Robert Bieri

My original interest in homological methods for infinite groups (cohomological dimension and Poincare duality for groups) shifted towards geometric and asymptotic methods. I analyze links between finiteness properties of groups and their modules to geometric and topological properties at infinity of G-spaces (e.g., the SIGMA-invariants which among other things provided a necessary conditions for finte presentability of G; but they also contributed a central polyhedrality result to the much later emerging Tropical Geometry). The focus was on familiar groups like metabelian, soluble, free and linear ones, or fundamental groups of 3-manifolds, but also Thompson's groups F, T and V.My most recent work started with the observation that Thurston's description of V can be interpreted in terms of rearranging the tiles of an ideal triangle-tessellation of the hyperbolic plane; then it analyzes the corresponding groups of rearrangements of the Euclidean tessellation Rn by unit cubes. This opens the door to new group theory related to more general Euclidean and hyperbolic tessellations.