Title: Hyperbolicity vs. word hyperbolicity for right-angled reflection groups in dimension 4
Speaker: Ryan Greene
Abstract: Andreev's theorem characterizes Coxeter groups that act geometrically on 3-dimensional hyperbolic space in a combinatorial way. Moussong showed that an analogous condition characterizes word hyperbolicity of general Coxeter groups. In the right-angled case, these conditions involve a simplicial complex associated to a Coxeter group called its nerve, the flag condition of Gromov, and a "no-square" condition. One might ask: To what extent does Moussong's criterion characterize right-angled Coxeter groups that act on 4-dimensional hyperbolic space?
We answer this question by example: We give a construction of an infinite family of right-angled word hyperbolic Coxeter groups that act cocompactly on a contractible 4-manifold. We show that none of them act discretely on 4-dimensional hyperbolic space, except one which coincides with the well-known right-angled 120-cell group. We also show that the examples that arise from applying the special subdivision procedure of Przytycki-Swiatkowski to triangulations of the 3-sphere do not act geometrically on 4-dimensional hyperbolic space.
