  We compute the spatial-volume dependence of the spectrum of 4D SU(3 <= N <=
6) gauge theories by lattice Monte-Carlo techniques. Setting the scale with the
string tension, the spatial volume is L^3 with 0.78fm <= L <= 2.3fm. The
Euclidean `time' direction is kept large enough to be considered infinite and
the boundary conditions are periodic in all four dimensions. We study the
mixing of torelon pairs with the scalar and tensor glueballs, using a 2x2
Hamiltonian based on large-N counting rules. Looking to the other symmetry
channels, finite-volume effects on the glueball spectrum are already
surprisingly small in SU(3), and they become rapidly smaller as N is increased:
several low-lying SU(6) states have no finite-volume corrections at the 1-2%
level, at least down to L=0.9fm. We discuss the relation of this work with
analytic calculations in small and intermediate volume, and with Eguchi-Kawai
reduction in the planar limit.
