On hyperbolic once-punctured-torus bundles.

*(English)*Zbl 1015.57501Among the early examples of hyperbolic \(3\)-manifolds were certain bundles over the circle with surface fiber. The universal cover of the fiber is the hyperbolic disk \(H^2\), and the inclusion of the fiber lifts to an equivariant imbedding of \(H^2\) into \(H^3\). In an unpublished but highly influential 1989 preprint, the first author and Thurston showed that when the fiber is a closed surface, this inclusion extends to an equivariant surjective map from the circle at infinity to the sphere at infinity.

When the fiber of the surface bundle \(M\) is not closed, each fiber in \(H^3\) has a spine which is a tree. C. T. McMullen [Invent. Math. 146, No. 1, 35-91 (2001; Zbl 1061.37025)] showed that when the fiber of \(M\) is a once-punctured torus, the ends of this tree reach every point of the sphere at infinity. Deleting a single edge from this tree leaves two complementary subtrees. For the case when \(M\) is the figure-8 knot, R. C. Alperin, W. Dicks, and J. Porti [Topology Appl. 93, No. 3, 219-259 (1999; Zbl 0926.57008)] showed that the points in the sphere at infinity which are reached by the ends of one of these two subtrees form complementary closed disks. On the Jordan curve which is their intersection, the set of cusps is dense, in fact the curve is made up of two arcs with common endpoints such that the set of cusps in each arc is dense and forms a single orbit of a finitely generated semigroup of MĂ¶bius transformations. In the paper under review, these results are extended to all hyperbolic once-punctured torus bundles.

The authors use a very algebraic approach to give a description of the set \({\mathcal E}\) of ends of the tree, and to develop a factorization \({\mathcal E}\to S^1\to S^2\), where \(S^2\) is a topological \(2\)-sphere. A result of B. H. Bowditch [The Cannon-Thurston map for punctured-surface groups, preprint 2002] shows that this \(S^2\) is equivariantly homeomorphic to the \(2\)-sphere at infinity. The authors obtain additional information about the exceptional set (the points in \(S^2\) which are the image of two or more points in \(S^1\)). In particular, the Hausdorff dimension of its preimage in \(S^1\) is \(0\). The paper is very precisely and carefully written.

When the fiber of the surface bundle \(M\) is not closed, each fiber in \(H^3\) has a spine which is a tree. C. T. McMullen [Invent. Math. 146, No. 1, 35-91 (2001; Zbl 1061.37025)] showed that when the fiber of \(M\) is a once-punctured torus, the ends of this tree reach every point of the sphere at infinity. Deleting a single edge from this tree leaves two complementary subtrees. For the case when \(M\) is the figure-8 knot, R. C. Alperin, W. Dicks, and J. Porti [Topology Appl. 93, No. 3, 219-259 (1999; Zbl 0926.57008)] showed that the points in the sphere at infinity which are reached by the ends of one of these two subtrees form complementary closed disks. On the Jordan curve which is their intersection, the set of cusps is dense, in fact the curve is made up of two arcs with common endpoints such that the set of cusps in each arc is dense and forms a single orbit of a finitely generated semigroup of MĂ¶bius transformations. In the paper under review, these results are extended to all hyperbolic once-punctured torus bundles.

The authors use a very algebraic approach to give a description of the set \({\mathcal E}\) of ends of the tree, and to develop a factorization \({\mathcal E}\to S^1\to S^2\), where \(S^2\) is a topological \(2\)-sphere. A result of B. H. Bowditch [The Cannon-Thurston map for punctured-surface groups, preprint 2002] shows that this \(S^2\) is equivariantly homeomorphic to the \(2\)-sphere at infinity. The authors obtain additional information about the exceptional set (the points in \(S^2\) which are the image of two or more points in \(S^1\)). In particular, the Hausdorff dimension of its preimage in \(S^1\) is \(0\). The paper is very precisely and carefully written.

Reviewer: Darryl McCullough (Norman)