Davis Complex: CAT(0) Model for Coxeter Groups

• Davis Complex is a piecewise Euclidean CAT(0) cell complex constructed from the spherical subsets of a Coxeter group's generators, ensuring contractibility and unique geodesics.
• Its structure is defined by the combinatorics of the nerve and group presentation, enabling explicit computations in L2-(co)homology and orbifold Euler characteristics.
• The complex underpins diverse applications including topological rigidity, uniform lattice classification, and extensions to hyperbolic geometry and operator theory.

The Davis complex is a piecewise Euclidean CAT(0) cell complex associated to a Coxeter group, serving as a canonical geometric model with profound applications in geometric group theory, topology, and combinatorics. It provides a proper, cocompact, and isometric action for the group and underlies constructions of aspherical manifolds and orbicomplexes. Its structure is tightly determined by the group presentation and the combinatorics of the nerve, and its features are central to problems ranging from $L^2$-cohomology to rigidity phenomena and explicit geometric realizations.

1. Construction and Key Properties

Given a finitely generated Coxeter group $W$ with generators $S$ and relations of the form $(s_is_j)^{m_{ij}} = 1$ ($m_{ij} \in \{2,3,4,\dots,\infty\}$), the Davis complex $\Sigma$ is built as the geometric realization of the poset of spherical subsets of $S$. Vertices correspond to spherical subsets (those generating finite subgroups), and simplices are indexed by chains of inclusions among these subsets. The fundamental chamber $K$ is typically modeled as the cone on the barycentric subdivision of the nerve $L$ (the simplicial complex recording relations among generators). The complex is defined as the quotient $(W \times K) / \sim$ by a group action that glues chambers along mirrors associated to generators.

The Davis complex is a piecewise Euclidean CAT(0) space (Fu et al., 2017), ensuring contractibility, unique geodesics, and nonpositive curvature. Its one-skeleton can be identified with the Cayley graph of $W$. The proper, cocompact group action allows $\Sigma$ to serve as a classifying space $K(W,1)$ for torsion-free $W$, and as an orbifold model when torsion is present.

2. Generalization and The Flag Complex Conjecture

The Charney–Davis Flag Complex Conjecture originally asserted a sign for a linear combination:

$$(-1)^n \sum_{\sigma \in S} \left(\frac{-1}{2}\right)^{\dim \sigma + 1} \geq 0$$

where $S$ is a flag complex triangulating a $(2n-1)$-generalized homology sphere and the sum runs over all simplices (Cesnavicius, 2010). Interpreting the $1$-skeleta of simplices as Coxeter graphs, the paper introduces a Generalized Flag Complex Conjecture by extending edge weights beyond 2, associating to each simplex $\sigma$ a finite Coxeter group $W_\sigma$. It defines the weighted sum

$$\omega(S) = \sum_{\sigma \in S} \frac{(-1)^{\dim \sigma + 1}}{|W_\sigma|}$$

and proves the equivalence:

$(-1)^n \omega(S) \geq 0$

for complexes where each $\sigma$ has finite $W_\sigma$. Importantly, the equivalence means that topological invariants (e.g., the orbifold Euler characteristic) of the Davis complex for arbitrary finite Coxeter types can be studied via right-angled cases, streamlining analysis of its combinatorial and topological features.

3. Weighted $L^2$-(Co)homology and the Fattened Complex

A central problem is computing $L^2$-Betti numbers of the Davis complex, important for deep conjectures (Singer, Hopf-Thurston) (Mogilski, 2015). Weighted $L^2$-(co)homology assigns to $\Sigma$ a family of Hilbert chain complexes, constructed by deforming the group algebra inner product via a multiparameter $q = (q_s)_{s\in S}$, producing a weighted measure on cells:

$$\langle f,g \rangle_q = \sum_{\sigma} f(\sigma)g(\sigma)\mu_q(\sigma)$$

where $\mu_q(\sigma)$ quantifies the $q$-weight for cell $\sigma$. The boundary/coboundary operators are adjusted for adjointness. The $L^2_q$-Betti number is then

$$L^2_qb_i(\Sigma) = \dim_{\mathcal{N}_q(W)} L^2_qH_i(\Sigma)$$

using the von Neumann algebra dimension. For specific nerves, explicit formulas are obtained; for instance, if $L$ is the complete graph $K_n(3)$:

$$L^2_qb_2(\Sigma) = 1 - \frac{nq}{1+q} + \frac{n(n-1)q^3}{2(1+2q+2q^2+q^3)}$$

The construction of a fattened Davis complex $\mathcal{U}(W, K^f)$—a locally compact homology manifold with boundary that deformation retracts onto $\Sigma$—enables the use of Mayer–Vietoris and Poincaré duality techniques that are otherwise unavailable for the original complex.

