The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K)

The poset $P:=WS\cup \{\mathrm{\varnothing }\}$ with $\mathrm{\varnothing }$ below every coset [F1]: for $p=w{W}_T$ the principal down-set $P_{\le p}$ consists of $\mathrm{\varnothing }$ and the cosets $w^{\mathrm{\prime }}{W}_U\subseteq w{W}_T$, which by [F2] are exactly the cosets $w^{\mathrm{\prime }}{W}_U$ with $U\subseteq T$ and $w^{\mathrm{\prime }}\in w{W}_T$; it is finite because $W_T$ and the set of subsets of the finite $T$ are finite. The map $w^{\mathrm{\prime }}{W}_U\mapsto \mathrm{conv}((w^{-1}{w}^{\mathrm{\prime }})W_U{x}_T)$ is a bijection from $P_{\le p}\setminus \{\mathrm{\varnothing }\}$ onto the nonempty faces of $C_T$ by [F3], it is order-preserving and reflecting by [F2] and [F3], and extending it by $\mathrm{\varnothing }\mapsto \mathrm{\varnothing }$ sends the formal least element to the empty face; hence $P_{\le p}$ is isomorphic to the face poset of $C_T$.

Gluing data. For $p=\dot{p}{W}_U\le q=\dot{q}{W}_T$, [F2] gives $U\subseteq T$ and [F7] gives $g:={\dot{q}}^{-1}\dot{p}\in W_T$. Put $C_p:=C_U$, $C_q:=C_T$ and $h_{p,q}:={\phi }_g^{T,U}$; by [F3] this is an isometry from $C_p$ onto the face of $C_q$ indexed by $p$. If $p\le q\le r$ with types $U\subseteq V\subseteq T$, then [F3] gives $h_{q,r}\circ h_{p,q}={\phi }_{{\dot{r}}^{-1}\dot{q}}^{T,V}\circ {\phi }_{{\dot{q}}^{-1}\dot{p}}^{V,U}={\phi }_{({\dot{r}}^{-1}\dot{q})({\dot{q}}^{-1}\dot{p})}^{T,U}={\phi }_{{\dot{r}}^{-1}\dot{p}}^{T,U}=h_{p,r}$, so the cocycle condition of [F4] holds. The unique representatives in [F7] make every map independent of notation for coset representatives; no choice of representatives is made.

Intersection condition. Let $p=w{W}_T$, $q=w^{\mathrm{\prime }}{W}_{T^{\mathrm{\prime }}}$ be cosets. For $y\in C_p$, define its address $r\le p$ to be the unique spherical coset indexing the face whose relative interior contains $y$; existence and uniqueness are the face decomposition of the polytope $C_T$ in [F3]. If $p\le q$, write $p=\dot{p}{W}_U$, $q=\dot{q}{W}_V$, and $r=\dot{p}u{W}_J$. The map $h_{p,q}={\phi }_{{\dot{q}}^{-1}\dot{p}}^{V,U}$ sends the face of $C_U$ indexed by $u{W}_J$ to the face of $C_V$ indexed by $({\dot{q}}^{-1}\dot{p})u{W}_J$ [F3], which has global label $\dot{q}({\dot{q}}^{-1}\dot{p})u{W}_J=r$; hence it preserves the address. For a point $y\in C_p$ with address $r$, set $\xi (y):=h_{r,p}^{-1}(y)\in C_r$. If $p\le q$, the cocycle gives $h_{r,q}=h_{p,q}\circ h_{r,p}$, so $\xi$ is unchanged by the generating identification $y\sim h_{p,q}(y)$; it is also unchanged by the inverse identification. Therefore equivalent points have the same address and the same $\xi$. Conversely, if $y\in C_p$ and $y^{\mathrm{\prime }}\in C_q$ have the same address $r$ and the same coordinate $\xi$, each is identified with that common point of $C_r$, so $y\sim y^{\mathrm{\prime }}$. Thus equivalence is exactly equality of address and $\xi$. In particular each map $C_p\to X$ is injective, since equal classes from $C_p$ have the same address and coordinate and $h_{r,p}$ is injective. If two cell images meet, their common class has an address $r\le p,q$, hence $r\le p\wedge q$ and lies in the image of $C_{p\wedge q}$; conversely every point of $C_{p\wedge q}$ lies in both images, with the empty meet interpreted as the empty set. This is the intersection condition of [F4].

