\(
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The Tits cone
Definition 52.2 Let \((W,S)\) be a Coxeter system with Gram form \(G\) on \(V \definedas \RR^S\) and simple roots \(\theset{\alpha_s}_{s\in S}\), and let \(W\) act on the dual space \(V^*\) by the contragredient of its action on \(V\). The closed fundamental chamber is \[
\overline{C} \definedas \theset{\, f\in V^* \mid f(\alpha_s)\geq 0 \text{ for all } s\in S \,}
,
\] and the Tits cone of \((W,S)\) is \[
U \definedas \Union_{w\in W} w\!\left(\overline{C}\right) \iscontainedin V^*
.
\] When \(G\) is nondegenerate it identifies \(V\) with \(V^*\), and \(\overline C\) becomes the cone \(\theset{v\in V \mid \beta(v,\alpha_s)\geq 0}\) over the closure of a Weyl chamber in the sense of Definition 49.13.
Theorem 52.3 (Properties of the Tits cone) Let \((W,S)\), \(\overline C\) and \(U\) be as in Definition 52.2, and for \(f\in \overline C\) write \(I(f) \definedas \theset{s\in S \mid f(\alpha_s) = 0}\).
\(U\) is a convex cone, and \(\overline{C}\) is a strict fundamental domain for the action of \(W\) on \(U\): every \(W\)-orbit in \(U\) meets \(\overline{C}\) in exactly one point.
The stabilizer of \(f\in\overline{C}\) in \(W\) is the standard parabolic subgroup \(W_{I(f)}\) of Definition 50.12.
A point \(f\in\overline{C}\) lies in the interior of \(U\) if and only if \(W_{I(f)}\) is finite, that is, if and only if the subdiagram \(\Sigma_{I(f)}\) is elliptic.
\(W\) acts properly discontinuously on the interior of \(U\).
These are theorems of Tits, recorded in (Bourbaki 2008; Humphreys 1990).
Vertices of a Coxeter polytope
Notation 52.5 Let \(P\iscontainedin\HH^n_L\) be a Coxeter polytope (Definition 49.11) with walls \(H_s = \alpha_s^{\perp}\) indexed by a finite set \(S\), and Coxeter–Vinberg diagram \(\Sigma\). For \(I\iscontainedin S\) write \[
A_I \definedas \spanof_\RR\theset{\, \alpha_s \mid s\in I \,} \iscontainedin L_\RR
.
\] The Gram matrix \(G[I,I]\) is the Gram matrix of a spanning set of \(A_I\), so its rank equals the rank of \(\restrictionof{\beta_L}{A_I}\); that common value is the rank of \(\Sigma_I\) in the sense of Definition 50.12.
Proposition 52.6 (Vertices and subdiagrams)
Let \(I\iscontainedin S\).
If \(\Sigma_I\) is elliptic of rank \(n\), then \(\dim A_I = n\), the orthogonal complement \(A_I^{\perp L_\RR}\) is a line spanned by a vector of positive norm, and \[
\Intersect_{s\in I} H_s
\] is a single point of \(\HH^n_L\), an ordinary vertex of \(P\) when it lies in \(\overline{P}\).
If \(\Sigma_I\) is parabolic of rank \(n-1\), then \(\dim A_I = n\), the form \(\restrictionof{\beta_L}{A_I}\) is negative semidefinite with radical a line \(\RR\eta\) spanned by an isotropic vector \(\eta\), one has \(A_I^{\perp L_\RR} = \RR\eta\), and \[
\Intersect_{s\in I}\overline{H_s}
\] meets the closure of \(\HH^n_L\) in the single boundary point \([\eta]\in\partial\HH^n_L\), an ideal vertex of \(P\) when it lies in \(\overline P\). In this case the negative-definite lattice \(\eta^{\perp L}/\eta\) has rank \(n-1\), and the stabilizer of \([\eta]\) in \(W\) is \(W_I\), acting on \((\eta^{\perp L}/\eta)\tensor_\ZZ\RR \cong \EE^{\,n-1}\) as the euclidean reflection group with diagram \(\Sigma_I\).
