I am trying to understand the proof of 3.2 in the paper "Coxeter groups are biautomatic" by Osajda and Prytycki [OP]: https://arxiv.org/abs/2206.07804
I will reformulate the statements with a slightly different notation. To not confuse walls and the Coxeter groups itself, I will denote the wall corresponding to a reflection $r \in W $ by $\Gamma_r$.
[OP, Definition 3.1] Let (W,S) be a Coxeter system. We say that the corresponding walls $ \Gamma_r, \Gamma_q$ intersect, if one is wall is NOT contained in a half space of the other wall. They state that this is equivalent to the group $\langle r,q \rangle$ being finite.
This is not trivial and my first question is, how this equivalence can be seen?
Further they call $r,q$ sharp-angled if they don´t commute and are simutaniously conjugated into S.
Now [OP, Lemma 3.2] says: Suppose that reflections $r,q \in W$ are sharp-angled and that $g \in W$ lies in a geometric fundamental domain for $\langle r,q \rangle$. Assume there is a wall U seperating $g$ from $\Gamma_r$ or from $\Gamma_q$. Let $\Gamma'$ be a wall distinct from $\Gamma_r$ and $\Gamma_q$ that is the translate of $\Gamma_r$ or $\Gamma_q$ under an element of $\langle r,q \rangle$. Then there is a wall $U'$ seperating g from $\Gamma'$.
In their proof they reduce this problem to a rank 3 Coxeter group and then consider different cases. I am happy if I can at least fully understand one case. Let $S = \{r,q,s \} $. WLOG we can assume $m_{sr} \geq 3$. They state: If $m_sq \geq 3$, then $\Gamma_s$ is disjoint from $\Gamma'$.
My question is: how do i show this?
My guess was to try and show, that the corresponding reflections of the walls has infinite order. That would make the respective group generated by those reflections infinite and thus the walls would be parallel by the equivalence we saw in the first paragraph.
I can elaborate further, if someone decides to try and help me.