The standing hypotheses. $X$ is connected: the $1$-cells $C_{w{W}_{\{s\}}}=\mathrm{conv}(\{x_{\{s\}},s{x}_{\{s\}}\})$ join the $0$-cells with addresses $w{W}_{\mathrm{\varnothing }}=\{w\}$ and $ws{W}_{\mathrm{\varnothing }}=\{ws\}$ (which are faces of those $1$-cells by [F3]), so the image of the disjoint union contains a copy of the connected Cayley graph of $(W,S)$ on the vertices $w$; and every cell $C_{w{W}_T}$ contains the $0$-cell of $w$ as the face indexed by $w{W}_{\mathrm{\varnothing }}\le w{W}_T$ [F3], so every cell is attached to that graph and $X$ is connected. It is locally finite: by [step 2.1] the cells whose image contains the class of a point $y$ of address $r=w_r{W}_{T(r)}$ are exactly the cells $C_q$ with $q\ge r$, and by [F2] a coset containing $w_r{W}_{T(r)}$ has the form $w_r{W}_{T^{\mathrm{\prime }}}$ with $T^{\mathrm{\prime }}\supseteq T(r)$; these are finitely many because $S$ is finite. There are finitely many shapes because the cells are the $C_T$, $T\in \mathbb{S}$, and $\mathbb{S}$ is finite.

Conclusion of (1). By [step 1.1] the shape $P$ has finite down-sets isomorphic to face posets of the cells $C_T$; by [step 1.2] the maps $h_{p,q}$ are the face isometries of a gluing satisfying the cocycle condition; by [step 2.1] the intersection condition holds; by [step 3.1] the hypotheses (H1)-(H3) hold. Hence the metric theorem [F10] applies: the chain metric $d$ is a metric on $X$ inducing the weak topology, and every closed $d$-bounded subset of $X$ is compact, so $(X,d)$ is complete and proper. The canonical map $f\text{\hspace{0.17em}⁣}:\mathrm{\mid }WS\mathrm{\mid }\to X$ of [F9] is a bijection and is affine on each simplex of $\mathrm{\mid }WS\mathrm{\mid }$; it is continuous because $\mathrm{\mid }WS\mathrm{\mid }$ carries the weak topology and each restriction to a closed simplex is affine; and its inverse is described on each closed cell of $X$ by the carrier-simplex coordinates of [F9], hence is also affine on each simplex of the subdivision and continuous. So $f$ is a homeomorphism $\mathrm{\Sigma }\cong X$, and by [F9] it carries the subposet $W{S}_{\le w{W}_T}$ onto the barycentric subdivision of the cell $C_T$.

Clause (2), incidence. Under the homeomorphism $\mathrm{\Sigma }\cong X$ the cells of $\mathrm{\Sigma }$ are the images of the cells $C_{w{W}_T}$ of dimension $\mathrm{\mid }T\mathrm{\mid }$ by [F3] and [step 4.1]; $W$ acts on the set of cosets of each fixed type transitively by [F2], so there is one orbit per spherical $T$; there are finitely many shapes since $\mathbb{S}$ is finite; and every closed cell meets only finitely many cells: the cells meeting the closed cell $w{W}_T$ are the $v{W}_V$ with $v{W}_V\cap w{W}_T\mathrm{\ne }\mathrm{\varnothing }$, equivalently $v\in w{W}_T{W}_V$ by [F2]; as $V$ ranges over the finitely many spherical subsets and $W_V$ and $W_T$ are finite, these are finitely many cells. The relative interiors of the cells partition $\mathrm{\Sigma }$ because two cells meet in the image of $C_{p\wedge q}$ [step 2.1], and within one cell the relative interiors of its faces partition it [F3]. The setwise stabilizer of the cell $w{W}_T$ is $w{W}_T{w}^{-1}$: $v\cdot w{W}_T=vw{W}_T$ equals $w{W}_T$ iff $w^{-1}vw\in W_T$ by [F2].

