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Property (T)

The research, as envisioned, can be broadly located at the intersection of computational and geometric group theory, convex optimization and real algebraic geometry.

The study of groups through their geometric actions has a long history which has started with the fundamental work of Dehn on group presentations. In 1967 Kazhdan discovered a property of groups (known as property (T)) defined in the language of unitary actions on Hilbert spaces.

Despite somewhat obscure definition, the property proved to have intriguing consequences in various branches of mathematics (algebraic groups, ergodic theory, geometric group theory),and computer science (resilient networks using expanders, group-theoretic hash functions).

Our primary objective is to prove that Mapping Class Groups have property (T). This would solve a long-standing open problem which has acquired some notoriety in the Geometric Group Theory community. We intend to do this by combining combinatorial and representation-theoretic methods with the power of modern computers, similarly to the way we established property (T) for$$\operatorname{Out}(F_n)$$.

Our secondary objectives are: to obtain lower bounds for Kazhdan constants of Chevalley groups; to remove the computer-assisted aspects of the current proof of property (T) for$$\operatorname{Out}(F_n)$$; and to combine the previous two points in order to prove property (T) for Chevalley groups over a large variety of rings.

## Team Members

PhD Marek Kaluba
Researcher
Karlsruhe Institute of Technology (KIT)
marek.kaluba(at)kit.edu

Dr. Dawid Kielak