Decompositions of Coxeter groups into trees of groups

Question

In Chapter 8.8 of Davis' "The geometry and topology of Coxeter groups" the smallest class $\mathcal{G}$ of Coxeter groups which contains all spherical Coxeter groups and which is closed under taking amalgamated free products of the form $W_1 \ast _{W_0} W_2 $ with $W_1, W_2 \in \mathcal{G}$ with common spherical special subgroup $W_0$, has been explored. Obviously, a Coxeter group is contained in $\mathcal{G}$ if and only if it decomposes as a tree of groups where each vertex group and each edge group is spherical. It was shown that a Coxeter group $W$ is contained in $\mathcal{G}$ if and only if $W$ is virtually free ().

I'm wondering what happens if we consider the somewhat similar class generated by all spherical and affine Coxeter groups that is closed under taking amalgamated free products over common spherical special subgroups. Is it possible that this class contains all Coxeter groups? If not, is there a characterization of groups that are contained in this class?

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Davis, Michael W., The geometry and topology of Coxeter groups., London Mathematical Society Monographs Series 32. Princeton, NJ: Princeton University Press (ISBN 978-0-691-13138-2/hbk). xiv, 584 p. (2008). ZBL1142.20020.

Answer 1

No. Indeed, every Coxeter group with no $\infty$ edge has Serre's Property FA and hence cannot be written as a nontrivial amalgam. Hence if it's not spherical or affine, it does not belong to the class you're defining. This notably applies to many Coxeter groups on 3 Coxeter generators.

(A group generated by a finite subset $S$ such that every element of the form $s$ or $st$ for $s,t\in S$ has finite order, has Serre's Property FA. This is somewhere in Serre's Astérisque "Arbres, amalgames, $\mathrm{SL}_2$" or its subsequent English translation "Trees".)