Finiteness conditions for 3-manifolds with boundary.

*(English)*Zbl 0642.57001The paper is intended as a refinement of an unpublished work “On Ahlfors finiteness theorem” by R. Kulkarni and P. Shalen (1984). First, with no hypothesis on the 3-manifold M, the authors obtain sharp bounds (in terms of the number of elements in a minimal generating set for \(\pi_ 1(M))\) for the maximal size of a collection of boundary components of M that are tori, Klein bottles, open MĂ¶bius bands, and pairwise nonhomotopic noncontractible annuli (For Kleinian 3-manifolds, these yield D. Sullivan’s results - Acta Math. 147, 289-299 (1981; Zbl 0502.57004)).

Secondly, the authors give sharp bounds for the topology of the boundary of M (the estimates of L. Bers in the case of Kleinian 3-manifolds - Am. J. Math. 89, 1078-1082 (1967; Zbl 0167.070)) with the additional (and necessary) hypothesis of intact boundary. This condition, needed to obtain these finiteness theorems, is essentially due to R. Kulkarni and P. Shalen (Lemma 5.1 and (5.2.1) in their above work): M has “intact boundary” iff the following condition holds: Suppose V is any simply-connected codimension-zero submanifold of M which has as its boundary a single properly-imbedded 2-disc D. Then D lies in a compact component of \(\partial V\) (necessarily a 2-sphere).

The second important approach to prove these finiteness theorems is the idea of using a core, i.e. a compact submanifold X of M such that \(X\subset M\) induces an isomorphism of fundamental groups [see G. P. Scott, J. Lond. Math. Soc., II. Ser. 6, 437-440 (1973; Zbl 0254.57003); ibid. 7, 246-250 (1973; Zbl 0266.57001)], which seems to have originated with A. Marden [Ann. Math., II. Ser. 99, 383-462 (1974; Zbl 0282.30014)], R. Kulkarni and P. Shalen (see above) and W. Abikoff [Proc. Am. Math. Soc. 97, 593-601 (1986; Zbl 0606.30046)]. In particular, using the relative core theorem [the second author, Q. J. Math., Oxf. II. Ser. 37, 299-307 (1986; Zbl 0628.57008)] and the approach of Kulkarni-Shalen, the authors give quick proofs of Bers’ estimate and Sullivan’s theorem.

Secondly, the authors give sharp bounds for the topology of the boundary of M (the estimates of L. Bers in the case of Kleinian 3-manifolds - Am. J. Math. 89, 1078-1082 (1967; Zbl 0167.070)) with the additional (and necessary) hypothesis of intact boundary. This condition, needed to obtain these finiteness theorems, is essentially due to R. Kulkarni and P. Shalen (Lemma 5.1 and (5.2.1) in their above work): M has “intact boundary” iff the following condition holds: Suppose V is any simply-connected codimension-zero submanifold of M which has as its boundary a single properly-imbedded 2-disc D. Then D lies in a compact component of \(\partial V\) (necessarily a 2-sphere).

The second important approach to prove these finiteness theorems is the idea of using a core, i.e. a compact submanifold X of M such that \(X\subset M\) induces an isomorphism of fundamental groups [see G. P. Scott, J. Lond. Math. Soc., II. Ser. 6, 437-440 (1973; Zbl 0254.57003); ibid. 7, 246-250 (1973; Zbl 0266.57001)], which seems to have originated with A. Marden [Ann. Math., II. Ser. 99, 383-462 (1974; Zbl 0282.30014)], R. Kulkarni and P. Shalen (see above) and W. Abikoff [Proc. Am. Math. Soc. 97, 593-601 (1986; Zbl 0606.30046)]. In particular, using the relative core theorem [the second author, Q. J. Math., Oxf. II. Ser. 37, 299-307 (1986; Zbl 0628.57008)] and the approach of Kulkarni-Shalen, the authors give quick proofs of Bers’ estimate and Sullivan’s theorem.

Reviewer: B.N.Apanasov

##### MSC:

57M05 | Fundamental group, presentations, free differential calculus |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

57S30 | Discontinuous groups of transformations |