## 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.