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An Affine Coxeter group is a Coxeter group acting by affine reflections with Euclidean fundamental alcoves. An irreducible affine Coxeter group has a positive-semidefinite Coxeter Gram matrix with a one-dimensional radical.

The walls occur in three families of parallel lines, and a choice of halfspaces in one family is determined by an integer cut. Lines from different families cross, so the three cuts can be chosen independently. It follows directly from the dual cube complex of a wallspace construction thatand that $C$ is the standard cubulation of ${\mathbb{R}}^3$.

Choose affine coordinates $x_1,x_2,x_3$ for the three wall families so that the original Euclidean plane is $x_1+x_2+x_3=0$ and the walls are the integer level sets. On the cut coordinates $(n_1,n_2,n_3)\in {\mathbb{Z}}^3$, the translation subgroup of the Affine Coxeter group $W$ adds vectors $(m_1,m_2,m_3)$ satisfying $m_1+m_2+m_3=0$, while its finite reflection subgroup permutes the three coordinates. Thereforeis constant on every $W$-orbit. Since $h(n,0,0)=n$ is unbounded, there are infinitely many vertex orbits. A cocompact cubical action on this locally finite cube complex would have only finitely many cube orbits, so the action of $W$ on $C$ is not cocompact.

At $t=1$, the diagonal entries of $U-L$ are $2$, while every off-diagonal $(i,j)$ entry isHence $U-L$ is exactly the Coxeter Gram matrix $G(W)$, and part c givesFor a Finite Coxeter group the Gram matrix is positive definite, and for a Hyperbolic Coxeter group it is nondegenerate with Lorentzian signature. In either case $\det (I-C)\ne 0$, so $1$ is not an eigenvalue of $C$ and the Coxeter element fixes no nonzero vector.

For an Affine Coxeter group, the Gram form has a nonzero radical. If $0\ne v\in \mathrm{rad}G(W)$, then $⟨v,e_i⟩=0$ for every $i$, and every generating reflection satisfiesTheir product $\sigma (c)$ therefore fixes $v$. Thus every affine Coxeter element has a nonzero fixed vector.