Following C.T.C. Wall, we say that a group $G$ is of type $F_n$ if it has a classifying space (a $K(G,1)$) whose n-skeleton is finite. When n=1 (resp. n=2) one recovers the condition of finite generation (resp. finite presentation). The study of examples of groups which are of type $F_{n-1}$ but not of type $F_n$ has a long history (Stallings, Bestvina-Brady, ...). One says that these examples of groups have exotic finiteness properties. In this talk I will explain how to use complex geometry to build new examples of groups with exotic finiteness properties. This is part of a joint work with F. Nicolas, which generalizes earlier works by Dimca, Papadima and Suciu, Llosa Isenrich and Bridson and Llosa Isenrich.