Margulis proved that a Cartan-Hadamard manifold X of strictly negative
curvature admitting cocompact lattices has a volume function
v(x,R) = vol(B(x,R)) which is asymptotically equivalent to a function
m(x)e^{hR}.
The function m(x) is called the Margulis' function of X.
Is this still true for manifolds with non-cocompact lattices?
We investigate the asymptotic behaviour of the volume function of
Cartan-Hadamard manifold of strictly negative curvature admitting a
non-cocompact lattice G. For 1/4-pinched spaces, we prove that the volume
growth is always purely exponential (i.e. v(x,R) is bounded between two
positive constants). However, we show that for \alpha-pinched
spaces, with arbitrary \alpha<1/4, the growth function
v(x,R) can be exponential, sub-exponential or (at present, still
conjecturally) even super-exponential, depending on the critical exponent
of the parabolic subgroups and on the finiteness of the Bowen-Margulis
measure.