Point stabilizers, vertex residues, and rank-two boundary words

Put ${\mathrm{\Phi }}_v:=\{\alpha \in \mathrm{\Phi }:B(v,\alpha )\in \mathbb{Z}\}$ and $V_v:=\mathrm{span}{\mathrm{\Phi }}_v$. This is finite and spans $V_v$. If $\alpha ,\beta \in {\mathrm{\Phi }}_v$, then $s_{\alpha }\beta =\beta -B(\beta ,{\alpha }^{\vee })\alpha \in \mathrm{\Phi }$ and $B(v,s_{\alpha }\beta )=B(v,\beta )-B(\beta ,{\alpha }^{\vee })B(v,\alpha )\in \mathbb{Z}$; hence $s_{\alpha }\beta \in {\mathrm{\Phi }}_v$. The reducedness and crystallographic conditions restrict from $\mathrm{\Phi }$, so ${\mathrm{\Phi }}_v$ is a reduced crystallographic root system in $V_v$. Its root hyperplanes translated through $v$ are exactly the walls in $\mathcal{A}(v)$.

If $\mathrm{\Phi }=\mathrm{\varnothing }$, put $V=\{0\}$ and $A=\bar{A}=\{0\}$. Otherwise, for each irreducible component $i$ let $V_i$ be the vertex list of the simplex $\overline{A_i}$ from [F13] and [F27], put $V:={\prod }_i{V}_i$, and regard $V$ as a finite subset of $E={⨁}_i{E}_i$. By [F27], each component point has barycentric coordinates in its listed vertices. For $x_i={\sum }_j{t}_{ij}{v}_{ij}\in \overline{A_i}$, the product weights ${\prod }_i{t}_{i,j_i}$ on tuples $(v_{i,j_i}{)}_i\in V$ are nonnegative, sum to $1$, and their convex combination is $(x_i{)}_i$; hence $\bar{A}\subseteq \mathrm{co}(V)$. Conversely, $\bar{A}$ is a convex product of simplices and contains $V$, so [F24] gives $\mathrm{co}(V)\subseteq \bar{A}$. Thus $\bar{A}=\mathrm{co}(V)$. For the empty system the same equality holds with $V=\{0\}$. For any alcove $C$, choose $y_0\in C$. By [F17] it is open, so choose $R>0$ with $B_E(y_0,R)\subseteq C$. By [F20] and [F21], choose $0<\rho <R$ such that $K_0:={\bar{B}}_E(y_0,\rho )$ is metric-compact, hence compact in the norm topology by [F28]. Since $\rho <R$, $K_0$ is contained in $B_E(y_0,R)$ and hence in $C$; put $O_C:=B_E(y_0,\rho )\subseteq K_0$. The set $K_0$ is convex by the triangle inequality in [F14]. Each singleton $\{v\}$ for $v\in V$ is metric-compact by [F26], hence compact in the norm topology by [F28], and is convex by [F24]. The finitely many singletons together with $K_0$ are therefore nonempty compact convex sets. Since $E$ with its norm topology is a real topological vector space by [F22], [F23] makes $K_C:=\mathrm{co}(V\cup K_0)$ compact and convex. It contains $\bar{A}=\mathrm{co}(V)$ and $O_C$; hence it contains every segment joining a point of $A$ to a point of $O_C$. By [F25], only finitely many walls meet $K_C$.

Let $O\subseteq E$ be nonempty and open, and let $L_1,\dots ,L_N$ be a finite family of proper affine subspaces. If $N=0$, or if $E=\{0\}$ (when every proper affine subspace is empty), any $p\in O$ avoids their union. Otherwise, write $L_j=a_j+D_j$ with $D_j$ a proper linear subspace by [F30]. By [F12], choose a direction $d\in E$ outside ${\bigcup }_j{D}_j$; then $d\mathrm{\ne }0$. Choose $p\in O$ and, by [F15], $r>0$ with $B_E(p,r)\subseteq O$. For $\mathrm{\mid }t\mathrm{\mid }<r\mathrm{/}\mathrm{\parallel }d{\mathrm{\parallel }}_B$, homogeneity in [F14] gives $\mathrm{\parallel }td{\mathrm{\parallel }}_B<r$, so $p+td\in O$. Each line $p+\mathbb{R}d$ meets each $L_j$ in at most one point because $d\mathrm{\notin }{D}_j$. Removing these finitely many parameters from this nonempty interval leaves an allowed $t$, so $p+td\in O\setminus {\bigcup }_j{L}_j$. This finite avoidance makes no use of the axiom of choice.

