What are the word hyperbolic affine Coxeter groups? It is well-knwon that all affine (irreducible) Coxeter systems can be classified by their Coxeter graphs, see Wikipedia. The corresponding diagrams are $(\tilde{A}_n)_{n \geq 1}$, $(\tilde{B}_n)_{n \geq 3}$, $(\tilde{C}_n)_{n \geq 2}$, $(\tilde{D}_n)_{n \geq 4}$, $(\tilde{E}_n)_{6 \leq n \leq 8}$, $\tilde{F}_4$, $\tilde{G}_2$ and $\tilde{I}_1$. It is further well-known that a Coxeter group is word hyperbolic (in the sense of Gromov) if and only if  $\mathbb{Z}+\mathbb{Z}\nsubseteq W$. This result is duo to Moussong. I guess there should be a (possibly even finite) list of all (irreducible) affine, word hyperbolic Coxeter groups. But I couldn't find any reference for that. Also I do not know enough about (affine) Coxeter groups to investigate on such a list myself.
So: What are the (Coxeter diagrams of the) affine word hyperbolic Coxeter systems?
 A: The infinite dihedral (diagram $\circ\stackrel{\infty}{—}\circ$) is the only one.
More precisely: if a Coxeter group $(W,S)$ is both virtually abelian and infinite hyperbolic then it's given by the above Coxeter system, possibly up to disjoint union with a Coxeter system of finite type.
Indeed, being hyperbolic and virtually abelian, it's virtually cyclic. It's known (Wall 1967) that a infinite virtually cyclic group is always finite-by-$\mathbf{Z}$ or finite-by-$D_\infty$. For a group generated by torsion elements, the first case is excluded.
For a group that is finite-by-$D_\infty$, there are two kinds of elements of order $2$: those in the finite kernel (which is the unique maximal finite normal subgroup), and those mapping to an element of order $2$ in the $D_\infty$ quotient. Call them of type (a) and (b) respectively.
Claim: every Coxeter generator of type (a) commutes with every Coxeter generator of type (b).
Granting the claim: those Coxeter generators of type (a) form a union of components in the Coxeter graph, and hence these form a finite direct factor. Hence we can reduce to the case when all Coxeter generators are of type (b). In $D_\infty$, any two distinct elements of order $2$ have product of infinite order. At least two Coxeter generators, say $u,v$ have distinct images in $D_\infty$. Suppose by contradiction that there is a third one $w$. If its image in $D_\infty$ is not the same as that of $u$ or $v$, then all $uv$, $uw$, $vw$ have infinite order. So $u,v,w$ form a triangle with $\infty$ edges, and this forms a group with a non-abelian free subgroup of index $2$. Otherwise, say $w$ has the same image as $u$. Then $u,v,w$ forms a triangle with two $\infty$ edges, and the corresponding subgroup is a free product $D_n\ast C_2$ with $D_n$ dihedral of order $\ge 4$, so again this has a non-abelian free subgroup of finite index. 
For the claim: consider $u$ of type (b), not commuting with $w$ of type (a). Again, there exists $v$ of type (b), with an $\infty$ edge between $u$ and $v$. So we have a triangle with labels $\infty$, $n\ge 3$, and $m\ge 2$. This Coxeter group is an amalgamated product $D_{2n}\ast_{C_2} D_{2m}$ over a common generator. Since the amalgamated subgroup has index $\ge 2$ in one and index $\ge 3$ in the other, it also has a non-abelian free subgroup of finite index.
A: None of these groups are word hyperbolic.  These groups act geometrically on a finite dimensional Euclidean space of dimension at least two and so are quasi-isometric to spaces which are not Gromov hyperbolic.