Proof. Suppose first that \(\Sigma_I\) is elliptic of rank \(n\). Then \(G[I,I]\) is negative definite of rank \(n\), so the \(\alpha_s\) with \(s\in I\) are linearly independent and \(\dim A_I = n\). Since \(\restrictionof{\beta_L}{A_I}\) is nondegenerate, \(L_\RR = A_I\perp A_I^{\perp}\), so \(A_I^{\perp}\) is a line of signature \((1,0)\), spanned by a vector \(v\) with \(v^2 > 0\). Now \(\Intersect_{s\in I}H_s\) is the image of \(A_I^\perp\) in \(\HH^n_L\), a single point.
Suppose instead that \(\Sigma_I\) is parabolic of rank \(n-1\). Each connected component of \(\Sigma_I\) has a one-dimensional kernel by Definition 49.15, so \(\restrictionof{\beta_L}{A_I}\) is negative semidefinite and degenerate. Its radical is \(A_I\intersect A_I^{\perp}\), a totally isotropic subspace of \(L_\RR\), hence of dimension at most \(\min(n_+,n_-) = 1\); being nonzero it is a line \(\RR\eta\) with \(\eta^2 = 0\). Then \(\dim A_I = \rank(\restrictionof{\beta_L}{A_I}) + 1 = n\), so \(\dim A_I^{\perp} = (n+1) - n = 1\), and \(\RR\eta\iscontainedin A_I^\perp\) forces \(A_I^{\perp} = \RR\eta\). The intersection \(\Intersect_{s\in I}\overline{H_s}\) is the image of \(A_I^\perp\) in \(\PP(L_\RR)\), the single point \([\eta]\), which lies on \(\partial\HH^n_L\) because \(\eta\) is isotropic. Since \(\eta\) is isotropic and \(L\) has signature \((1,n)\), the form induced on \(\eta^{\perp}/\eta\) is negative definite of rank \(n-1\); and \(A_I\iscontainedin\eta^{\perp}\), with \(A_I/\RR\eta\) of dimension \(n-1\), so \(W_I\) acts on \(\eta^{\perp}/\eta\) as the reflection group of \(\Sigma_I\), which is euclidean by Definition 50.8.
Corollary 52.7 Sending an ideal vertex \([\eta]\) of \(P\) to the set \(I\) of all walls of \(P\) through \([\eta]\) is a bijection from the ideal vertices of \(P\) onto a set of parabolic subdiagrams of \(\Sigma\) of rank \(n-1\), and each such \(\Sigma_I\) is maximal among the parabolic subdiagrams of \(\Sigma\), in the sense of Definition 49.16.
Proof. Let \([\eta]\) be an ideal vertex and \(I\) the set of walls through it, so \(A_I \iscontainedin \eta^{\perp}\) and \(\Sigma_I\) is parabolic of rank \(n-1\) by Proposition 52.6. Suppose \(\Sigma_J\) is parabolic with \(J\containing I\). Writing \(r\leq 1\) for the dimension of the radical of \(\restrictionof{\beta_L}{A_J}\), one has \(\dim A_J - r = \rank(\restrictionof{\beta_L}{A_J})\leq n_-(L_\RR) = n\), so \(\dim A_J\leq n+1\); and \(\dim A_J = n+1\) would make \(A_J = L_\RR\), whose form is nondegenerate of signature \((1,n)\) rather than negative semidefinite. Hence \(\dim A_J\leq n = \dim A_I\), so \(A_J = A_I\) and every wall indexed by \(J\) contains \(A_I^{\perp} = \RR\eta\), that is, passes through \([\eta]\). By the choice of \(I\) this gives \(J = I\).
