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**Degenerations of hyperbolic structures. III: Actions of 3-manifold groups on trees and Thurston’s compactness theorem.**
*(English)*
Zbl 0661.57004

This paper is the last in a series of three papers [for parts I and II see ibid. 120, 401-476 (1984; Zbl 0583.57005) and ibid. 127, 403-456 (1988; Zbl 0656.57003)]. The main result of this paper is the following purely topological theorem.

Theorem. Let \({\mathcal M}\) be a compact, orientable, connected, irreducible 3-manifold. Let \(\Gamma =\pi_ 1({\mathcal M})\). Suppose that \(\Gamma\) \(\times T\to T\) is an action of \(\Gamma\) on an \({\mathbb{R}}\)-tree such that for any non-degenerate segment \(A\subset T\) the subgroup of \(\Gamma\) fixing A does not contain a free group of rank 2. Let \(\Sigma\) \(\subset {\mathcal M}\) be the characteristic submanifold. Then for any component C of \({\mathcal M}\setminus \Sigma\) the subgroup \(\pi_ 1(C)\subset \Gamma\) has a fixed point in T.

This theorem can be easily proved by standard 3-manifold topology in the case when T is a simplicial tree. To give a similar proof in the general case, one should replace surfaces by measured laminations in such a proof. This is the main idea of the proof, but its realisation requires a lot of work. It is based on the theory of measured laminations in 3- manifolds developed in the second paper in the series. As the main application of this theorem, the authors give a new proof of the following theorem of W. Thurston [Ann. Math. II. Ser. 124, 203-246 (1986)].

Theorem. If \({\mathcal M}\) is a compact, orientable, irreducible 3-manifold, if \(\partial {\mathcal M}\) is incompressible and if (\({\mathcal M},\partial {\mathcal M})\) contains no essential annuli or tori, then the subspace of characters of discrete and faithful representations of \(\pi_ 1({\mathcal M})\) into \(SL_ 2({\mathbb{C}})\) or \(PGL_ 2({\mathbb{C}})\) is compact.

To deduce this theorem from the previous one, the authors use an abstract theory of spaces of characters developed in the first part. This theorem is an important piece (but only a piece) of the yet to be published proof of Thurston’s famous hyperbolization theorem. While Thurston’s proof is based on some new concepts in hyperbolic geometry, the approach used by authors is based on more familiar ideas from 3-manifold topoloy and foliation theory. So this series of paper can be considered as an important step in understanding and absorbing Thurston’s ideas for a wider circle of mathematicians.

To finish the review, I would like to cite the following statement of the authors. “The way in which we approached this work was pervasively influenced by the mathematical world view of two people - Dennis Sullivan and William Thurston”.

Theorem. Let \({\mathcal M}\) be a compact, orientable, connected, irreducible 3-manifold. Let \(\Gamma =\pi_ 1({\mathcal M})\). Suppose that \(\Gamma\) \(\times T\to T\) is an action of \(\Gamma\) on an \({\mathbb{R}}\)-tree such that for any non-degenerate segment \(A\subset T\) the subgroup of \(\Gamma\) fixing A does not contain a free group of rank 2. Let \(\Sigma\) \(\subset {\mathcal M}\) be the characteristic submanifold. Then for any component C of \({\mathcal M}\setminus \Sigma\) the subgroup \(\pi_ 1(C)\subset \Gamma\) has a fixed point in T.

This theorem can be easily proved by standard 3-manifold topology in the case when T is a simplicial tree. To give a similar proof in the general case, one should replace surfaces by measured laminations in such a proof. This is the main idea of the proof, but its realisation requires a lot of work. It is based on the theory of measured laminations in 3- manifolds developed in the second paper in the series. As the main application of this theorem, the authors give a new proof of the following theorem of W. Thurston [Ann. Math. II. Ser. 124, 203-246 (1986)].

Theorem. If \({\mathcal M}\) is a compact, orientable, irreducible 3-manifold, if \(\partial {\mathcal M}\) is incompressible and if (\({\mathcal M},\partial {\mathcal M})\) contains no essential annuli or tori, then the subspace of characters of discrete and faithful representations of \(\pi_ 1({\mathcal M})\) into \(SL_ 2({\mathbb{C}})\) or \(PGL_ 2({\mathbb{C}})\) is compact.

To deduce this theorem from the previous one, the authors use an abstract theory of spaces of characters developed in the first part. This theorem is an important piece (but only a piece) of the yet to be published proof of Thurston’s famous hyperbolization theorem. While Thurston’s proof is based on some new concepts in hyperbolic geometry, the approach used by authors is based on more familiar ideas from 3-manifold topoloy and foliation theory. So this series of paper can be considered as an important step in understanding and absorbing Thurston’s ideas for a wider circle of mathematicians.

To finish the review, I would like to cite the following statement of the authors. “The way in which we approached this work was pervasively influenced by the mathematical world view of two people - Dennis Sullivan and William Thurston”.

Reviewer: N.V.Ivanov

### MSC:

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

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

57R30 | Foliations in differential topology; geometric theory |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |

20F99 | Special aspects of infinite or finite groups |