Almost identical imitations of (3,1) - dimensional manifold pairs and the branched coverings

Abstract

By a good (3, \)-manίfold pair (M, L) (or a good 1-manifold L in a 3-manifold M)y we mean that M i s a compact connected oriented 3-manifold and L is a compact proper smooth 1-submanifold of M such that any 2-sphere component of the boundary dM meets L with at least three points. For a compact connected oriented 3-manifold Ey let d0E be the union of all tori in dE and 81E=dE— dQE. Let int E=E— dE and int0 E=E—d0E. A compact connected oriented 3-manifold E is said to be hyperbolic if int £ (when dιE=ψ) or the double D (int0 E) pasting along dγE (when dλE 4= 0) has a complete Riemannian structure of constant curvature — 1. Then we define the volume Yo\E of E to be the hyperbolic volume Yo\(mtE) (when dλE=0) or the half hyperbolic volume Vol (D(int0 E))β (when d1E=^0)y and the isometry group Isom E of E to be the hyperbolic isometry group Isom (int E) (when dλE=0) or the quotient by T of the following subgroup {/elsom (D(int o £)) |/τ=τ/} (when dλE^0)y where T denotes the unique isometry of D(into2?) induced from the involution of D(int0 E) interchanging the two copies of into£'(cf. [22]). By Mostow rigidity theorem (cf. [23], [24]), Voli? is a topological invariant of E and IsomZ? is a unique (up to conjugations) finite subgroup of the difΐeomorphism group Diff E. Furthermore, there is a natural isomorphism IsomE^Outπ1(E)=Autπ1(E)l Inn πι(E) and for any finite subgroup G of Diff E there is a natural monomorphism G^Out πι(E), so that G is isomorphic to a subgrpup of Isom E. In a previous paper [8], for each good (3,l)-manifold pair (My L), we have constructed an infinite family of almost identical imitations (M, L*) of (M, L) such that the exterior £(L*, M) of L* in M is hyperbolic. In this paper, we shall strengthen this result from the viewpoint of regular branched coverings.*