The angular link of a vertex of the Davis complex is the large metric flag nerve

Fix $T\in \mathbb{S}$ and $s\in T$. The vectors $v_t^{(T)}$ ($t\in T$) are the $B$-dual basis of $V_T$ and $B_T$ is an inner product, so $B(x_T,e_s)={\sum }_{t\in T}{d}_{t}B(v_t^{(T)},e_s)=d_s$; the reflection formula of [F1] gives $\rho (s)x_T=x_T-2B(x_T,e_s)e_s$, that is $s{x}_T-x_T=-2d_s{e}_s\mathrm{\ne }0$. By [F4] the point $x_T$ is a vertex of the compact convex polyhedral cell $C_T$ of dimension $\mathrm{\mid }T\mathrm{\mid }$, and the nonempty faces of $C_T$ are exactly the sets $\mathrm{conv}(u{W}_U{x}_T)$, one for each coset $u{W}_U$ ($u\in W_T$, $U\subseteq T$).

By the inclusion criterion of [F5] a face $\mathrm{conv}(u{W}_U{x}_T)$ contains $x_T$ exactly when $\{x_T\}=\mathrm{conv}(W_{\mathrm{\varnothing }}{x}_T)\subseteq \mathrm{conv}(u{W}_U{x}_T)$, that is exactly when $W_{\mathrm{\varnothing }}\subseteq u{W}_U$, i.e. $1\in u{W}_U$, which forces $u{W}_U=W_U$. Hence the faces of $C_T$ through $x_T$ are precisely the sets $\mathrm{conv}(W_U{x}_T)$, $U\subseteq T$, so the faces of type $\{s\}$ through $x_T$ are precisely the segments $\mathrm{conv}(W_{\{s\}}{x}_T)=[x_T,\rho (s)x_T]$, the straight segment from $x_T$ to $\rho (s)x_T$ because $\rho (s)x_T-x_T=-2d_s{e}_s$. In particular $[x_T,s{x}_T]$ is the only $1$-face of $C_T$ through $x_T$ of type $\{s\}$, as asserted in (1).

The reflection formula in [F1] gives, for $x,y\in V$ and $a=B(x,e_r)$, $b=B(y,e_r)$, $B(\rho (r)x,\rho (r)y)=B(x-2a{e}_r,y-2b{e}_r)=B(x,y)-2ab-2ab+4ab=B(x,y)$, since $B(e_r,e_r)=1$. Every $u\in W$ is a finite word in the generators and $\rho$ is a homomorphism, so composition proves $B(\rho (u)x,\rho (u)y)=B(x,y)$. The facets of $C_T$ through $x_T$ are the faces $F_s:=\mathrm{conv}(W_{T\setminus \{s\}}{x}_T)$ for $s\in T$, by 1.1, 1.2 and the dimension statement in [F4]. Fix $s\in T$. Each generator $r\in T\setminus \{s\}$ fixes the quotient of $V_T$ by $V_{T\setminus \{s\}}$: the reflection formula gives $\rho (r)e_s-e_s\in V_{T\setminus \{s\}}$ and $\rho (r)V_{T\setminus \{s\}}=V_{T\setminus \{s\}}$. Thus for every $u\in W_{T\setminus \{s\}}$, $\rho (u{)}^{-1}{e}_s\in e_s+V_{T\setminus \{s\}}$ and $\rho (u{)}^{-1}{e}_t\in V_{T\setminus \{s\}}$ for $t\mathrm{\ne }s$. Pairing with the dual basis gives $B(\rho (u)v_s^{(T)},e_t)=B(v_s^{(T)},\rho (u{)}^{-1}{e}_t)={\delta }_{st}$ for every $t\in T$, so nondegeneracy of $B_T$ implies $\rho (u)v_s^{(T)}=v_s^{(T)}$. Consequently $B(v_s^{(T)},u{x}_T)=B(v_s^{(T)},x_T)=c_s$ for every such $u$, and $F_s$ lies in the supporting hyperplane $B(v_s^{(T)},y)=c_s$; [F5] puts $C_T$ in the half-space $B(v_s^{(T)},y)\le c_s$. Since $F_s$ has dimension $\mathrm{\mid }T\mathrm{\mid }-1$, it is exactly the equality face, and its inward normal is proportional to $-v_s^{(T)}$. By [F6], $T_{x_T}{C}_T=\{\xi \in V_T:B(v_s^{(T)},\xi )\le 0\text{ for every }s\in T\}$. Writing $\xi ={\sum }_{t\in T}{a}_t{e}_t$ gives $B(v_s^{(T)},\xi )=a_s$, so this is $\mathrm{cone}\{-e_s:s\in T\}$; these generators are extreme because each nonnegative combination ${\sum }_{t\mathrm{\ne }s}{\lambda }_t(-e_t)$ has $B(\cdot ,e_s)$-pairing ${\sum }_{t\mathrm{\ne }s}{\lambda }_t(-B(e_t,e_s))\ge 0$, whereas $B(-e_s,e_s)=-1$.