If $\mathrm{\Phi }=\mathrm{\varnothing }$, then $E=\{0\}$ by [F1], there are no walls, and we may take $r_0=1$. Otherwise fix $z\in E$. For each of the finitely many roots $\alpha$, put $c_{\alpha }:=B(z,\alpha )$ and let ${\delta }_{\alpha }:=1$ if $c_{\alpha }\in \mathbb{Z}$, while otherwise let ${\delta }_{\alpha }:=\min \{c_{\alpha }-\lfloor c_{\alpha }\rfloor ,\lfloor c_{\alpha }\rfloor +1-c_{\alpha }\}>0$. If $\mathrm{\parallel }u-z{\mathrm{\parallel }}_B<{\delta }_{\alpha }\mathrm{/}(2\mathrm{\parallel }\alpha {\mathrm{\parallel }}_B)$, Cauchy--Schwarz [F14] gives $\mathrm{\mid }B(u,\alpha )-c_{\alpha }\mathrm{\mid }<{\delta }_{\alpha }\mathrm{/}2$; therefore $B(u,\alpha )$ is not an integer unless $c_{\alpha }$ is an integer and $B(u,\alpha )=c_{\alpha }$. Taking the minimum of these positive radii over $\mathrm{\Phi }$ gives $r_z>0$ such that $B_E(z,r_z)$ meets only walls through $z$.

If $E=\{0\}$, then $\mathrm{\Phi }=\mathrm{\varnothing }$ by [F1], there are no walls, and $W_a$ is generated by the empty family by [F2]; hence there is one alcove, one sector, and $K_v=R_v=\{1\}$. Assume now $E\mathrm{\ne }\{0\}$. For $H=H_{\alpha ,k}\in \mathcal{A}(v)$ one has $k=B(v,\alpha )$ and $r_{\alpha ,k}(v+u)=v+s_{\alpha }(u)$. Thus, after translating $v$ to $0$, $R_v:=⟨r_H:H\in \mathcal{A}(v)⟩$ is the Weyl group of ${\mathrm{\Phi }}_v$ on $V_v$ and acts trivially on $V_v^{\perp }$. By [F6] it is finite, and by [F7] it acts simply transitively on the chambers of the local root arrangement; these chambers times $V_v^{\perp }$ are exactly the sectors at $v$. If ${\mathrm{\Phi }}_v=\mathrm{\varnothing }$, then $V_v=\{0\}$, $R_v=\{1\}$, and there is one sector, namely $E$, which is connected because any two points are joined by a straight path continuous by [F22, F32] and hence connected by [F29].

Fix any alcove $C$, choose $x\in A$, and use $O_C,K_C$ from step 1.2. Let ${\mathcal{H}}_C$ be the finite set of distinct walls meeting $K_C$. For every intersecting pair of distinct walls $H,H^{\mathrm{\prime }}\in {\mathcal{H}}_C$, their intersection $P=H\cap H^{\mathrm{\prime }}$ has codimension two: by [F31], distinct affine hyperplanes that intersect have independent normals, since proportional normals would make them equal. Also $x\mathrm{\notin }P$ because $x\in A$ by [F13]. Writing $D_P$ for the direction space of $P$, choose $p\in P$ and put $w_P:=x-p\mathrm{\notin }{D}_P$. The affine subspace $L_P:=p+(D_P+\mathbb{R}{w}_P)$ is proper: $D_P$ has codimension two and adjoining the independent vector $w_P$ raises its dimension by one. Apply the finite-avoidance argument of step 1.3 to this finite family in the open set $O_C$ to choose $y$ outside every $L_P$; if the family is empty, choose any $y\in O_C$. If $[x,y]$ met an intersection $P$, then $y\in L_P$, contrary to this choice; thus the segment meets no two distinct walls at the same point. It lies in $K_C$, so it meets only walls from ${\mathcal{H}}_C$ and crosses each at most once because $x$ lies on no wall. At a crossing point $q$ on a wall $H$, choose a compact closed ball about $q$ using [F20] and [F21]. By [F25] only finitely many walls meet that ball, and none of the other walls in that list contains $q$. For each such wall $H_{\beta ,l}$, the positive radius $\mathrm{\mid }B(q,\beta )-l\mathrm{\mid }\mathrm{/}(2\mathrm{\parallel }\beta {\mathrm{\parallel }}_B)$ gives a ball about $q$ missing it, by Cauchy--Schwarz [F14]. Taking the minimum of these finitely many radii and the original ball radius (or the original radius if there are no other walls) yields a neighborhood meeting the arrangement only in $H$. Each half-ball is convex, hence path-connected by [F32] with continuity from [F22], and connected by [F29]; it lies in an alcove, and the two alcove closures share an open patch of $H$, so they are adjacent. Therefore the segment yields a finite gallery from $A$ to $C$; by [F4], each successive alcove is obtained from the preceding one by a wall reflection in $W_a$. In particular every alcove lies in $W_a\cdot A$.

