On the volumes of cusped, complex hyperbolic manifolds and orbifolds (Q1974853)

The author studies volumes of hyperbolic manifolds and orbifolds, specifically he only considers cusped (that is, noncompact, finite volume) complex hyperbolic manifolds and orbifolds. He proves the following: The smallest volume of a cusped complex hyperbolic 2-manifold is \(8\pi^2/3\). Furthermore, the author improves the complex hyperbolic version of Shimizu's lemma [cf. \textit{S. Kamiya}, Hiroshima Math. J. 21, 1-21 (1991; Zbl 0721.30032) and \textit{J. R. Parker}, Int. J. Math. 3, 291-308 (1992; Zbl 0761.32014)]. By making use of this Shimizu-type lemma, the author gives some results on the lower bound of volumes of cusped hyperbolic manifolds and orbifolds. For instance, be proves the following: Let \(M\) be a finite-volume, 2-dimensional complex orbifold with \(k\)-ends. Then there are disjoint, embedded cusp neighborhoods each with volume 1/4. In particular, \(\text{Vol} (M)\geq k/4\).