If distinct $s,t\in T$, then $W_{\{s,t\}}\le W_T$ is finite, so the order $m(s,t)$ of $st$ is finite. The unit vectors $-e_s,-e_t$ in the positive-definite plane $V_{\{s,t\}}\subseteq V_T$ have inner product $B(e_s,e_t)=-\cos (\pi \mathrm{/}m(s,t))$. Their local spherical edge length is therefore $\arccos (-\cos (\pi \mathrm{/}m(s,t)))=\pi -\pi \mathrm{/}m(s,t)\ge \pi \mathrm{/}2$, by [F1], [F2], [F6], [F14].

For spherical $T$, $B_T$ is an inner product by [F7], and the cell $w{W}_T$ of the Davis complex is identified with $C_T$ so that $w{W}_{\mathrm{\varnothing }}=\{w\}$ corresponds to $x_T$ and the $1$-cell $w{W}_{\{s\}}$, $s\in T$, corresponds to the unique $1$-face of type $\{s\}$ through the vertex; the direction of this $1$-cell at $w$ is therefore the ray ${\mathbb{R}}_{\ge 0}(\rho (s)x_T-x_T)={\mathbb{R}}_{\ge 0}(-e_s)$ by 1.1.

By [F6] the angular link of the vertex is ${\mathrm{Lk}}_{C_T}(x_T)=T_{x_T}{C}_T\cap S(V_T)$, the unit sphere in $(V_T,B_T)$ (Real and complex inner-product spaces and their induced length); by 1.3 and 1.4 its vertices are the unit vectors $-e_s$ with pairwise products $B(-e_s,-e_t)=B(e_s,e_t)$, that is with Gram matrix $B_T$. By the Gram construction [F8] the spherical simplex $\mathrm{\Sigma }(B_T)$ built on the unit vertices realizing $B_T$ is unique up to a linear isometry of its ambient inner product space; a linear isometry carrying its vertices to the $-e_s$ maps it onto ${\mathrm{Lk}}_{C_T}(x_T)$, and it preserves the angular distances $\arccos ⟨\cdot ,\cdot ⟩$, so it is an isometry of angular links [F15]. Moreover, for $U\subseteq T$ the face $\mathrm{conv}(W_U{x}_T)$ has vertex $x_T$, and by the same computation applied to the finite-type system $(W_U,U)$ its link at $x_T$ is $\mathrm{\Sigma }(B_U)$, the face of $\mathrm{\Sigma }(B_T)$ spanned by the vertices $-e_s$, $s\in U$, with the vertex map $-e_s\mapsto -e_s$.