Suppose exactly two distinct walls through $v$ have normals $\alpha ,\beta$. Their normals are not parallel, since distinct parallel hyperplanes through one point coincide. If $B(\alpha ,\beta )\mathrm{\ne }0$, then $\gamma :=s_{\alpha }\beta$ is a root distinct from $\pm \alpha ,\pm \beta$, and $B(v,\gamma )=B(v,\beta )-B(\beta ,{\alpha }^{\vee })B(v,\alpha )\in \mathbb{Z}$. The wall normal to $\gamma$ through $v$ is therefore a third distinct wall, a contradiction. Hence $B(\alpha ,\beta )=0$ and the two walls are orthogonal.

Let $P$ be the span of two nonparallel normals in ${\mathrm{\Phi }}_v$ and set $\mathrm{\Psi }:={\mathrm{\Phi }}_v\cap P$. This is a finite reduced crystallographic root system spanning $P$: reflections in its roots preserve $P$, and the root-system conditions restrict from step 1.1. Its Weyl group $W(\mathrm{\Psi })$ is finite by [F6] and acts simply transitively on its chambers by [F7]. By [F31], the mirror arrangement in $P$ consists of finitely many lines through the origin, so its sectors occur cyclically and the sector-adjacency graph is connected. Let $r$ and $s$ be the reflections in the two boundary lines of one sector $D$. Each sends $D$ to its neighbor across that line. Inductively, if $h(D)$ is reached for some $h\in ⟨r,s⟩$, the reflections across its two boundary lines are $hr{h}^{-1}$ and $hs{h}^{-1}$, so both neighboring sectors are also in the orbit. Thus $⟨r,s⟩$ is transitive on sectors; simple transitivity of $W(\mathrm{\Psi })$ then gives $⟨r,s⟩=W(\mathrm{\Psi })$. If $m$ is the order of $rs$, the two line reflections generate a dihedral group of order $2m$. Simple transitivity therefore gives exactly $2m$ sectors, and transitivity by orthogonal maps makes their angles equal, hence each angle is $\pi \mathrm{/}m$. Choose inward-pointing root normals $\alpha ,\beta$ to the boundary lines of $D$ and put $e_1:=\alpha \mathrm{/}\mathrm{\parallel }\alpha \mathrm{\parallel }$, $e_2:=\beta \mathrm{/}\mathrm{\parallel }\beta \mathrm{\parallel }$, $c_1:=\mathrm{\parallel }\alpha \mathrm{\parallel }$, and $c_2:=\mathrm{\parallel }\beta \mathrm{\parallel }$. Their Gram matrix has diagonal entries $1$ and off-diagonal entry $-\cos (\pi \mathrm{/}m)$, so it is the rank-two Coxeter form with label $m$; the scaled roots $c_1{e}_1=\alpha$ and $c_2{e}_2=\beta$ have integral Cartan numbers by [F1]. Thus these data form a positive-definite crystallographic scaling, and [F8] gives $m\in \{2,3,4,6\}$. Thus every rank-two local mirror arrangement is the stated dihedral arrangement.

If $\mathrm{\Phi }=\mathrm{\varnothing }$, then $E=\{0\}$ and the unique sector and alcove are both $\{0\}$. Otherwise fix $v$ and let $r_v$ be from step 1.4. Each local sector $S$ is a sign-pattern intersection of open half-spaces by [F31], hence convex; it is a cone with apex $v$, so $v\in \bar{S}$ and $S\cap B_E(v,r_v)$ is nonempty and convex. It is path-connected by straight segments under [F32], which are continuous by [F22], and hence connected by [F29]. Since the ball meets no walls except those through $v$, this set lies in one global alcove, call it $C_S$, and $v\in \overline{C_S}$. Conversely, if $v\in \bar{C}$ for a global alcove $C$, then $C\cap B_E(v,r_v)$ is nonempty by closure and convex by [F17] and [F15], so it is path-connected by [F32] and connected by [F22, F29]. It avoids every wall through $v$, so it lies in one local sector $S$. The two connected sets overlap, hence $C=C_S$. If $C_S=C_T$, the connected set $C\cap B_E(v,r_v)$ meets both sectors, which forces $S=T$ because distinct sectors are distinct connected components of the local complement. Thus local sectors at $v$ correspond bijectively to alcoves whose closures contain $v$.

