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Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $\varphi\in(0,\pi)\setminus\mathbb{Q}\pi$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_\theta$ of singular hyperbolic metrics on a torus with a single cone point of angle $\theta=2(\pi-\varphi)$, and answer an analogous question for the holonomy map $\rho_\xi$ of such a hyperbolic structure $\xi$. In an appendix by X.~Gao, concrete examples of $\theta$ and $\xi\in\mathfrak{C}_\theta$ are given where the image of each $\rho_\xi$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds., Comment: 32 pages. To appear in Math. Z

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.

and it is easy to see that this is constant on conjugacy classes. In this setting, it can be shown that η(s) and ζ(s) have extensions as meromorphic functions to the entire complex plane. The proof relies on non-commutative harmonic analysis and the functions are studied via the spectral properties of the Laplace-Beltrami operator. In this note we shall consider an analogous situation where we replace H by an R-tree. R-trees are a class of metric spaces which generalize the more familiar simplicial trees. In recent years there has been much interesting work on group actions on R-trees (for a good survey see [10]) which can, in part, be viewed as a generalization of the now classical Bass-Serre theory of group actions on trees [16]. In particular, Morgan and Shalen have shown that the fundamental groups Γ of compact surfaces M with Euler characteristic strictly less than −1 act freely on R-trees. More precisely, they show that given a hyperbolic structure on M , there

For any two closed orientable surfaces, there is a map of nonzero degree from one to another. This result is based on the fact that the orientable closed surface are classified completely. There is no such a classification in dimension 3. In this paper, all closed aspherical 3-manifolds are divided into eight classes. A complete answer of the existences of nonzero degree maps between 3-manifolds in the different classes is obtained in a reasonable sense. The existence of nonzero degree maps between 3-manifolds in the same class are also discussed. There are several results of independent interest. It is noted that, the existence of a map of nonzero degree between two 3-manifolds is a very restrictive condition posed on those two 3-manifolds, which is much different from the case of dimension 2. The shape of our work is certainly influenced by Thurston. However, both the statements and the proofs of our theorems do not involve the Thurston's geometries. We understand that if we state the results in the category of Haken manifolds and Thurston's geometrical manifolds and invoke the G r o m o v Norm of Hyperbol ic 3-manifold (see Chap. 6 of r T l ] and Chap. 4 of [-Gro]), the proof of some result in the paper can be simplified. We present our work in the way below for two reasons: (a) we prefer to prove all results in a uniform elementary way in the sense of using only the fundamental groups and the incompressible surfaces of 3-manifolds; (b)since the Thurston's most famous conjecture that every atoroidal closed irreducible 3-manifold has a hyperbolic structure is still open, we get the results in a theoretically wider category. There are 5 sections after the introduction. All manifolds and maps under consideration are piece-wise linear. (There is nothing lost under this assumption, see Chap. 1 of [Hi.) In the first four sections of these five, all 3-manifolds are orientable and irreducible. Section 1 states the theorems proved in the paper. Section 2 provides results which are used to prove the theorems. Section 3 and Sect. 4 are the proofs of the theorems. Various examples are given in Sect. 4.

Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast to uniform hyperbolicity, singular hyperbolicity does not immediately imply robust topological properties, such as the transitivity. In this paper, we prove that on an open and dense subset of the space of $$C^1$$ vector fields of a compact manifold, any singular hyperbolic attractors is robustly transitive.