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This article is cited in 6 scientific papers (total in 6 papers)
Volume and fundamental group of a manifold of nonpositive curvature
S. V. Buyalo
Abstract:
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.
Received: 12.03.1983
Citation:
S. V. Buyalo, “Volume and fundamental group of a manifold of nonpositive curvature”, Mat. Sb. (N.S.), 122(164):2(10) (1983), 142–156; Math. USSR-Sb., 50:1 (1985), 137–150
Linking options:
https://www.mathnet.ru/eng/sm2281https://doi.org/10.1070/SM1985v050n01ABEH002737 https://www.mathnet.ru/eng/sm/v164/i2/p142
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Abstract page: | 322 | Russian version PDF: | 101 | English version PDF: | 12 | References: | 54 | First page: | 2 |
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