The aim of this project is to gather mathematicians working in complex and algebraic geometry on the one hand and in geometric group theory and topology on the other hand. The interactions between these fields are fruitful, have a long history and have seen remarkable recent developments. The members of this project are united by a common interest for group theory and almost all of them have already worked on topics related to fundamental groups of kähler and algebraic varieties or groups of automorphisms and of birational transformations of algebraic varieties. We aim at developing a common background in geometric group theory, complex differential geometry and algebraic geometry. We plan to organize a yearly workshop to gather the members of the project, as well as their phd students. Doctoral training will be a key element of the project. We also wish to hire postdoctoral researchers, to foster collaborations between members of the group and to organize one international conference at the end of the project. Among other things, we plan to approach the following research topics: (1) the study of group of birational transformations in arbitrary dimension, notably via their action on nonpositively curved spaces, the study of birational structures on algebraic varieties. (2) the study of the Kodaira problem for the fundamental group, and its link with the minimal model program in the kähler context. (3) the study of complex hyperbolic lattices, as well as the study of toroidal compactifications of finite-volume noncompact quotients of the unit ball of C^n. This family of examples combines a large part of the research topics of the members of this project (complex hyperbolicity, small cancellation theory in geometric group theory, finiteness properties of groups).