Drawing of the eight Thurston geometries? - MathOverflow

Drawing of the eight Thurston geometries?

2010-05-14T03:09:01Z

Do you know of a picture, drawing, or other concise visual representation of the eight three-dimensional Thurston geometries?

I am imagining something akin to the standard picture (of a sphere, plane, and saddle) used to illustrate the three constant curvature geometries in dimension two. Of course, it takes more doing to illustrate representative three-manifolds, and there are more choices for natural examples, but I was surprised when I couldn't find such a picture. Another option would be to depict or indicate some of the geometries in less direct ways, for instance via the structure of stabilizers.

Answer by Agol for Drawing of the eight Thurston geometries?

2010-05-14T05:50:50Z

I gave a talk describing some of the geometries, which has some figures picturing the geometries. These are mostly based on the descriptions in Thurston's book, which has some nice pictures. The shape of space also has nice pictures, but I don't think it describes all 8 geometries. In some sense, all but hyperbolic geometry may be pictured as 1-dimensional bundles over surfaces, or surface bundles over the circle. Hyperbolic geometry may be thought of as glass with varying index of refraction, and spherical geometry may also be thought of this way (I computed the conformal factor once, but I don't know it off the cuff).

I don't know of a figure that collates pictures of the geometries into one.

Answer by Sam Nead for Drawing of the eight Thurston geometries?

2010-07-07T10:42:36Z

Here is a nice cyclic ordering of the eight geometries:

HxR, SxR, E^3, Sol, Nil, S^3, PSL, H^3

derived from staring at Peter Scott's table of Seifert fibered geometries. The table is organized by Euler characteristic of the base 2-orbifold and Euler class of the bundle. (See his BAMS article.) The cyclic ordering also has a bit of antipodal symmetry.

I didn't come up with geometric pictures of the eight geometrics but I have thought about "icons" to represent them. Here are my suggestions - I'm interested to hear what other people think/suggest.

HxR -- triangular prism (where the triangle is slim ie ideal)
SxR -- cylinder
E^3 -- cube
Sol -- tetrahedron with one pair of opposite edges truncated
Nil -- annulus with a segment of a spiral (representing a Dehn twist)
S^3 -- circle
PSL -- trefoil knot
H^3 -- figure eight knot (or possibly a slim tetrahedron)

I think it is also reasonable to ask for a "prototypical" three-manifold for each of the eight geometries. Here is an attempt:

HxR -- punctured torus cross circle
SxR -- two-sphere cross circle
E^3 -- three-torus
Sol -- mapping cylinder of [[2,1],[1,1]]
Nil -- mapping cylinder of [[1,1],[0,1]]
S^3 -- three-sphere
PSL -- trefoil complement
H^3 -- figure eight complement

Notice that all of the examples are either surface bundles over circles or circle bundles over surfaces, or both (ie products).