Groups with exotic finiteness properties from complex Morse theory

Abstract Recent constructions have shown that interesting behaviours can be observed in the finiteness properties of Kähler groups and their subgroups. In this work, we push this further and exhibit, for each integer , new hyperbolic groups admitting surjective homomorphisms to and to , whose kernel is of type but not of type . By a fibre product construction, we also find examples of non‐normal subgroups of Kähler groups with exotic finiteness properties.