Webrelaunch 2020

Geometric Group Theory II (Winter Semester 2023/24)

Lecture: Thursday 11:30-13:00 20.30 SR 2.59
Friday 8:00-9:30 20.30 SR 2.66
Problem class: Wednesday 15:45-17:15 20.30 SR 2.59

Geometric Group Theory studies the interactions between finitely generated groups and geometric spaces, creating connections between algebra and geometry. In this course we will study advanced topics in Geometric Group Theory with a particular focus on hyperbolic and non-positively curved groups. Introduced by Gromov in the 1980s hyperbolic groups form an active topic of research. A hyperbolic group is a group whose Cayley graph admits a (delta-)hyperbolic metric, where we call a space hyperbolic if all of its geodesic triangles are "very thin". Gromov observed that this thin triangle condition captures many strong properties that are shared by fundamental groups of closed manifolds with negative sectional curvature (such as real hyperbolic spaces). For instance, hyperbolic groups are always finitely presented, they do not contain higher rank free abelian subgroups, they have solvable word problem and they satisfy a strong version of the so-called Tits alternative. We will start this course with a discussion of notions of non-positive curvature in metric spaces. We will then introduce hyperbolic groups and spaces and then show that they satisfy all of these strong properties. Finally, we then plan to move on to other advanced topics in Geometric Group Theory, such as more general non-positively curved groups.

Further information on the course can be found on the ILIAS page.