Coxeter systems with two-dimensional Davis-Vinberg complexes

Maximal torsion-free subgroups of certain lattices of hyperbolic buildings and Davis complexes

We give an explicit construction of a maximal torsion-free finite-index subgroup of a certain type of Coxeter group. The subgroup is constructed as the fundamental group of a finite and non-positively curved polygonal complex. First we consider the special case where the universal cover of this polygonal complex is a hyperbolic building, and we construct finite-index embeddings of the fundamental group into certain cocompact lattices of the building. We show that in this special case the fundamental group is an amalgam of surface groups over free groups. We then consider the general case, and …

Existence, covolumes and infinite generation of lattices for Davis complexes

Let $Σ$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $Σ$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The Coxeter group W may be regarded as a uniform lattice in G. We show that many such G also admit a nonuniform lattice $Γ$, and an infinite family of uniform lattices with covolumes converging to that of $Γ$. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice $Γ$ is not finitely generated. Examples of $Σ$ to which ou…

Coxeter群在融合环上的表示与超平面补集

Representations of Coxeter groups over fusion rings and hyperplane complements

本文研究Coxeter群在融合环上的忠实实现及其Vinberg系统，证明其诱导超平面补集嵌入，从而为Artin–Tits群间的强可容许同态提供几何实现。

Reflection subgroups of Coxeter groups

We use geometry of Davis complex of a Coxeter group to prove the following result: if G is an infinite indecomposable Coxeter group and $H\subset G$ is a finite index reflection subgroup then the rank of H is not less than the rank of G. This generalizes results of math/0305093. We also describe some properties of the nerves of the group and the subgroup in the case of equal ranks.

在Davis复形积中的简单晶格

Simple Lattices in Products of Davis Complexes

本文研究了在特定Davis复形积中构造简单一致晶格的方法，引入了Burger-Mozes通用群的类比，并提供了一个局部标准来判断顶点传递群在通用群中的稠密性。

Divergence in right-angled Coxeter groups

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.

Rigidity of two-dimensional Coxeter groups

A Coxeter system is called two-dimensional if its associated Davis complex is two-dimensional (equivalently, every spherical subgroup has rank less than or equal to 2). We prove that given a two-dimensional system (W,S) and any other system (W,S') which yields the same reflections, the diagrams corresponding to these systems are isomorphic, up to the operation of diagram twisting defined by N. Brady, J. McCammond, B. Muhlherr, and W. Neumann. As a step in the proof of this result, certain two-dimensional groups are shown to be reflection rigid in the sense defined by the above authors, and a r…

Simplicite de groupes d'automorphismes d'espaces a courbure negative

We prove that numerous negatively curved simply connected locally compact polyhedral complexes, admitting a discrete cocompact group of automorphisms, have automorphism groups which are locally compact, uncountable, non linear and virtually simple. Examples include hyperbolic buildings, Cayley graphs of word hyperbolic Coxeter systems, and generalizations of cubical complexes, that we call even polyhedral complexes. We use tools introduced by Tits in the case of automorphism groups of trees, and Davis-Moussong's geometric realisation of Coxeter systems.

A family of uniform lattices acting on a Davis complex with a non-discrete set of covolumes

Let $(W,S)$ be a Coxeter system with Davis complex $Σ$. The polyhedral automorphism group $G$ of $Σ$ is a locally compact group under the compact-open topology. If $G$ is a discrete group (as characterised by Haglund--Paulin), then the set $\mathcal V_u(G)$ of uniform lattices in $G$ is discrete. Whether the converse is true remains an open problem. Under certain assumptions on $(W,S)$, we show that $\mathcal V_u(G)$ is non-discrete and contains rationals (in lowest form) with denominators divisible by arbitrarily large powers of any prime less than a fixed integer. We explicitly construct our…

CAT(0) cubical complexes for graph products of finitely generated abelian groups

We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it is related to the embedding into a right-angled Coxeter group described by Davis and Januszkiewicz. We compare the approaches and also adapt the argument that the action extends to finite index supergroup that is a graph product of finite groups.

Geometrization of 3-dimensional Coxeter orbifolds and Singer's conjecture

Associated to any Coxeter system $(W,S)$, there is a labeled simplicial complex $L$ and a contractible CW-complex $Σ_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $Σ_L$ admits a cellulation under which the nerve of each vertex is $L$. It follows that if $L$ is a triangulation of $\mathbb{S}^{n-1}$, then $Σ_L$ is a contractible $n$-manifold. In this case, the orbit space, $K_L:=Σ_L/W$, is a \emph{Coxeter orbifold}. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits along Euclidean suborbifolds into t…

Systolizing buildings

We introduce a construction turning some Coxeter and Davis realizations of buildings into systolic complexes. Consequently groups acting geometrically on buildings of triangle types distinct from $(2,4,4)$, $(2,4,5)$, $(2,5,5)$, and various rank $4$ types are systolic.

边细分与直角 Coxeter 群的 $L^2$-同调

Edge subdivisions and the $L^2$-homology of right-angled Coxeter groups

本文研究直角 Coxeter 群 Davis 复形的 $L^2$-同调消失问题，给出边细分下保持 Singer 猜想成立的条件，验证了若干重心细分情形，并构造了挠增长版本猜想的反例。

粗粒度立方体刚性

Coarse cubical rigidity

研究显示，许多右角艺术和科克塞特群的所有紧致立方体划分在粗粒度上看起来相同，诱导相同的粗粒度中位结构。该研究为非超几何群提供了此类性质的第一个例子。