Property (T)

该项目的目的是对几何群论中重要的群，即Chvalley群、映射类群和外(F_N)通过等距作用于Hilbert空间的方式获得新的见解。更具体地说，我们将研究Kazhdan性质(T)的存在性，我们将提供Kazhdan常数的下界；研究该问题的定性和定量方面将使我们能够描绘上述群的刚性景观的详细图景。我们的主要目标是证明映射类群具有性质(T)。这将解决一个长期悬而未决的问题，这个问题在几何群论社区中已经获得了一些恶名。我们打算通过将组合和表示论方法与现代计算机的能力相结合来实现这一点，类似于我们为OUT(F_N)建立性质(T)的方法。我们的次要目标是：获得整数上Chvalley群的性质(T)的新的、统一的证明(这也将计算Kazhdan常数的下界)；去掉当前关于OUT(F_N)的性质(T)的计算机辅助方面的证明；并结合前面的两点来证明许多种类的环上的Chvalley群的性质(T)。

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.