  Some time ago, it was found that the never-ending oscillatory chaotic
behaviour discovered by Belinsky, Khalatnikov and Lifshitz (BKL) for the
generic solution of the vacuum Einstein equations in the vicinity of a
spacelike ("cosmological") singularity disappears in spacetime dimensions $D=
d+1>10$. Recently, a study of the generalization of the BKL chaotic behaviour
to the superstring effective Lagrangians has revealed that this chaos is rooted
in the structure of the fundamental Weyl chamber of some underlying hyperbolic
Kac-Moody algebra. In this letter, we show that the same connection applies to
pure gravity in any spacetime dimension $\geq 4$, where the relevant algebras
are $AE_d$. In this way the disappearance of chaos in pure gravity models in $D
> 10$ dimensions becomes linked to the fact that the Kac-Moody algebras $AE_d$
are no longer hyperbolic for $d > 9$.
