z(W)=Z(W_{S\setminus\tilde{S}})$ and $Z(W_{\tilde{S}})=1$, where…

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…

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…

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…

Let $(W,S)$ be a Coxeter system, let $G$ be a group of symmetries of $(W,S)$ and let $f : W \to \GL (V)$ be the linear representation associated with a root basis $(V, \langle .,. \rangle, \Pi)$.We assume that $G \subset \GL (V)$, and that…

Associated to any Coxeter system $(W,S)$, there is a labeled simplicial complex $L$ and a contractible CW-complex $\Sigma_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $\Sigma_L$ admits a cellulation under which the…

Coxeter and Dynkin diagrams classify a wide variety of structures, most notably finite reflection groups, lattices having such groups as symmetries, compact simple Lie groups and complex simple Lie algebras. The simply laced or "ADE" Dynkin…

For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and…

Let $\Sigma$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $\Sigma$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The…

One possible way to obtain the quasicrystallographic structures is the projections of the higher dimensional lattices into 2D or 3D subspaces. In this work we introduce a general technique applicable to any higher dimensional lattice. We…

We define the Coxeter cochain complex of a Coxeter group (G,S) with coefficients in a Z[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We…

We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce…

In this fourth part, (with the notations of the preceding parts) we make the following hypothesis: $(W,S)$ is a Coxeter system, irreducible, $2$-spherical and $S$ is finite. Let $R:W\to GL(M)$ be a reducible reflection representation of…

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We…

A Coxeter group acts properly and cocompactly by isometries on the Davis complex for the group; we call the quotient of the Davis complex under this action the Davis orbicomplex for the group. We prove the set of finite covers of the Davis…

We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The…

The study of the set of limit roots associated to an infinite Coxeter group was initiated by Hohlweg, Labb\'{e} and Ripoll and further developed by Dyer, Hohlweg, P\'eaux and Ripoll. The Davis complex associated to a finitely generated…

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such…

Let $(W,S)$ be a Coxeter system of type $A$, so that $W$ can be identified with the symmetric group $\mathrm{Sym}(n)$ for some positive integer $n$ and $S$ with the set of simple transpositions $\{\,(i,i+1)\mid 1\leqslant i\leqslant…

We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to…