Examples of aspherical hyperbolic simplicial complexes. An application of small cancellation for rotation families of groups

23 May 2011
Remi Coulon
The goal of this talk is to construct new examples of hyperbolic aspherical complexes. More precisely, given an aspherical simplicial complex P and a subcomplex Q of P, we are looking for conditions under which the complex obtained by attaching a cone of base Q on P remains aspherical. If Q is a set of loops of a 2-dimensional complex, J.H.C. Whitehead proved that this new complex is aspherical if and only if the elements of the fundamental group of P represented by Q do not satisfy any identity. To deal with higher dimensional subcomplexes we use small cancellation theory and extend the geometric point of view developed by T. Delzant and M. Gromov to rotation families of groups. In particular we obtain hyperbolic aspherical complexes obtained by attaching a cone over the "real part" of a hyperbolic complex manifold.