Infinite reflection subgroups of affine Coxeter groups

Question

Let $(W,S)$ be an irreducible affine Coxeter system of rank $n \geq 3$ (affine for instance as in the sense of Chapter 4 of Humphreys "Reflection groups and Coxeter groups").

Let $t_1,\ldots, t_n$ be a set of (not necessarily simple) reflections such that $W= \langle t_1, \ldots , t_n \rangle$. Further assume that $\langle t_1, \ldots , t_{n-2} \rangle$ is finite.

Does there exist a reflection $r \in \langle t_{n-1}, t_n \rangle $ such that $\langle t_1, \ldots, t_{n-2},r \rangle$ is finite as well?

An easy example: $(W, \{s_1,s_2,s_3\})$ affine of type $\widetilde{A}_2$. Then obviously we have $W=\langle s_1, s_2s_3s_2,s_2 \rangle$ and $\langle s_1 \rangle$ is finite. While $\langle s_1, s_2s_3s_2 \rangle$ is infinite, we have that $\langle s_1,s_2 \rangle$ is finite.

It is well known that a Coxeter group is infinite iff it contains an infinite dihedral group. So equivalently, the question is whether there exists a reflection $r \in \langle t_{n-1}, t_n \rangle $ such that $\langle t_1, \ldots, t_{n-2},r \rangle$ does not contain an infinite dihedral subgroup?

Answer 1

In the Euclidean plane, take

• $t_1=$ reflection about the axis $\{x=0\}$
• $t_2=$ reflection about the axis $\{y=0\}$
• $t_3=$ reflection about the axis $\{x=1\}$
• $t_4=$ reflection about the axis $\{y=1\}$

Then $\langle t_1,t_2\rangle$ is finite, but

• $\langle t_1,t_2,t_3\rangle$ contains $t_1t_3=$ translation by $(2,0)$
• $\langle t_1,t_2, t_4\rangle$ contains $t_1t_4=$ translation by $(0,2)$
• $\langle t_1,t_2, t_3t_4\rangle$ contains $t_3t_4 t_1 t_4 t_3=$ reflection about the axis $\{x=2\}$, hence $t_3t_4 t_1 t_4 t_3 t_1=$ translation by $(4,0)$.

Edit: This is not a counter-example because this Coxeter system is not irreducible.