Covolume, compactness, and finite volume
Definition 52.9 Let \(\Gamma\leq\Isom(\HH^n_L)\) be a discrete subgroup. Its covolume is the hyperbolic volume \(\vol(\HH^n_L/\Gamma)\) of the quotient orbifold. If \(\Gamma\) is generated by the reflections in the walls of a Coxeter polytope \(P\), then \(P\) is a strict fundamental domain and \(\vol(\HH^n_L/\Gamma) = \vol(P)\).
Theorem 52.10 (Finite volume and compactness) Let \(P\iscontainedin\HH^n_L\) be a Coxeter polytope with finitely many walls and Coxeter–Vinberg diagram \(\Sigma\).
\(\vol(P) < \infty\) if and only if, in the projective model of Definition 49.9, the closure of \(P\) is the convex hull of finitely many points of \(\overline{\HH^n_L}\) — equivalently, of the ordinary and ideal vertices supplied by Proposition 52.6.
\(P\) is compact if and only if \(\vol(P)<\infty\) and \(P\) has no ideal vertex; equivalently, if and only if the closure of \(P\) in the projective model is the convex hull of finitely many ordinary vertices. By Corollary 52.7 this is the condition that no parabolic subdiagram of \(\Sigma\) of rank \(n-1\) is cut out by a set of walls meeting in \(\overline{P}\).
\(P\) has infinite volume if and only if its closure in the projective model has a vertex outside \(\overline{\HH^n_L}\).
These are Vinberg’s criteria (Vinberg 1967; Vinberg 1985).
Corollary 52.12 (The simplex case)
Suppose \(\abs{S} = n+1\) and \(P\) is a simplex, so that its \(n+1\) vertices are cut out by the \(n+1\) maximal proper subdiagrams \(\Sigma_{S\sm\theset{s}}\), \(s\in S\). Then
\(P\) is compact if and only if every \(\Sigma_{S\sm\theset{s}}\) is elliptic;
\(\vol(P) < \infty\) if and only if every \(\Sigma_{S\sm\theset{s}}\) is elliptic or parabolic, the ideal vertices being those \(s\) for which \(\Sigma_{S\sm\theset{s}}\) is parabolic;
\(\vol(P) = \infty\) if and only if some \(\Sigma_{S\sm\theset{s}}\) is hyperbolic.
Example 52.14 (Triangle groups) Let \(2\leq p\leq q\leq r\leq\infty\) and let \(\Sigma\) be the rank-three diagram with \(m_{12} = p\), \(m_{13} = q\), \(m_{23} = r\). A Coxeter polytope realizing \(\Sigma\) is a triangle with angles \(\pi/p\), \(\pi/q\), \(\pi/r\), an angle \(0\) being read as an ideal vertex, and the type of \(\Sigma\) is decided by the angle sum: \[
\frac1p + \frac1q + \frac1r
\;\begin{cases}
> 1 & \text{spherical, and the triangle lies in } S^2;\\
= 1 & \text{euclidean, and the triangle lies in } \EE^2;\\
< 1 & \text{hyperbolic, and the triangle lies in } \HH^2.
\end{cases}
\] In the hyperbolic case the triangle has area \(\pi\left(1 - \tfrac1p - \tfrac1q - \tfrac1r\right)\) by the Gauss–Bonnet theorem, so every hyperbolic triangle group has finite covolume; it is cocompact exactly when \(p,q,r\) are all finite, and each entry equal to \(\infty\) contributes one ideal vertex. The spherical solutions are \((2,2,r)\), \((2,3,3)\), \((2,3,4)\) and \((2,3,5)\), which are the rank-three entries \(I_2(r)\times A_1\), \(A_3\), \(B_3\) and \(H_3\) of Theorem 50.17; the euclidean solutions are \((2,3,6)\), \((2,4,4)\) and \((3,3,3)\), which are \(\tilde G_2\), \(\tilde C_2\) and \(\tilde A_2\).