Almost identical imitations of (3,1)-dimensional manifold pairs and the branched coverings
Let \((M,L)\) be a smooth (3,1)-manifold pair, i.e. a pair consisting of a smooth 3-manifold \(M\) and a properly embedded smooth 1-submanifold \(L\) of \(M\). An imitation of \((M,L)\) is the fixed point set of an involution \(\alpha\) of \((M,L)\times[-1,1]\) which interchanges \(M\times\{1\}\) and \(M\times\{-1\}\), and is thus some kind of approximation to \((M,L)\). There are several variants of this notion, which are obtained by imposing on \(\alpha\) various degrees of `similarity' to the standard involution \((x,t)\mapsto(x,-t)\). For a general theory of imitations see the author's paper in [ibid. 26, No. 3, 447-464 (1989; Zbl 0701.57014)]. The author [ibid. No. 4, 743-758 (1989; Zbl 0701.57015)] has proved that every closed (3,1)-manifold pair \((M,L)\) has an infinite family of almost identical imitations \((M^*,L^*)\) (this is the strongest variant of imitation, and its definition implies that \(M^*\approx M)\) such that the exterior of \(L^*\) in \(M^*\) is hyperbolic. In the present paper he improves on this result by first slightly strengthening the definition of almost identical imitation and secondly by proving that the \((M^*,L^*)\) can be chosen so that not only the exterior of \(L^*\) in \(M^*\) but also any finite regular covering space of the exterior of any (open and closed) part of \(L^*\) branched along the rest of \(L^*\) is hyperbolic. (Actually, the author allows compact manifolds with boundary in both papers, but for nonclosed manifolds the statement is slightly more complicated.) The main theorem contains some additional statements on possible choices of the family of the imitations \((M^*,L^*)\); these statements concern the volumes and isometry groups of the hyperbolic 3-manifolds in question but are too complicated to state here. The same goes for several additional theorems and corollaries.
scientific article; zbMATH DE number 219874