Examples of acylindrical 3-manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:00:16Z http://mathoverflow.net/feeds/question/98833 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98833/examples-of-acylindrical-3-manifolds Examples of acylindrical 3-manifolds HW 2012-06-05T02:05:37Z 2012-06-05T14:56:06Z <p>Let $C$ be the compact cylinder $S^1\times [0,1]$. A 3-manifold $M$ with incompressible boundary is called <em>acylindrical</em> 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 $\partial M$.</p> <blockquote> <p>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 $g$.</p> </blockquote> <p>I'm sure such things should be well known to the experts.</p> <p>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.</p> http://mathoverflow.net/questions/98833/examples-of-acylindrical-3-manifolds/98835#98835 Answer by Igor Rivin for Examples of acylindrical 3-manifolds Igor Rivin 2012-06-05T02:36:09Z 2012-06-05T02:36:09Z <p>I believe Bob Brooks constructs really cool examples in this paper:</p> <p>MR0860677 (88b:32050) Brooks, Robert(1-UCLA) Circle packings and co-compact extensions of Kleinian groups. Invent. Math. 86 (1986), no. 3, 461–469. </p> <p>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.</p> http://mathoverflow.net/questions/98833/examples-of-acylindrical-3-manifolds/98837#98837 Answer by Misha for Examples of acylindrical 3-manifolds Misha 2012-06-05T02:57:32Z 2012-06-05T02:57:32Z <p>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 $K$. </p> http://mathoverflow.net/questions/98833/examples-of-acylindrical-3-manifolds/98847#98847 Answer by Bruno Martelli for Examples of acylindrical 3-manifolds Bruno Martelli 2012-06-05T06:50:55Z 2012-06-05T06:50:55Z <p>You are looking for </p> <blockquote> <p>compact 3-manifolds that admit a hyperbolic metric with geodesic boundary</p> </blockquote> <p>equivalently</p> <blockquote> <p>compact 3-manifolds that do not contain any essential surface with $\chi \geq 0$</p> </blockquote> <p>with the additional requirement that every boundary component has the same genus $g$.</p> <p>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 <a href="http://www.ms.unimelb.edu.au/~snap/orb.html" rel="nofollow">orb</a> to check whether the complement has a hyperbolic structure with geodesic boundary.</p> <p>An alternative construction uses ideal triangulations, extending Thurson's original "knotted y" example from <a href="http://library.msri.org/books/gt3m/" rel="nofollow">his notes</a>. 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 <i>à la Casson</i> which excludes any normal surface with $\chi \geq 0$). </p> <p>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 $g=2$.</p> <p>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 <a href="http://www.ams.org/mathscinet/search/publdoc.html?arg3=&amp;co4=AND&amp;co5=AND&amp;co6=AND&amp;co7=AND&amp;dr=all&amp;pg4=AUCN&amp;pg5=TI&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=&amp;s5=Volumes%20of%20hyperbolic%20manifolds%20with%20geodesic%20boundary&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=2&amp;mx-pid=1293303" rel="nofollow">result of Myiamoto</a>) and smallest Matveev complexity: we have investigated these manifolds <a href="http://arxiv.org/abs/math.GT/0206156" rel="nofollow">here</a>. There are many such manifolds because there are many triangulations with one vertex and one edge: their number grow more than exponentially in $g$.</p> http://mathoverflow.net/questions/98833/examples-of-acylindrical-3-manifolds/98878#98878 Answer by Richard Kent for Examples of acylindrical 3-manifolds Richard Kent 2012-06-05T14:48:10Z 2012-06-05T14:56:06Z <p><img src="http://www.math.wisc.edu/~rkent/Suzuki.jpg" alt="alt text"></p> <p>The exterior of Suzuki's Brunnian graph on $n$-edges, here pictured with $n=7$, is irreducible, atoroidal, boundary incompressible, and acylindrical. See </p> <p>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 </p> <p>or </p> <p>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.</p> <p>(I think these manifolds may be contained in Bruno's list also.)</p>