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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?

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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.

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  • $\begingroup$ Thanks for the example. But it is not exactly what I'm looking for. The number of reflections generating the group should match the rank, so in the plane this should be three. Also $t_3t_4$ is not a reflection. $\endgroup$
    – P. Wegener
    Oct 8, 2021 at 9:56
  • $\begingroup$ But $\langle t_1,t_2,t_3,t_4\mid (t_i t_{i+1})^2 \rangle$ is an affine Coxeter system of rank $4$, right? I guess the issue is that it is not irreducible. Maybe you can give precisions on your terminology? $\endgroup$ Oct 8, 2021 at 10:45
  • $\begingroup$ irreducible = Coxeter graph is connected. Yes, sorry, so "irreducible" is the problem here (as your group is not irreducible) $\endgroup$
    – P. Wegener
    Oct 8, 2021 at 11:55

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