Clause (4). Let $\pi \text{\hspace{0.17em}⁣}:WS\to \mathbb{S}$ be the type map. On each simplex of $\mathrm{\Sigma }$ belonging to a chain $q_0\subset \cdots \subset q_k$, map the vertex $q_i$ to $\pi (q_i)$ and extend affinely; this defines a continuous map $p\text{\hspace{0.17em}⁣}:\mathrm{\Sigma }\to K$ because the type map preserves inclusions [F2] and $\mathrm{\Sigma }$ has the weak topology. It satisfies $p\circ j={\mathrm{i}\mathrm{d}}_K$ and is $W$-invariant because $\pi (vq)=\pi (q)$ [F2]. Every simplex is a left translate of a simplex of $K$: if $q_0={\dot{q}}_0{W}_{T_0}$, then for each $i$, [F2] gives $q_i={\dot{q}}_0{W}_{T_i}$, so left multiplication by ${\dot{q}}_0^{-1}$ takes the chain to $W_{T_0}\subseteq \cdots \subseteq W_{T_k}$. Thus the orbit projection $q_{\mathrm{\Sigma }}\text{\hspace{0.17em}⁣}:\mathrm{\Sigma }\to W\mathrm{\backslash }\mathrm{\Sigma }$ restricts to a surjection on $K$. If $x,x^{\mathrm{\prime }}\in K$ and $v\cdot x=x^{\mathrm{\prime }}$, then $W$-invariance gives $x=p(x)=p(v\cdot x)=p(x^{\mathrm{\prime }})=x^{\mathrm{\prime }}$, so $K$ meets each orbit exactly once. The induced map $\bar{p}\text{\hspace{0.17em}⁣}:W\mathrm{\backslash }\mathrm{\Sigma }\to K$ is continuous because $\bar{p}\circ q_{\mathrm{\Sigma }}=p$ and the quotient topology makes $q_{\mathrm{\Sigma }}$ a quotient map: for open $V\subseteq K$, $q_{\mathrm{\Sigma }}^{-1}({\bar{p}}^{-1}(V))=p^{-1}(V)$ is open. The restriction $q_{\mathrm{\Sigma }}{\mathrm{\mid }}_K$ is continuous and surjective, so $W\mathrm{\backslash }\mathrm{\Sigma }$ is compact as a continuous image of compact $K$ [F6]. Since $K$ is Hausdorff, [F6] makes $\bar{p}$ a homeomorphism. Hence $K$ is a strict fundamental domain.