For distinct $s,t\in S$, every word in $s,t$ reduces using $s^2=t^2=1$ to an alternating word; if $m(s,t)<\mathrm{\infty }$, each such word represents one of at most $2m(s,t)$ elements of the form $(st{)}^k$ or $s(st{)}^k$, so $W_{\{s,t\}}$ is finite. If $m(s,t)=\mathrm{\infty }$, the element $st$ has infinite order by the Coxeter-matrix convention, so $W_{\{s,t\}}$ is infinite. Hence $\{s,t\}$ is an edge of $L$ exactly when $m(s,t)<\mathrm{\infty }$. In that case its local edge-cell length is the value in 1.4; the direction of the $1$-cell $w{W}_{\{s\}}$ at $w$ is ${\mathbb{R}}_{\ge 0}(-e_s)$ by 1.5. These rays and their local edge lengths are independent of the positive tuple $(d_s)$. No global shortest-path equality in the whole link is used.

Fix $w\in W$. The cells of $\mathrm{\Sigma }$ containing $w$ are the cells $w{W}_T$, $T\in \mathbb{S}$, and the link of $w$ in $w{W}_T$ is ${\mathrm{Lk}}_{C_T}(x_T)$, identified with $\mathrm{\Sigma }(B_T)$ in 1.6; for $U\subseteq T$ the face $w{W}_U$ corresponds to the face $\mathrm{conv}(W_U{x}_T)$ and its direction face at $w$ is $\mathrm{\Sigma }(B_U)$. The cellulation glues these spherical simplices along those faces by the identity vertex maps $-e_s\mapsto -e_s$, compatible for all inclusions; [F17] gives a chain metric on each connected component, with the weak topology, and auxiliary path distance $+\mathrm{\infty }$ between components. Truncating at $\pi$ gives a metric on the whole link with the same weak topology: balls of radius less than $\pi$ are exactly the componentwise intrinsic balls. The angular-link gluing of [F9] identifies this glued link with ${\mathrm{Lk}}_{\mathrm{\Sigma }}(w)$. By the cell statement in [F7], its cells are exactly the spherical subsets $T\in \mathbb{S}$, with Gram matrices $B_T$, glued along their common faces, so it is the finite spherical complex $X=\mathrm{\mid }L{\mathrm{\mid }}_B$. The direction of $w{W}_{\{s\}}$ corresponds to vertex $s$ of $L$, and the cellwise link isometries induce the isometry of the glued polyhedral complexes and their truncated angular metrics by [F18].

By 2.1 every edge of $X$ has local length $\pi -\pi \mathrm{/}m(s,t)\ge \pi \mathrm{/}2$, so $X$ is large. For a pairwise adjacent set $T$ of vertices of $L$, the cosine test matrix in [F10] is $B_T$, including the empty matrix when $T=\mathrm{\varnothing }$. By [F7], $B_T$ is positive definite exactly when $W_T$ is finite, and by [F2] that is exactly when $T$ spans a simplex of $L$. Thus $X$ is metric flag. The almost-negative matrix of [F10] equals $B$: on edges its entry is $\cos (\pi -\pi \mathrm{/}m(s,t))=B(e_s,e_t)$ by 2.1 and [F1], while on non-edges $m(s,t)=\mathrm{\infty }$ by 2.1 and the entry is $B(e_s,e_t)=-1$. Therefore $X$ is a finite large metric flag complex with associated matrix $B$.