4. Lattices, Covolumes, and Group Actions

The automorphism group $G$ of the Davis complex, defined as the set of isometries preserving its polyhedral structure (under the compact-open topology), is locally compact (Sercombe, 2015). Uniform lattices in $G$ (i.e., discrete subgroups with compact quotient) are crucial for arithmetic and topological classification. Prior work showed that if $G$ is discrete, the set $\mathcal{V}_u(G)$ of covolumes of uniform lattices is discrete. The converse does not always hold: for certain Coxeter systems with free generators or cyclic nerves, $\mathcal{V}_u(G)$ is non-discrete, containing rationals with denominators divisible by arbitrarily large powers of primes below a fixed integer. Explicit complexes of groups, developed to guarantee the universal cover is $\Sigma$, enable the construction of such uniform lattices, with covolume computed via Serre's formula as sums over reciprocal orders of vertex groups.

5. Topological Rigidity and Orbicomplexes

Topological rigidity is the property that homeomorphism type is determined by the fundamental group within a class of spaces. For Davis orbicomplexes—quotients of the Davis complex by a Coxeter group—this rigidity can fail (Stark, 2016). Examples are provided where finite-sheeted covers of the orbicomplex have isomorphic fundamental groups but are not homeomorphic. This non-rigidity impedes abstract commensurability classification and demonstrates that algebraic invariants alone do not suffice to distinguish topological types in this setting. Conversely, by restricting to subclasses defined by additional combinatorial constraints on the nerve (such as 3-convexity and non-repetitivity) and structure of singular sets, infinite families of topologically rigid quotients can be constructed (Wu, 2022).

6. Extensions and Generalizations

The Davis–Moussong complex is seen as a special case in a broader construction for "Dyer groups"—a generalized class encompassing Coxeter, right-angled Artin, and graph products of cyclic groups (Soergel, 2022). For any such group, a CAT(0) cell complex $\Sigma$ is constructed, combining Coxeter polytopes, cubes, and branched stars, with structure prescribed by scwols (simple categories without loops) and complexes of groups. This complex unifies and extends the geometric models provided by Davis and Salvetti complexes, with links analyzed via combinatorial and metric flag conditions to ensure nonpositive curvature.

7. Connections to Limit Roots, Imaginary Cones, and Convergence Theory

For infinite Coxeter groups, the Davis complex plays a role in understanding limit roots and imaginary cones (Fu et al., 2017). A natural $W$-equivariant embedding is defined from the Davis complex into the normalized imaginary cone $Z$, furnishing access to asymptotic properties of root systems. The convex hull of the set of limit roots is shown to correspond to the projection of $Z$, and the Davis complex—through its combinatorial and geometric structure—provides key insight into accumulation phenomena and group dynamics.

Within numerical analysis and optimization, the "Davis complex" analogy is sometimes invoked in operator theory. The Davis–Wielandt shell (for a bounded operator $T$) is the set $$\mathrm{DW}(T) = \{ ((Tx,x), \|Tx\|^2) : \|x\|=1 \}$$ in $\mathbb{C} \times \mathbb{R}_+$, and recent work has sharpened bounds on its radius (Bhunia et al., 2020). In optimization, the fixed-point set of Davis–Yin splitting is termed the "Davis Complex" (Lee et al., 2022). Scaled relative graph (SRG) theory enables geometric convergence analysis by linking operator behavior to contraction factors in the complex plane; improved estimates for linear rates and parameter selection are derived via this geometric framework.

8. Explicit Constructions and Maximal Subgroups

In lattice theory, explicit polygonal complexes whose universal cover is the Davis complex allow realization of maximal torsion-free subgroups (Norledge et al., 2016). By building coverings between suitable complexes of groups, one constructs finite-index torsion-free embeddings whose indices attain lower bounds set by orders of local groups. These methods yield amalgams of surface groups over free groups and have broad applications in the structure and classification of discrete subgroups acting on CAT(0) spaces.

9. Four-dimensional Hyperbolic Geometry

The Davis hyperbolic four-manifold $\mathcal{D}$ exemplifies higher-dimensional manifestations, constructed via reflection in a right-angled 120-cell. Analysis utilizing adjunction inequalities, genus-two totally geodesic surfaces, and Seiberg–Witten moduli space computations leads to vanishing of all Seiberg–Witten invariants for $\mathcal{D}$ (Lin et al., 11 Mar 2025). This result connects the Davis manifold with conjectures in four-manifold topology, particularly the LeBrun conjecture, and has implications for the existence of almost-complex structures, surface bundle structures, and extensions to arithmetic manifolds.

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In summary, the Davis complex serves as a central object linking Coxeter group theory, CAT(0) geometry, weighted homology, representation theory, optimization, and manifold topology. Its combinatorial structure, geometric realizations, and deep connections to algebraic invariants support broad theory and computation across mathematics, while its generalizations, rigidity and non-rigidity phenomena, and explicit constructions continue to advance understanding in geometric group theory and related fields.