Clause (3), action and properness. For a coset $q=w{W}_T$ and $v\in W$, put $a(v,q):={\dot{vq}}^{-1}v\dot{q}\in W_T$ by [F2], [F7], and define $L_v\text{\hspace{0.17em}⁣}:C_q\to C_{vq}$ in the type-$T$ charts by $L_v=\rho (a(v,q)){\mathrm{\mid }}_{C_T}$. This is an isometry because $a(v,q)\in W_T$ permutes the orbit vertices of $C_T$. Also $a(1,q)=1$, so $L_1=\mathrm{i}\mathrm{d}$, and $a(u,vq)a(v,q)={\dot{uvq}}^{-1}uv\dot{q}=a(uv,q)$, so $L_u{L}_v=L_{uv}$; these are the left-action identities [F8]. If $p\le q$ has types $U\subseteq T$, let $g:={\dot{q}}^{-1}\dot{p}$, $a_q:=a(v,q)$, $a_p:=a(v,p)$, and $g^{\mathrm{\prime }}:={\dot{vq}}^{-1}\dot{vp}=a_{q}g{a}_p^{-1}$. For $x\in C_U$, [F3] says $a_p\in W_U$ fixes $z_{T,U}$, so $L_v{\mathrm{\mid }}_{C_q}(h_{p,q}(x))=\rho (a_{q}g)(x+z_{T,U})=\rho (g^{\mathrm{\prime }})\rho (a_p)(x+z_{T,U})=h_{vp,vq}(L_v{\mathrm{\mid }}_{C_p}(x))$. Thus $L_v$ preserves the equivalence relation defining $X$ and descends to an action by cellwise isometries. It preserves the chain metric because it sends each chain to one of the same length, and its inverse is $L_{v^{-1}}$. Each $L_v{\mathrm{\mid }}_{C_q}$ carries the vertices of $C_q$ to those of $C_{vq}$, hence sends their barycentres to each other; the affine barycentric maps on carrier simplices show that this action agrees, under [step 4.1], with the natural left action on $\mathrm{\Sigma }$. For properness, let $C\subseteq \mathrm{\Sigma }$ be compact. By [F14], every point has a neighborhood contained in a finite closed star, hence meeting only finitely many cells; finitely many such neighborhoods cover $C$, so $C$ meets only finitely many cells. For cells $u{W}_T$ and $v{W}_{T^{\mathrm{\prime }}}$, the set of $w$ with $w(u{W}_T)\cap v{W}_{T^{\mathrm{\prime }}}\mathrm{\ne }\mathrm{\varnothing }$ equals $v{W}_{T^{\mathrm{\prime }}}{W}_T{u}^{-1}$: [F2] says exactly that $(wu{)}^{-1}v\in W_T{W}_{T^{\mathrm{\prime }}}$. This set is finite because $W_T$ and $W_{T^{\mathrm{\prime }}}$ are finite. There are only finitely many pairs of cells meeting $C$, so only finitely many $w$ satisfy $wC\cap C\mathrm{\ne }\mathrm{\varnothing }$.

Clause (5). Let $R$ be the relation on $W\times K$ in the statement and put $S(x)=\{s:x\in K_s\}$. If the carrier chain of $x\in K$ is $T_0\subset \cdots \subset T_k$, then $x\in K_s=\mathrm{\mid }{\mathbb{S}}_{\ge \{s\}}\mathrm{\mid }$ exactly when every vertex of that carrier chain contains $s$, which is equivalent to $s\in T_0$. Thus $S(x)=T_0$ and the subgroup it generates is $W_{T_0}$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups). For $u\in W_{T_0}$, the chamber carrier chain $W_{T_0}\subset \cdots \subset W_{T_k}$ has each vertex fixed by $u$, since [F2] gives $u{W}_{T_i}=W_{T_i}$; hence $u$ fixes $j(x)$. The relation $R$ is an equivalence relation because on each fibre over $x$ it is equality of left cosets of the subgroup $W_{T_0}$, and $[w,x]\mapsto w\cdot j(x)$ is well defined by the fixed-point calculation. It is $W$-equivariant and surjective: a point with carrier chain $q_0\subset \cdots \subset q_k$ has, by [F2], the form ${\dot{q}}_0{W}_{T_0}\subset \cdots \subset {\dot{q}}_0{W}_{T_k}$; its barycentric coordinates give $x$ in the chamber simplex on $T_0\subset \cdots \subset T_k$, and its image is ${\dot{q}}_0\cdot j(x)$. It is injective: if $w\cdot j(x)=w^{\mathrm{\prime }}\cdot j(x^{\mathrm{\prime }})$, equality of carrier simplices and barycentric coordinates gives $x=x^{\mathrm{\prime }}$ and $w{W}_{T_0}=w^{\mathrm{\prime }}{W}_{T_0}$; therefore $w^{-1}{w}^{\mathrm{\prime }}\in W_{T_0}=W_{S(x)}$ and $[w,x]=[w^{\mathrm{\prime }},x^{\mathrm{\prime }}]$. For continuity, the map $F\text{\hspace{0.17em}⁣}:W\times K\to \mathrm{\Sigma }$, $F(w,x)=w\cdot j(x)$, is continuous: the slices $\{w\}\times K$ are open and its restriction to each is the continuous map $x\mapsto w\cdot j(x)$. It is constant on $R$-classes, so it descends continuously through the quotient projection by [F11]. For the inverse, on a simplex with chain $q_0\subset \cdots \subset q_k$, use the unique representative ${\dot{q}}_0$ and the affine barycentric-coordinate map to its chamber simplex in $K$, then send that point $x$ to $[{\dot{q}}_0,x]$. This is continuous on the simplex by the product and quotient topologies [F11]. These formulas agree on common faces: when the minimum coset rises to $q_j$, both representatives lie in $q_j$, so their quotient classes agree because ${\dot{q}}_0^{-1}{\dot{q}}_j\in W_{T_j}=W_{S(x)}$. The weak topology of $\mathrm{\Sigma }$ is simplexwise, so the inverse is continuous. The two maps are mutually inverse by the carrier-chain construction, proving the claimed homeomorphism.

