Geodesic
Volume and fundamental group of a manifold of nonpositive curvature
Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 137-150 
S. V. Buyalo. Volume and fundamental group of a manifold of nonpositive curvature. Sbornik. Mathematics, Tome 50 (1985) no. 1, pp. 137-150. http://geodesic.mathdoc.fr/item/SM_1985_50_1_a8/
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     author = {S. V. Buyalo},
     title = {Volume and fundamental group of a~manifold of nonpositive curvature},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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[4] Cheeger J., “Finiteness theorems for Riemannian manifolds”, Amer. J. Math., 92:1 (1970), 61–74 | DOI | MR | Zbl

[5] Buyalo S. V., “Mnogoobraziya nepolozhitelnoi krivizny s malym ob'emom”, Matem. zametki, 29:2 (1981), 243–252 | MR | Zbl

[6] Gromov M., “Manifolds of negative curvature”, J. Diff. Geom., 13:2 (1978), 223–230 | MR | Zbl

[7] Lawson H. S., Yau S. T., “Compact manifolds of nonpositive curvature”, J. Diff. Geom., 7 (1972), 211–228 | MR | Zbl

[8] Wolf J. A., “Homogeneity and bounded isometries in manifolds of negative curvature”, Illinois J. Math., 1964, no. 8, 14–18 | MR | Zbl

[9] Buyalo S. V., “Ob'em i fundamentalnaya gruppa mnogoobraziya nepolozhitelnoi krivizny”, Tezisy dokladov simpoziuma po geometrii v tselom i osnovaniyam teorii otnositelnosti, Novosibirsk, 1982, 20–21