Totally geodesic surfaces in fibered 3-manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:40:35Z http://mathoverflow.net/feeds/question/38188 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38188/totally-geodesic-surfaces-in-fibered-3-manifolds Totally geodesic surfaces in fibered 3-manifolds thku 2010-09-09T15:39:08Z 2010-09-09T20:43:47Z <p>Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface?</p> <p>(Of course such manifolds exist if the 'Virtually Fibered Conjecture' were correct, since a geodesic surface lifts to the fibered cover. But is there something more eplicit?) </p> http://mathoverflow.net/questions/38188/totally-geodesic-surfaces-in-fibered-3-manifolds/38189#38189 Answer by HW for Totally geodesic surfaces in fibered 3-manifolds HW 2010-09-09T16:05:11Z 2010-09-09T16:05:11Z <p>This example is neither particularly easy nor explicit, but it is at least a definite family of examples.</p> <p>It follows from a paper of <a href="http://www.math.jussieu.fr/~bergeron/Travaux_files/BHW%2528lastversion%2529.pdf" rel="nofollow">Bergeron--Haglund--Wise</a> and work of <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=AUCN&amp;pg6=PC&amp;pg7=ALLF&amp;pg8=ET&amp;review_format=html&amp;s4=agol%252C%2520i%2a&amp;s5=&amp;s6=&amp;s7=&amp;s8=All&amp;vfpref=html&amp;yearRangeFirst=&amp;yearRangeSecond=&amp;yrop=eq&amp;r=6&amp;mx-pid=2399130" rel="nofollow">Agol</a> that any 'standard' arithmetic hyperbolic 3-manifold virtually fibres. These examples contain lots of totally geodesic surfaces which, as you say, will lift to totally geodesic surfaces in the fibred cover.</p> http://mathoverflow.net/questions/38188/totally-geodesic-surfaces-in-fibered-3-manifolds/38206#38206 Answer by Bill Thurston for Totally geodesic surfaces in fibered 3-manifolds Bill Thurston 2010-09-09T17:32:59Z 2010-09-09T20:43:47Z <p>There are many specific known examples. Here is one construction:</p> <p>Start with the 3-torus \$T^3\$, parametrize in the standard way as \$R^3/Z^3\$. It fibers over the circle in many ways. Let \$a\$, \$b\$ and \$c\$ be three disjoint circles, coming form lines parallel to the x, y and z axes. For most fibrations, these three circles are transverse to the fibers. Form a branched cover of the torus with two-fold branching over all preimages of these 3 circles. The resulting manifold has a hyperbolic structure that can be constructed from right-angled hyperbolic dodecahedra, and is commensurable with the 4-fold branched cover of \$S^3\$ over the Borromean rings. You can think of it this way: you can take a unit cube as fundamental domain for the torus, and arrange that a, b and c lie on faces of the cube, each bisecting a pari of (glued together) opposite facce. This induces a subdividision of the boundary of the cube into what look like rectangles, but are really pentagons. </p> <p>The map (x,y,z) -> x+y+z gives a fibration over the torus, also works for any branched cover as described. The preimage of any face of the cube is an extended face plane of a dodecahedron, and is always a totally geodesic immersed surface, but it splits into two embedded surfaces for suitable branched covers of \$T^3\$ (perhaps the one you first come up with.) </p> <p>The tiling of hyperbolic space by right-angled dodecahedra has a cameo appearance in the video "Not Knot" we made at the Geometry Center, available together with "Outside In" on DVD from <a href="http://akpeters.com/searchresults.asp?type=quick&amp;keywords=Not+Knot&amp;btnSubmit.x=0&amp;btnSubmit.y=0" rel="nofollow">AKPeters</a>. In the 1984 Scientific American Article <em>The Mathematics of three-dimensional manifolds</em> that Jeff Weeks and I wrote, a manifold in this family (constructed from right-angled hyperbolic dodecahedra and having the properties you asked for) was described as the configuration space of a mechanical linkage. I don't think these particular properties were pointed out in Scientific American.</p> <p>This and other examples that are counterintuitive at first were a good part of my motivation when I raised the question whether all hyperbolic 3-manifolds virtually fiber over the circle, which at the time was a radical idea.</p>