  In recent papers, it has been shown that (i) the dynamics of theories
involving gravity can be described, in the vicinity of a spacelike singularity,
as a billiard motion in a region of hyperbolic space bounded by hyperplanes;
and (ii) that the relevant billiard has remarkable symmetry properties in the
case of pure gravity in $d+1$ spacetime dimensions, or supergravity theories in
10 or 11 spacetime dimensions, for which it turns out to be the fundamental
Weyl chamber of the Kac-Moody algebras $AE_d$, $E_{10}$, $BE_{10}$ or $DE_{10}$
(depending on the model). We analyse in this paper the billiards associated to
other theories containing gravity, whose toroidal reduction to three dimensions
involves coset models $G/H$ (with $G$ maximally non compact). We show that in
each case, the billiard is the fundamental Weyl chamber of the (indefinite)
Kac-Moody ``overextension'' (or ``canonical Lorentzian extension'') of the
finite-dimensional Lie algebra that appears in the toroidal compactification to
3 spacetime dimensions. A remarkable feature of the billiard properties,
however, is that they do not depend on the spacetime dimension in which the
theory is analyzed and hence are rather robust, while the symmetry algebra that
emerges in the toroidal dimensional reduction is dimension-dependent.
