Finite subgroups of a Coxeter group lie in spherical parabolics

Let $(S,m)$ be a Coxeter matrix with $S$ finite and $W$ the presented group (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). For $T\subseteq S$ put $W_T:=⟨s:s\in T⟩$ (Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups (1), The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups) and call $T$ spherical when $W_T$ is finite; write $\mathbb{S}$ for the set of spherical subsets. Let $\mathrm{\Sigma }$ be the Davis realization with cells $q=w{W}_T$ for $T\in \mathbb{S}$, the point-stabilizer formula and the chain metric of The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) and The Davis complex of a finite-rank Coxeter system is CAT(0) (Moussong's theorem). A spherical parabolic is a conjugate $w{W}_T{w}^{-1}$ with $T\in \mathbb{S}$. Assume the Axiom of Choice (The Axiom of Choice); this is required through The Davis complex of a finite-rank Coxeter system is CAT(0) (Moussong's theorem) for its CAT(0) conclusion. The proper-space branch of Circumcenters of bounded sets and fixed sets of isometries in complete CAT(0) spaces used for finite orbits in (1) requires no additional Choice; no Choice is used in (2) or (3).

(1) Fixed points of finite subgroups. Every finite subgroup $H\le W$ has a fixed point on $\mathrm{\Sigma }$: for any $x_0\in \mathrm{\Sigma }$ the orbit $H{x}_0$ is finite, hence bounded, and its center $c$ (Circumcenters of bounded sets and fixed sets of isometries in complete CAT(0) spaces (1),(2)) is fixed by every $h\in H$. Moreover the fixed set ${\mathrm{\Sigma }}^H={\bigcap }_{h\in H}\mathrm{Fix}(h)$ is nonempty, closed, convex, complete and CAT(0) in the induced metric, and contractible with a continuous geodesic contraction to each of its points (Circumcenters of bounded sets and fixed sets of isometries in complete CAT(0) spaces (3),(4), Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles (3)).

(2) Point stabilizers are spherical parabolics. Every point of $\mathrm{\Sigma }$ lies in the relative interior of exactly one cell $q=w{W}_T$ (The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (2)); let $\dot{q}$ be the unique minimum-length representative of $q$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (3)). Since $T$ is spherical, $(W_T,T)$ is a finite-type Coxeter system, and the chamber tiling of its reflection space partitions $V_T$ into relative interiors of chamber faces $w_0{C}_I^T$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (2), The canonical reflection homomorphism, roots, reflections, and the positive cone, The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset, The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere (1),(3),(4)). Thus if $y$ is in the relative interior of $q$ and has coordinate $y^{\mathrm{\prime }}\in \mathrm{relint}(w_0{C}_I^T)$ in the $\dot{q}$-chart, where $w_0\in W_T$ and $I\subseteq T$, then ${\mathrm{Stab}}_W(y)=(\dot{q}{w}_0)W_I(\dot{q}{w}_0{)}^{-1}$ (The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (2)). Because $I\subseteq T$ and $W_T$ is finite, $W_I\le W_T$ is finite; hence this point stabilizer is a spherical parabolic and is contained in the setwise stabilizer $w{W}_T{w}^{-1}$ of $q$.

(3) Containment in a spherical parabolic. Every finite subgroup $H\le W$ is contained in a spherical parabolic. Choose an $H$-fixed point $c$ by (1), let $q=w{W}_T$ be its unique carrier cell, and let $\dot{q}$, $y^{\mathrm{\prime }}$, $w_0$, $I$ be as in (2). Then $H\le {\mathrm{Stab}}_W(c)=(\dot{q}{w}_0)W_I(\dot{q}{w}_0{)}^{-1}\le \dot{q}{W}_T{\dot{q}}^{-1}=w{W}_T{w}^{-1};$ the last equality holds because $\dot{q}\in w{W}_T$ and $W_T$ is a subgroup. Since $W_T$ is finite, $w{W}_T{w}^{-1}$ is a spherical parabolic. One may take this parabolic to be the setwise stabilizer of the unique carrier cell of $c$.

(4) Scope and abstentions. The statements hold for every finite-rank Coxeter system, including infinite and noncrystallographic ones; the finite subgroup $H$, the conjugating element $w$ and the spherical type $T$ all exist without any finiteness or crystallographic hypothesis on $(W,S)$. No assertion is made here about the conjugacy classes or the number of finite subgroups, about virtual torsion-freeness or residual finiteness, about automaticity, flat subspaces, Moussong hyperbolicity or about the visual boundary, and no alternative proof route is used as a supplier in this item.

• The point-stabilizer formula is the chamber-face formula. The naive reading ${\mathrm{Stab}}_W(y)=w{W}_{T\cap S(y^{\mathrm{\prime }})}{w}^{-1}$ with $S(y^{\mathrm{\prime }})=\{s\in T:B(y^{\mathrm{\prime }},e_s)=0\}$ is false: in $A_2$ with $S=\{s,t\}$ let $y^{\mathrm{\prime }}=s{v}_s=\frac{2}{3}(e_t-e_s)$, so that $B(y^{\mathrm{\prime }},e_s)=-1$ and $B(y^{\mathrm{\prime }},e_t)=1$, whence $S(y^{\mathrm{\prime }})=\mathrm{\varnothing }$; but $\rho (t)v_s=v_s$ because $B(v_s,e_t)=0$, so $sts$ fixes $y^{\mathrm{\prime }}$ and ${\mathrm{Stab}}_{W_S}(y^{\mathrm{\prime }})=⟨sts⟩\mathrm{\ne }\{1\}$. The formula recorded in clause (2) is the chamber-face formula of The cellulation of the Davis complex: incidence, stabilizers and the model U(W,K) (2), which computes the correct conjugate $W_I$ through the chamber containing $y^{\mathrm{\prime }}$.