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Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being not word hyperbolic). Does this imply that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{F}_{2}$ as well? Here $\mathbb{F}_{2}$ denotes the free group in two generators. If this is not true: Is it at least true for $S$ large enough?

Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being word hyperbolic). Does this imply that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{F}_{2}$ as well? Here $\mathbb{F}_{2}$ denotes the free group in two generators. If this is not true: Is it at least true for $S$ large enough?

Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being not word hyperbolic). Does this imply that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{F}_{2}$ as well? Here $\mathbb{F}_{2}$ denotes the free group in two generators. If this is not true: Is it at least true for $S$ large enough?

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Copies of $\mathbb{Z}\oplus \mathbb{F}_2$ in non-affine, irreducible Coxeter groups

Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being word hyperbolic). Does this imply that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{F}_{2}$ as well? Here $\mathbb{F}_{2}$ denotes the free group in two generators. If this is not true: Is it at least true for $S$ large enough?