Volume and fundamental group of a manifold of nonpositive curvature

The relationship between the volume $v(M)$ and the fundamental group $\pi_1(M)$ of a closed manifold $M$ of nonpositive curvature $K_\sigma$, $-1\leqslant K_\sigma\leqslant0$, is studied. The main result asserts that if $\pi_1(M)$ does not contain nontrivial normal abelian subgroups, then
$$ v(M)\geqslant\beta_ne^{-\alpha_nD(M)}, $$
where $D(M)$ is the diameter of $M$ and $\alpha_n$, $\beta_n>0$ depend only on the dimension of $M$. From this it follows, in particular, that for given $n\geqslant2$ and $C>0$ there exist only finitely many pairwise nonhomeomorphic $n$-dimensional closed $M$ with $-1\leqslant K_\sigma\leqslant0$ and $D(M)\leqslant C$.
Figures: 1.
Bibliography: 9 titles.