T4: Groups and actions

The study of symmetry and space through the medium of groups and their actions has long been a central theme in modern mathematics, indeed one that cuts across a wide spectrum of research within the Cluster. There are two main constellations of activity in the Cluster that coalesce around groups and dynamics as basic objects of study, and these are captured in the three research units collected here.

Units "Groups, dynamics and C*-algebras" and "Entropy, probability and geometry of groups" both focus on aspects of groups and dynamics grounded in measure and topology in their most abstract sense. This research treats infinite discrete groups as geometric or combinatorial objects and employs tools from functional analysis, probability, and combinatorics. The unit "Algebraic groups and Lie groups" examines, in contrast to abstract or discrete groups, groups with additional structure that naturally arise in algebraic and differential geometry.

Groups, dynamics and C*-algebras

Investigators: Geffen, Kerr, Tent, Winter

In this unit the connection to operator algebras is fundamental, influencing many other mathematical areas, including the rigidity theory of Lie groups and their lattices, and the study of manifolds and geometry through invariants like K-theory. Despite the diversity of problems and research paths, a common philosophy unites them: unravelling structural relationships at a fundamental level within a broad landscape shaped by key concepts such as finite approximation and amenability, freeness and tree-like/hyperbolic geometry and property-(T)-style rigidity.

Entropy, probability and geometry of groups

Investigators: Deninger, Kerr, Mukherjee

This unit circles around three basic themes within the expanded field of group-geometric ergodic theory and probability: (i) the application of tools from Ornstein theory and geometric group theory to advance a general program around orbit equivalence and entropy, with a special focus on rigidity problems for Bernoulli actions, (ii) the development of noncommutative cyclotomy conditions that will determine when an algebraic action an amenable or sofic group has zero entropy, and (iii) the use of invariant percolation to investigate geometric and analytic phenomena in infinite groups such as property (T) and the Haagerup property, in particular through the operator-algebraic lens of Roe algebras.

Algebraic groups and Lie groups

Investigators: Böhm, Hartl, Hellmann, Kramer, Lourenço, Schneider, Viehmann

This unit examines, in contrast to abstract or discrete groups, groups with additional structure that naturally arise in algebraic and differential geometry: algebraic groups and Lie groups. One of the research directions here focusses on algebraic groups and their actions that arise in the context of the Langlands programme, where representations of $p$-adic Lie groups are one of the main objects of study. A second point of focus includes more general locally compact groups and their associated (Bruhat–Tits) buildings and actions of real reductive groups in the context of Riemannian geometry.