Take $g\in K_v$ and a local sector $S$. Because $g$ fixes $v$ and permutes the wall arrangement by [F2], $g(S)$ is a local sector. By simple transitivity of $R_v$ on sectors, choose $h\in R_v$ with $h(S)=g(S)$, using [F7] and step 2.1. Both maps fix $v$ and are isometries by [F3], so $h^{-1}g$ preserves $B_E(v,r_v)$ and the local sector $S$; it therefore stabilizes the unique incident alcove $C_S$ from step 3.1. Every alcove is a $W_a$-translate of $A$ by step 2.2, and its stabilizer is trivial by [F18] and conjugation to $A$. Thus $h^{-1}g=1$, so $g=h\in R_v$. Conversely every generator $r_H$ of $R_v$ fixes $v$ pointwise by [F2], hence $R_v\le K_v$. We conclude $K_v=R_v$, proving finiteness, generation by the reflections through $v$, and simple transitivity on sectors.

If $E=\{0\}$, then $\mathrm{\Phi }=\mathrm{\varnothing }$ by [F1] and $J=\mathrm{\varnothing }$ by its definition; there is one sector and one incident alcove with no facets, so the panel-type assertion is immediate. Otherwise, for an incident alcove $C$, let $I_C(v)\subseteq J$ be the types of its facets containing $v$. Any two distinct sectors of the finite central arrangement of walls through $v$ are connected by a gallery: choose regular points $x$ and $y_0$ in the two sectors and in the ball from step 1.4. The set $O$ given by the second sector intersected with this open ball is nonempty and open. For each pair of distinct local walls, their intersection $P$ has codimension two by the same affine-hyperplane argument in step 2.2, and $x\mathrm{\notin }P$. By the affine-span argument in step 2.2, the affine hull $L_P$ of $x$ and $P$ is a proper affine subspace. Apply the finite-avoidance argument of step 1.3 to these finitely many subspaces in $O$ to choose $y$ outside them (if there are no pairs, take $y=y_0$). The ball is convex by [F15], so $[x,y]$ stays in it; it crosses each local wall at most once and never crosses two at the same point. By the sector-to-alcove correspondence in step 3.1, these local galleries give galleries of incident global alcoves crossing only walls through $v$. At each crossing [F4] gives the adjacent alcove by reflection in a wall through $v$. This reflection fixes $v$ and carries every facet of the first alcove containing $v$ bijectively to a facet of the second containing $v$. By the full type equivariance in [F5], it preserves all these facet types, not just the shared-panel type. Hence $I_C(v)$ is independent of $C$; call it $I(v)$ and set $t(v):=J\setminus I(v)$. This proves the panel-type clause, including the case $I(v)=\mathrm{\varnothing }$. If $v$ is a vertex of $\overline{g(A)}$, the facets of that alcove containing $v$ are exactly its panels through $v$, so their types are $I(v)=J\setminus t(v)$.

Let $p\mathrm{\ne }q$ lie in $I(v)$ and choose any incident alcove $C=g(A)$. By the definition of $I(v)$, its facets of types $p,q$ both contain $v$; [F13] says they meet in a codimension-two face $F$ with $v\in \bar{F}$. Their wall reflections are $g{s}_p{g}^{-1}$ and $g{s}_q{g}^{-1}$ by [F19], so their product has the same finite order $m$ as $s_p{s}_q$. Let $u$ lie in the relative interior of $F$. Because $\bar{C}$ is a product of geometric simplices, the tangent cone of $\bar{C}$ at $u$ has lineality space $\mathrm{span}(F-F)$. The interior of $C$ lies on one side of every wall; therefore the defining linear form of any wall through $u$ has one sign on this tangent cone and must vanish on its lineality space. Hence every wall through $u$ contains $F$, and its normal lies in the two-dimensional normal plane. The p- and q-facets give two independent normals, so the local root subsystem has rank two. By the sector-to-alcove correspondence in step 3.1 its $2m$ local sectors correspond exactly to the $2m$ alcoves incident to $F$. Since the closure of $C$ is a product of geometric simplices, precisely its p- and q-facets contain $u$; step 4.2 therefore shows that the panel labels around this local cycle alternate p,q. Writing ${\sigma }_p,{\sigma }_q$ for the abstract generators of $W_{pq}$, the boundary word is $({\sigma }_p{\sigma }_q{)}^m$ or $({\sigma }_q{\sigma }_p{)}^m$. By [F9] the first is a defining relator, and the reverse is its conjugate by ${\sigma }_p$; [F10] then shows every homomorphism from a Coxeter group with this rank-two restriction sends the boundary word to $1$. No axiom of choice is used: all root, chamber, wall, and word lists here are finite.