Geometric and homological complexity of groups

• Applicant: Kevin Li, Ph.D.
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 570747020

The complexity of a group can be defined geometrically through its classifying space. Many different integer-valued invariants of groups arise in this way and admit homological counter-parts. Interesting special cases are the geometric dimension, amenable category, and Farber’s topological complexity. In this project, I will answer the following questions: When do the geometric and homological complexities agree and when do they differ? What are the values for Bestvina—Brady groups and right-angled Coxeter groups? Under which notion of quasi-isometry are the complexities invariant? To answer these questions, I will combine methods from different areas of mathematics: equivariant topology, homological algebra, and geometric group theory. This project makes progress towards long-standing open problems on the simplicial volume of manifolds and an algebraic characterisation of topological complexity.

• DFG Programme: WBP Position