Notes of Genevieve Walsh’s lecture

Right-angled Coxeter groups, polyhedral complexes, and acute triangulations

Joint work with Sang-Hyan Kim.

1. Coxeter groups

Given a simplicial graph {Gamma}, there is an associated right-angled Coxeter group, generated by vertices of {Gamma}, 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 {L} be a triangulation of the 2-sphere. The corresponding Coxeter group {C(L)} is one ended and word hyperbolic if and only if {L} is acute.

This group theory theorem has a purely combinatorial corollary.

Corollary 2 Let {L} be a triangulation of the 2-sphere. Then {L} 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 {Q}. {C(L)} coincides with the group generated by reflections in the faces of {Q}, which is one ended and hyperbolic.

3. Proof

From {L}, Davis and Moussong construct a cube complex {P_L}. It is a subcomplex of the cube {[-1,1]^n}. For each simplex {[v_i,ldots,v_j]} in {L}, add the faces parallel to the {v_i,ldots,v_j} coordinate planes. One easily sees that a 4-cycle produces a torus in {P_L}.

Useful: draw {P_L} when {L} 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 {P_L} is dual to {L}. So {P_L} is a 3-manifold.

3.1. Hyperbolic {Rightarrow} acute

Assume that {C(L)} is one ended and hyperbolic. Then {P_L} admits a {C(L)}-invariant hyperbolic metric. Since {C(L)} is generated by mirror symmetries, it has a fundamental domain {P_H} which is a right-angled polygon. Pick a point inside {P_H}, draw geodesics normal to each side. This gives rise to an acute triangulation of the sphere at infinity, which is combinatorially equivalent to {L}.

3.2. Acute {Rightarrow} hyperbolic

Conversely, from an acute triangulation, we construct a {CAT(-1)} complex on which {C(L)} acts geometrically. It will be made of hyperbolic polyhedra. In order to get {CAT(-1)}, it suffices to check that links are {CAT(1)}. 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 {P_H}. For a Euclidean polyhedron {P_E}, for each {xin P_E}, take the set of unit normals of supporting planes at {x}. This tessalates the 2-sphere. Similarly, for each vertex {xin P_H}, 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

  1. angles around vertices are {>2pi},
  2. lengths of closed geodesics are {>2pi}

We call such complexes strongly {CAT(1)}.

3.4. Sequel of proof

Start with an acute triangulation {T}. Construct a Euclidean polyhedron whose Gauss image is {T} as follows. For each vertex of {T}, ake the plane in {mathbb{E}^3} tangent to the sphere at that vertex, let {P_E} be the intersection of half-spaces defined by these spaces which contain the unit ball. Then {P_E} is strongly obtuse. Do the same thing in hyperbolic space from a very small ball, get a polyhedron {P_H} 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 {CAT(1)}.

Question. Do there exist acute triangulations of {S^n}, {ngeq 3} ? Examples exist for {n=3}.

Question. Study the space of acute triangulations of a given combinatorial type. Study the space of {CAT(-1)} polyhedral complexes with a given combinatorial structure.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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