Residues, the compact chamber quotient, and the finite Coxeter sphere versus the contractible Davis cell

A simplex in the simplicial link of $q=w{W}_{\mathrm{\varnothing }}=\{w\}$ is a chain $q<q_1<\cdots <q_k$ in $WS$; this is the link convention of [F19]. Write $q_i=v_i{W}_{T_i}$. By [F4], $q\subseteq q_i$ forces $q_i=w{W}_{T_i}$, and the same criterion shows that $q_1<\cdots <q_k$ exactly when $\mathrm{\varnothing }⊊T_1⊊\cdots ⊊T_k$; conversely every such chain of nonempty spherical types gives a simplex in the link. By [F5], these are precisely the simplices of $\mathrm{sd}L$. The coface residue of any cell $q$ is the order subcomplex induced by $W{S}_{\ge q}$, since an order-complex simplex is a chain; this gives the asserted residue.

By [F9] the $W$-action on $\mathrm{\Sigma }$ is proper. By [F10] its orbit quotient is homeomorphic to $K$ and $K$ is a strict fundamental domain. Since $S$ is finite, $K$ is finite by [F3], hence the quotient is compact.

Suppose $W$ is finite. Then $S\in \mathbb{S}$ and $W_S=W$ is the maximum coset, so [F7] identifies $\mathrm{\Sigma }$ with the barycentric subdivision of the unique top cell $C_S$. If $n=0$, $C_S$ and $\mathrm{\Sigma }$ are points and their boundary is $\mathrm{\varnothing }=S^{-1}$. If $n\ge 1$, [F20] makes $C_S$ a closed $n$-ball with boundary $S^{n-1}$, and [F7] gives the same topology for $\mathrm{\Sigma }$; in particular $\mathrm{\Sigma }$ is contractible. Its boundary cells are indexed by the proper spherical cosets $w{W}_I$ with $I⊊S$; [F6,F8] give their dimensions $\mathrm{\mid }I\mathrm{\mid }$. By [F21,F23], the same coset labels a Coxeter simplex of dimension $n-\mathrm{\mid }I\mathrm{\mid }-1$, and its face incidence reverses coset inclusion. Thus the boundary cellulation and Coxeter triangulation are dual. Their face-poset flags correspond by reversing each finite coset chain, so the barycentric subdivisions are isomorphic. By [F11,F22], the standard chamber section $C\cap S^{n-1}$ is a fundamental $(n-1)$-simplex of the Coxeter complex; $K$ is instead the $n$-dimensional cone on $\mathrm{sd}L$ by [F3].

For a finite rank-two system with $m(s,t)=3$ or $4$, choose $d_s=d_t$. [F13] gives a regular $2m(s,t)$-gon, so the $A_2$ and $B_2$ top cells are a hexagon and an octagon; [F7] identifies their Davis complexes with the barycentric subdivisions of these cells. For the all-right-angled three-generator case, [F14] gives $s^2=t^2=1$ and $(st{)}^2=1$, so $st=(st{)}^{-1}=ts$ for each pair. The homomorphism $W\to (\mathbb{Z}\mathrm{/}2{)}^3$ sending each generator to its basis vector is therefore well-defined, and the homomorphism back sending the basis vectors to $a,b,c$ is well-defined by commutativity and involutivity; their composites fix generators, so $W\cong (\mathbb{Z}\mathrm{/}2{)}^3$. By [F15] the Coxeter form is diagonal with $B(e_s,e_s)=1$; [F16,F17] make each simple reflection flip just its own coordinate. Then $v_s^{(S)}=e_s$, $x_S={\sum }_s{d}_s{e}_s$, and the orbit consists of all sign vectors ${\sum }_s{\epsilon }_s{d}_s{e}_s$. Its convex hull is the product ${\prod }_s[-d_s{e}_s,d_s{e}_s]$, a box and a cube when the distances agree, by [F6]. The finite Davis complex is its barycentric subdivision by [F7].

