Abstract: I will talk about my viewpoint regarding a method for proving property T developed by Dymara and Januszkiewicz. A group G has property T if every continuous affine isometric action of G on a real Hilbert space has a fixed point.
The original motivation of Dymara and Januszkiewicz came from groups acting on buildings, but the idea does not use anything more than angles between subspaces in a finite-dimensional Euclidian space. Their main result says that if a group G is generated by finite subgroups G_i and each pair generates a group with property T and sufficiently large Kazhdan constant then the whole group also has property T.
One can use this result to show that some groups like SL_n(F_p[t_1,…,t_k]) have property T, almost without using any representation theory. Another application allows us to compute the exact values of the Kazhdan constant and the spectral gap for the Laplacian for any finite Coxeter group with respect to its standard generating set.
Parts of the talk are based on the work of M. Ershov and A. Jaikin.

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