Subspace arrangements and property T

We reformulate and extend the geometric method for proving Kazhdan property Tdeveloped by Dymara and Januszkiewicz and used by Ershov and Jaikin. The mainresult says that a group G, generated by finite subgroups G_i, has property Tif the group generated by each pair of subgroups has property T andsufficiently large Kazhdan constant. Essentially, the same result was proven byDymara and Januszkiewicz, however our bound for "sufficiently large" issignificantly better. As an application of this result, we give exact bounds for the Kazhdanconstants and the spectral gaps of the random walks on any finite Coxeter groupwith respect to the standard generating set, which generalizes a result ofBacher and de la Hapre.