Examples of acylindrical 3-manifolds - MathOverflow

Examples of acylindrical 3-manifolds

2012-06-05T02:05:37Z

Let $C$ be the compact cylinder $S^1\times [0,1]$. A 3-manifold $M$ with incompressible boundary is called acylindrical if every map $(C,\partial C)\to (M,\partial M)$ that sends the components of $\partial C$ to essential curves in $\partial M$ is homotopic rel $\partial C$ into $\partial M$.

I'm looking, for each $g\geq 2$, for examples of compact, orientable, acylindrical, hyperbolic 3-manifolds $M_g$ with non-empty, incompressible boundary such that each component of $\partial M_g$ is homeomorphic to the surface of genus $g$.

I'm sure such things should be well known to the experts.

Here's a little motivation. Such examples would be useful because, given an arbitrary hyperbolic 3-manifold $N$ with incompressible boundary, you can glue copies of the $M_g$ to the non-toroidal boundary components of $N$ and the result, by Geometrization (for Haken 3-manifolds, so you only need Thurston, not Perelman), is a hyperbolic 3-manifold of finite volume.

Answer by Igor Rivin for Examples of acylindrical 3-manifolds

2012-06-05T02:36:09Z

I believe Bob Brooks constructs really cool examples in this paper:

MR0860677 (88b:32050) Brooks, Robert(1-UCLA) Circle packings and co-compact extensions of Kleinian groups. Invent. Math. 86 (1986), no. 3, 461–469.

The idea is that given a circle-packed hyperbolic surface (such are dense in teichmuller space, by an earlier theorem of Brooks) one can manufacture a hyperbolic manifold whose boundary consists of four copies of the surface.

Answer by Misha for Examples of acylindrical 3-manifolds

2012-06-05T02:57:32Z

See the proof of theorem 19.8 in my book. I explain two constructions, one via orbifold trick as Igor explained and the other using Meyers' theorem. Meyers' idea is: Take genus $g$ handlebody $H$ and take a knot $K\subset H$ which busts all essential annuli and disks in $H$. Then do a Dehn surgery on $H$ along $K$.

Answer by Bruno Martelli for Examples of acylindrical 3-manifolds

2012-06-05T06:50:55Z

You are looking for

compact 3-manifolds that admit a hyperbolic metric with geodesic boundary

equivalently

compact 3-manifolds that do not contain any essential surface with $\chi \geq 0$

with the additional requirement that every boundary component has the same genus $g$.

To construct such manifolds you may draw pictures of suffficiently knotted graphs in $S^3$ consisting of some copies of genus-$g$ graphs, and take their complements. Then you can use orb to check whether the complement has a hyperbolic structure with geodesic boundary.

An alternative construction uses ideal triangulations, extending Thurson's original "knotted y" example from his notes. Pick a bunch of tetrahedra and pair their faces so that every edge in the resulting triangulation has valence $> 6$. Then remove an open star at each vertex. Geometrization guarantees that the resulting manifold admits a hyperbolic metric with geodesic boundary (because you can put an angle structure à la Casson which excludes any normal surface with $\chi \geq 0$).

For example, you can take $g\geqslant 2$ tetrahedra and pair the faces in such a way that the resulting triangulation consists of one vertex and one edge only (which has thus valence $6g$). The resulting manifold is a hyperbolic 3-manifold with connected genus-$g$ geodesic boundary. Its hyperbolic structure is simply obtained by giving each tetrahedron the structure of a truncated regular hyperbolic tetrahedron with all dihedral angles of angle $\pi/(3g)$. Thurston's knotted y is obtained in this way for $g=2$.

The manifolds constructed in this way are "the simplest ones" among those having a connected genus-$g$ boundary, from different viewpoints: they have smallest volume (as a consequence of a result of Myiamoto) and smallest Matveev complexity: we have investigated these manifolds here. There are many such manifolds because there are many triangulations with one vertex and one edge: their number grow more than exponentially in $g$.

Answer by Richard Kent for Examples of acylindrical 3-manifolds

2012-06-05T14:48:10Z

The exterior of Suzuki's Brunnian graph on $n$-edges, here pictured with $n=7$, is irreducible, atoroidal, boundary incompressible, and acylindrical. See

Luisa Paoluzzi and Bruno Zimmermann. On a class of hyperbolic 3-manifolds and groups with one defining relation. Geom. Dedicata, 60(2):113–123, 1996

Akira Ushijima. The canonical decompositions of some family of compact orientable hyperbolic 3-manifolds with totally geodesic boundary. Geom. Dedicata, 78(1):21–47, 1999.

(I think these manifolds may be contained in Bruno's list also.)