Finite volume of the Gromov-truncated cosmological billiard in higher-dimensional gravity

We prove that the Gromov non-squeezing theorem, applied to the compact Kähler factor of the internal space in compactified gravity, furnishes a hard wall on each Kähler modulus that caps every cusp of the BKL cosmological billiard. In two dimensions, the Gauss–Bonnet theorem gives the billiard area exactly as π − θ. In the general case, every cusp of the Weyl chamber involves at least one compact modulus, each such cusp is intersected by the corresponding Gromov wall at a finite interior point with positive dihedral angle, and the resulting polytope is compact with finite volume. Every classical trajectory bounces infinitely often; the cosmological singularity is not attained.