Eliminating components in Quillen's conjecture

Abstract

We generalize an earlier result of Segev, which shows that some component in a minimal counterexample to Quillen's conjecture must admit an outer automorphism. We show in fact that every component must admit an outer automorphism. Thus we transform his restriction-result on components to an elimination-result: namely one which excludes any component which does not admit an outer automorphism. Indeed we show that the outer automorphisms admitted must include p-outers: that is, outer automorphisms of order divisible by p. This gives stronger, concrete eliminations: for example if p is odd, it eliminates sporadic and alternating components—thus reducing to Lie-type components (and typically forcing p-outers of field type). For p=2, we obtain similar but less restrictive results. We also provide some tools to help eliminate suitable components that do admit p-outers in a minimal counterexample.