If $W$ is infinite then $S\mathrm{\notin }\mathbb{S}$, so there is no top cell; its $0$-cells are indexed by all $w\in W$, so $\mathrm{\Sigma }$ is not a single finite polytope. For the universal Coxeter matrix, each distinct pair has label $\mathrm{\infty }$. Let $R$ be the set of finite words with no equal adjacent letters, including the empty word. For each $s\in S$, define a permutation ${\lambda }_s$ of $R$ by deleting an initial $s$ when present and otherwise prefixing $s$. Each ${\lambda }_s$ is an involution; because [F14] leaves only the relators $s^2$, the assignment extends to a homomorphism $W\to \mathrm{Sym}(R)$. With composition acting right-to-left, any word $s_1\cdots s_k\in R$ maps the empty word to $s_1\cdots s_k$, so no nonempty word in $R$ represents the identity. For distinct $s,t$, the alternating words $(st{)}^k$ lie in $R$ and map the empty word to distinct words of lengths $2k$; hence every subgroup generated by at least two generators is infinite and is not spherical. By [F1], the only spherical types are $\mathrm{\varnothing }$ and the singletons, so the only cells are vertices and edges; by [F8,F12], $\mathrm{\Sigma }$ is the Cayley graph. A closed path with no immediate backtracking has adjacent distinct edge labels, so its label is a nonempty word in $R$ representing $1$, impossible by the action just constructed. The Cayley graph is connected because $S$ generates $W$, hence it is a tree. For $\mathrm{\mid }S\mathrm{\mid }=2$ it is the bi-infinite line; for $\mathrm{\mid }S\mathrm{\mid }\ge 3$, distinct generators give distinct neighbors at each vertex, so it is a regular tree of valence $\mathrm{\mid }S\mathrm{\mid }$.

Fix the vertex $w{W}_{\mathrm{\varnothing }}=\{w\}$. Every incident cell is $q=w{W}_T$ by [F4]; in its $\dot{q}$-chart this vertex is $\rho (a)x_T$ with $a={\dot{q}}^{-1}w\in W_T$. At $x_T$, [F25] puts every orbit difference $\rho (u)x_T-x_T$ in $-V_{T,+}:=\{-{\sum }_{s\in T}{c}_s{e}_s:c_s\ge 0\}$, while $\rho (s)x_T-x_T=-2d_s{e}_s$ for every $s\in T$ by [F6, F15, F16, F17]. Thus the nonnegative hull of $C_T-x_T$, its tangent cone, is exactly $-V_{T,+}$. Because the $e_s$ are independent and $d_s>0$, its nonzero rays have the cross-section $\{-{\sum }_s{c}_s{e}_s:c_s\ge 0,{\sum }_s{c}_s=1\}$, a simplex with vertices labelled by $T$; radial normalization identifies this cross-section with the spherical link. The face indexed by $W_U$, $U\subseteq T$, has tangent cone $-V_{U,+}$ by the same argument for its orbit hull [F6, F25], so it contributes precisely the simplex face on $U$. Applying the isometry $\rho (a)$ gives the same labelled link at $\rho (a)x_T$. The empty type contributes the empty simplex. By [F24] two incident cells $w{W}_T,w{W}_{T^{\mathrm{\prime }}}$ meet in $w{W}_{T\cap T^{\mathrm{\prime }}}$, and the face isometries in [F7] identify their links along precisely the face on $T\cap T^{\mathrm{\prime }}$. These simplices are exactly the nerve $L$ by [F1]. The simplicial link from step 1.1 is its barycentric subdivision, in agreement with [F18].

Clauses (i)–(iv) follow from steps 1.1, 2.1, 1.2, 1.3, 1.4 and 1.5. All constructions are explicit and use only finite-dimensional coordinate calculations and finite case distinctions; no selection from an arbitrary family is used, so the Axiom of Choice is not needed.