Similar fillings and isolation of cusps of hyperbolic 3-manifolds (Q952911)

In the paper under review, the author gives a further analysis of a certain class of hyperbolic 3-manifolds introduced by \textit{R. Frigerio, B. Martelli} and \textit{C. Petronio} [J. Differ. Geom. 64, No.~3, 425--455 (2003; Zbl 1073.57010)]. For given positive integers \(g\) and \(k\), the class consists of compact oriented 3-manifolds having an ideal triangulation with \(g+k\) tetrahedra and with a closed surface of genus \(g\) and \(k\) tori as boundary. Actually each element was shown to admit a complete finite-volume hyperbolic structure with geodesic boundary. The author studies three particular properties for the manifolds in the class: Isolation of cusps, Nonisolation of the boundary, Similar fillings. First, it is shown that, for each element of the class having \(k\) cusps, any \(h\)-tuple of cusps (\(h \leq k\)) is geometrically isolated from the others, i.e., any small deformation of the hyperbolic structure on the manifold induced by Dehn filling on the latter cusps does not affect the Euclidean structure at the former cusps. This gives new phenomena which cannot be explained by known reasons described in \textit{D. Calegari} [Proc. Am. Math. Soc. 129, No.10, 3109-3119 (2001; Zbl 0971.57022 )]. Next, it is shown that, for each element of the class, an infinite set of Dehn fillings on the manifold gives hyperbolic 3-manifolds with mutually non-isometric totally geodesic boundaries. Finally, it is shown that, for any \(k > 0\), there exists an element \(X_k\) in the class with \(k\) cusps having the following property: There exists a finite set of slopes on each of the \(k\) boundary tori such that the manifolds obtained by Dehn filling along the slopes not included in the described finite sets on \(h\) tori (\(h \leq k\)) are all hyperbolic, and for each of them, at least \(\frac{k! 3^h}{h! (k - h)!}\) hyperbolic Dehn fillings on \(X_k\) give pairwise non-homeomorphic but geometrically similar manifolds, i.e., the manifolds share the same value of hyperbolic volume, cusp volume, cusp shape, length of the shortest return paths, complex length of the shortest closed geodesics, the first homology groups, Heegaard genus, Turaev-Viro invariants. Again this gives new phenomena which cannot be explained by known reasons described in \textit{C. D. Hodgson, R. G. Meyerhoff} and \textit{J. R. Weeks} [Topology '90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 195--206 (1992; Zbl 0767.57007)].