Séance Séminaire

Séminaire Gaston Darboux

Do negatively curved manifolds with (non-uniform) lattices have a Margulis function?

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.