Simple quotients of hyperbolic 3-manifold groups

We show that hyperbolic -manifolds have residually simple fundamental group.

     

Degree-one maps between hyperbolic 3-manifolds with the same volume limit

Suppose that are degree-one maps between closed hyperbolic 3-manifolds with

Then, our main theorem, Theorem 2, shows that, for all but finitely many , is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston's rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.

     

Vol3 and other exceptional hyperbolic 3-manifolds

D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional hyperbolic manifolds in their proof that a manifold which is homotopy equivalent to a hyperbolic manifold is hyperbolic. These families are each conjectured to consist of a single manifold. In fact, an important point in their argument depends on this conjecture holding for one particular exceptional family. In this paper, we prove the conjecture for that particular family, showing that the manifold known as in the literature covers no other manifold. We also indicate techniques likely to prove this conjecture for five of the other six families.

     

3-manifolds that admit knotted solenoids as attractors

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.

     

The 3-manifold recognition problem

We introduce a natural Relative Simplicial Approximation Property for maps from a 2-cell to a generalized 3-manifold and prove that, modulo the Poincaré Conjecture, 3-manifolds are precisely the generalized 3-manifolds satisfying this approximation property. The central technical result establishes that every generalized 3-manifold with this Relative Simplicial Approximation Property is the cell-like image of some generalized 3-manifold having just a 0-dimensional set of nonmanifold singularities.

     

Geodesic knots in closed hyperbolic 3-manifolds

We consider the existence of simple closed geodesics or “geodesic knots” in finite volume orientable hyperbolic 3-manifolds. Every such manifold contains at least one geodesic knot by results of Adams, Hass and Scott in (Adams et al. Bull. London Math. Soc. 31: 81–86, 1999). In (Kuhlmann Algebr. Geom. Topol. 6: 2151–2162, 2006) we showed that every cusped orientable hyperbolic 3-manifold in fact contains infinitely many geodesic knots. In this paper we consider the closed manifold case, and show that if a closed orientable hyperbolic 3-manifold satisfies certain geometric and arithmetic conditions, then it contains infinitely many geodesic knots. The conditions on the manifold can be checked computationally, and have been verified for many manifolds in the Hodgson-Weeks census of closed hyperbolic 3-manifolds. Our proof is constructive, and the infinite family of geodesic knots spiral around a short simple closed geodesic in the manifold.      

An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let be a hyperbolic 3-manifold obtained by identifying the faces of convex ideal polyhedra . If the faces of are glued to , then can be decomposed into ideal tetrahedra by subdividing the 's.

     

Divergence in 3-manifold groups

The divergence of the fundamental group of compact irreducible 3-manifolds satisfying Thurston's geometrization conjecture is calculated. For every closed Haken 3-manifold group, the divergence is either linear, quadratic or exponential, where quadratic divergence occurs precisely for graph manifolds and exponential divergence occurs when a geometric piece has hyperbolic geometry. An example is given of a closed 3-manifoldN with a Riemannian metric of nonpositive curvature such that the divergence is quadratic and such that there are two geodesic rays in the universal coverN whose divergence is precisely quadratic, settling in the negative a question of Gromov's.Partially supported by NSF grant DMS-9200433.     

Height in splittings of hyperbolic groups

SupposeH is a hyperbolic subgroup of a hyperbolic groupG. Assume there existsn > 0 such that the intersection ofn essentially distinct conjugates ofH is always finite. Further assumeG splits overH with hyperbolic vertex and edge groups and the two inclusions ofH are quasi-isometric embeddings. ThenH is quasiconvex inG. This answers a question of Swarup and provides a partial converse to the main theorem of [23].     

An infinitely generated intersection of geometrically finite hyperbolic groups

Two discrete, geometrically finite subgroups of the isometries of hyperbolic n-space () are defined whose intersection is infinitely generated. This settles, in dimensions 4 and above, a long-standing question in Kleinian and hyperbolic groups reiterated at a problem session chaired by Bernard Maskit at the AMS meeting 898, March 3-5, 1995, a conference in honor of Bernard Maskit's 60th birthday.

     

The visual core of a hyperbolic 3-manifold

In this note we introduce the notion of the visual core of a hyperbolic 3-manifold , and explore some of its basic properties. We investigate circumstances under which the visual core of a cover of N embeds in N, via the usual covering map . We go on to show that if the algebraic limit of a sequence of isomorphic Kleinian groups is a generalized web group, then the visual core of the algebraic limit manifold embeds in the geometric limit manifold. Finally, we discuss the relationship between the visual core and Klein-Maskit combination along component subgroups. Received: 16 March 1999 / Revised version: 14 May 2001 / Published online: 19 October 2001     

Tits alternative for 3-manifold groups

Let M be an irreducible, orientable, closed 3-manifold with fundamental group G. We show that if the pro-p completion of G is infinite then G is either soluble-by-finite or contains a free subgroup of rank 2. Both authors are partially supported by “Bolsa de produtividade de pesquisa” from CNPq, Brazil. Received: 16 February 2006     

Recognition of subgroups of direct products of hyperbolic groups

We give examples of direct products of three hyperbolic groups in which there cannot exist an algorithm to decide which finitely presented subgroups are isomorphic.

     

Symplectic tori in homotopy 's

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic -manifold , obtained by knot surgery on the rational elliptic surface with the left-handed trefoil knot . has the simplest homotopy type among simply-connected symplectic -manifolds known to exhibit such a property.

     

Complete flat surfaces with two isolated singularities in hyperbolic 3-space

We construct examples of flat surfaces in H³ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in H³ with only one end and at most two isolated singularities.     

A sharper tits alternative for 3-manifold groups

The following theorem is proven. LetM be a closed, orientable, irreducible 3-manifold such that rankH ₁(M, ℤ/pℤ)≥3 for some primep. Then either π₁(M) is virtually solvable or it contains a free group of rank 2.     

Estimates for the extinction time for the Ricci flow on certain -manifolds and a question of Perelman

We show that the Ricci flow becomes extinct in finite time on any Riemannian -manifold without aspherical summands.