Between buildings and free factor complexes

Abstract

The main achievement of this thesis is the construction of a new family of simplicial complexes interpolating between Tits buildings and free factor complexes. For every finite graph G, we obtain a simplicial complex CC associated to the outer automorphism group of the right-angled Artin group AG. These complexes are defined using the intersection patterns of cosets of parabolic subgroups. Each of them is homotopy Cohen–Macaulay and in particular homotopy equivalent to a wedge of d-spheres. The dimension d can be read off from the defining graph G and provides a new invariant for the automorphism group of AG. In order to deduce this and further properties of CC, we introduce new methods for studying the topology of coset complexes and coset posets, refine the decomposition sequence for automorphism groups of right-angled Artin groups established by Day–Wade and study the asymptotic geometry of Culler– Vogtmann Outer space. In particular, we show that the simplicial boundary of the Outer space of the free group Fn can be described in terms of complexes of free factors of Fn and study the connectivity properties of these complexes.