Closed geodesics on non-simply-connected manifolds

The article contains the next
Theorem. If a Riemannian manifold $m^N$ is closed and there is a normal Abelian subgroup $G\subset\pi_1(M^n)$ of nonzero finite rank such that the factor-group $\pi_1(M^n)/G$ is aperiodic, i.e., it contains elements of infinite order then $N(t)\geqslant C_t/\ln t$, where $N(t)$ the number of geometrically distinct geodesies of length at most t and $C$ is a positive constant.
The theorem implies an analogous estimate for the growth of $N(t)$for closed manifolds with almost solvable fundamental groups.