Let $U\in \mathbb{S}$ and $w\in W$. For a spherical $T\supseteq U$, use the $w$-chart of its cell $w{W}_T$, so $w{W}_U$ corresponds to $F_U:=\mathrm{conv}(W_U{x}_T)$. Its direction space is $V_U:=\mathrm{span}\{e_u:u\in U\}$: orbit differences lie in $V_U$ by the reflection formula, and the differences $u{x}_T-x_T=-2d_u{e}_u$ span it. The facets containing $F_U$ are exactly the $F_s$ of step 1.3 with $s\mathrm{\notin }U$, by the face-inclusion criterion [F5]. Thus at a relative interior point of $F_U$, [F6] gives $T_{F_U}{C}_T=\{\xi :B(v_s^{(T)},\xi )\le 0(s\in T\setminus U)\}=V_U+\mathrm{cone}\{-e_s:s\in T\setminus U\}$. Let $P_U$ be the $B_T$-orthogonal projection onto $V_U$. Intersecting this cone with $V_U^{\perp }$ gives exactly the cone generated by $-(e_s-P_U{e}_s)$: projection gives one inclusion, and subtracting the $V_U$ component of any such nonnegative combination gives the other. Their normalized Gram matrix is the Schur complement along $U$ from [F18], exactly the normal link of the face $U$ in $\mathrm{\Sigma }(B_T)$. In these $w$-charts the face maps of Finite Coxeter orbit polytopes, face isometries and their cocycle (3),(4) are $v\mapsto v+z_{T^{\mathrm{\prime }},T}$ for $U\subseteq T\subseteq T^{\mathrm{\prime }}$; their derivatives are inclusions, and the projection onto $V_U$ agrees on $V_T$. Thus they preserve the projected rays, and their cocycle makes the identifications agree on common cofaces. Consequently the link of $w{W}_U$ is obtained by gluing these spherical face links for $T\supseteq U$; its vertices are exactly the $s\mathrm{\notin }U$ for which $U\cup \{s\}$ is spherical, its cells are the $T^{\mathrm{\prime }}$ with $U\cup T^{\mathrm{\prime }}$ spherical, and its simplex Gram matrices are the iterated normalized Schur complements of $B$ along $U$, by [F10] and [F18]. This is the face link ${\mathrm{Lk}}_X(U)$. For $U=\mathrm{\varnothing }$ it is $X$, already finite large metric flag by step 3.1. For nonempty $U$, each link simplex Gram matrix is positive definite by the Schur-complement identity. Its off-diagonal entries are nonpositive: eliminating one vertex $v$ from a current positive-definite Gram matrix with nonpositive off-diagonal entries replaces an off-diagonal entry $c_{st}$ by $c_{st}-c_{sv}{c}_{tv}\le 0$; its new diagonal entries $1-c_{sv}^2$ are positive by positive definiteness, and normalization by their positive square roots preserves signs. Repeating over the vertices of $U$ proves largeness of every link. To check metric flagness, let $Q$ be any pairwise adjacent set in ${\mathrm{Lk}}_X(U)$. Then $U\cup Q$ is pairwise adjacent in $X$. The link cosine matrix on $Q$ is the positive-diagonal normalization of the Schur complement of $B_U$ in the matrix $B_{U\cup Q}$; this follows entrywise from orthogonal projection off $V_U$ and the link formula [F18]. Since $B_U$ is positive definite, the block Schur-complement identity makes this link matrix positive definite exactly when $B_{U\cup Q}$ is. By step 3.1 the latter is positive definite exactly when $U\cup Q$ spans a simplex of $X$, which is exactly when $Q$ spans a simplex of the link. The empty set passes the empty-matrix convention in [F10]. Thus every cell link is again finite large metric flag. Finally, [F12] identifies a sufficiently small ball at a point in the relative interior of $w{W}_U$ with the corresponding ball about $(0,o)$ in ${\mathbb{R}}^{\mathrm{\mid }U\mathrm{\mid }}\times C({\mathrm{Lk}}_{\mathrm{\Sigma }}(w{W}_U))$; the link computation is independent of $(d_s)$ because its directions are the rays in step 2.1.

By step 3.1 and step 4.1, $X$ and every face link are finite large metric flag complexes. Under the stated AC assumption, [F11] therefore makes all these links CAT(1). Its compact short-loop argument uses the untruncated intrinsic geodesic metrics of the components supplied by [F17]. A pair at angular distance $<\pi$ and every triangle of perimeter $<2\pi$ lie in one such component; all triangle sides are $<\pi$ by [F13]. Distances between side points are at most half that perimeter, hence also $<\pi$, so the spherical comparison tests agree before and after truncation. This reconciles clause (6) with the completed supplier. AC is used only through [F17] and [F11]; the direct calculations make no further choice selections. Together with the preceding calculations this proves clauses (1)–(6), with the scope of (7).