Clause (2), point stabilizers. Let $q=w{W}_T$ and let $y$ lie in the relative interior of its cell, with coordinate $y^{\mathrm{\prime }}\in C_T$ in the $\dot{q}$-chart. Let $w_0\in W_T$, $I\subseteq T$ be determined by $y^{\mathrm{\prime }}\in w_0{C}_I^T$ (relative interior of a chamber face, [F5]). Let $z$ be the point of the copy $C_{W_T}=C_T$ corresponding to $y^{\mathrm{\prime }}$ under the barycentric subdivision of [step 4.1]. For every $v\in W_T$, $a(v,W_T)=v$, so [step 6.1] makes the action on this copy exactly $\rho (v)$, and $y$ corresponds to $\dot{q}\cdot z$ under $\mathrm{\Sigma }\cong X$. The subdivision pairs each vertex $u{W}_U$ with the face $\mathrm{conv}(u{W}_U{x}_T)$ by [step 1.1], and $\rho (v)$ carries this face to $\mathrm{conv}(vu{W}_U{x}_T)$; as an isometry it carries each face barycentre and its barycentric coordinates to the corresponding ones. Thus ${\mathrm{Stab}}_W(z)\cap W_T={\mathrm{Stab}}_{W_T}(y^{\mathrm{\prime }})=w_0{W}_I{w}_0^{-1}$ by [F5], using the stabilizer definition [F12]. Any element fixing $z$ preserves the cell whose relative interior contains $z$; the cell is unique by [step 5.1], and its setwise stabilizer is $W_T$ by [step 5.1]. Hence ${\mathrm{Stab}}_W(z)=w_0{W}_I{w}_0^{-1}$ and ${\mathrm{Stab}}_W(y)=\dot{q}{\mathrm{Stab}}_W(z){\dot{q}}^{-1}=(\dot{q}{w}_0)W_I(\dot{q}{w}_0{)}^{-1}$. This is a conjugate of the spherical parabolic $W_I$, and it lies in the setwise stabilizer $w{W}_T{w}^{-1}$ because $w_0{W}_I{w}_0^{-1}\le W_T$ and $\dot{q}\in w{W}_T$.

The clauses are proved: (1) is [step 4.1], (2) is [step 5.1] with [step 7.1], (3) is [step 6.1], (4) is [step 5.2] and (5) is [step 6.2]. No Choice is used: coset charts use the unique minimum-length representatives of [F7], all chamber and cell models are finite, and the topological and metric arguments use no selection from an arbitrary family.