Right-angled Coxeter groups, polyhedral complexes, and acute triangulations
Joint work with Sang-Hyan Kim.
1. Coxeter groups
Given a simplicial graph , there is an associated right-angled Coxeter group, generated by vertices of , and two generators commute iff the corresponding vertices are joined by an edge.
2. Acute triangulations
An acute triangulation is a combinatorial triangulation of the 2-sphere which can be realized by geodesic triangles with acute angles.
Theorem 1 Let be a triangulation of the 2-sphere. The corresponding Coxeter group is one ended and word hyperbolic if and only if is acute.
This group theory theorem has a purely combinatorial corollary.
Corollary 2 Let be a triangulation of the 2-sphere. Then is acute if and only if it has no separating 3 or 4-cycles.
Indeed, Davis and Moussong show that 3 and 4-cycles produce spheres and tori in the Cayley graph. Conversely, Andreev’s theorem implies that, when there are no 3 and 4-cycles, the triangulation is achieved by a hyperbolic polyhedron . coincides with the group generated by reflections in the faces of , which is one ended and hyperbolic.
From , Davis and Moussong construct a cube complex . It is a subcomplex of the cube . For each simplex in , add the faces parallel to the coordinate planes. One easily sees that a 4-cycle produces a torus in .
Useful: draw when is a triangulation of the 1-sphere. Get a polyherdon combinatorially equivalent to the reflection tiling of a right-angled polygon in hyperbolic plane.
The link of a vertex in is dual to . So is a 3-manifold.
3.1. Hyperbolic acute
Assume that is one ended and hyperbolic. Then admits a -invariant hyperbolic metric. Since is generated by mirror symmetries, it has a fundamental domain which is a right-angled polygon. Pick a point inside , draw geodesics normal to each side. This gives rise to an acute triangulation of the sphere at infinity, which is combinatorially equivalent to .
3.2. Acute hyperbolic
Conversely, from an acute triangulation, we construct a complex on which acts geometrically. It will be made of hyperbolic polyhedra. In order to get , it suffices to check that links are . For this, we shall use a result of C. Hodgson and I. Rivin.
3.3. Gauss images
Hodgson and Rivin introduce the Gauss image of a hyperbolic polyhedron . For a Euclidean polyhedron , for each , take the set of unit normals of supporting planes at . This tessalates the 2-sphere. Similarly, for each vertex , take the polar dual, glue them together. This produces a spherical complex which is not isometric to the 2-sphere.
Theorem 3 (Hodgson-Rivin) A spherical complex is the Gauss image of a hyperbolic polyhedron iff
- angles around vertices are ,
- lengths of closed geodesics are
We call such complexes strongly .
3.4. Sequel of proof
Start with an acute triangulation . Construct a Euclidean polyhedron whose Gauss image is as follows. For each vertex of , ake the plane in tangent to the sphere at that vertex, let be the intersection of half-spaces defined by these spaces which contain the unit ball. Then is strongly obtuse. Do the same thing in hyperbolic space from a very small ball, get a polyhedron which is again strongly obtuse. Do as if it were right angled, and build a complex from it. The links are combinatorially equivalent to the octahedron triangulation. According to Theorem 2, triangles in links are larger than octants, so links are .
Question. Do there exist acute triangulations of , ? Examples exist for .
Question. Study the space of acute triangulations of a given combinatorial type. Study the space of polyhedral complexes with a given combinatorial structure.