Property (T)

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The goal of the project is to obtain new insights into the ways in which groups of key importance in Geometric Group Theory, namely Chevalley groups, Mapping Class Groups, and Out(F_n), act on Hilbert spaces by isometries. More specifically, we will investigate the presence of Kazhdan's property (T), and we will provide lower bounds for Kazhdan constants; studying both the qualitative and quantitative aspects of the problem will allow us to paint a detailed picture of the rigidity landscape for the above groups.Our main goal 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 Out(F_n).Our secondary objectives are: to obtain a new, unified proof of property (T) for Chevalley groups over the integers (which will also compute lower bounds for Kazhdan constants); to remove the computer-assisted aspects of the current proof of property (T) for 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.