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We present a procedure of group cubization: it results in a group whose features resemble some of those of a given group and which acts without fixed points on a cubical complex. As a main application, we establish the lack of Kazhdan’s property (T)for Burnside groups.
Suppose that is a complex, reductive algebraic group. A real form of is an antiholomorphic involutive automorphism , so is a real Lie group. Write for the Galois cohomology (pointed) set . A Cartan involution for is an involutive holomorphic automorphism of , commuting with , so that is a compact real form of . Let be the set , where the action of the nontrivial element of is by . By analogy with the Galois group, we refer to as the Cartan cohomology of with respect to . Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism , where is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism .
We apply this result to give simple proofs of some well-known structural results: the Kostant–Sekiguchi correspondence of nilpotent orbits; Matsuki duality of orbits on the flag variety; conjugacy classes of Cartan subgroups; and structure of the Weyl group. We also use it to compute for all simple, simply connected groups and to give a cohomological interpretation of strong real forms. For the applications it is important that we do not assume that is connected.
Suppose that is a Schramm–Loewner evolution () in a smoothly bounded simply connected domain and that is a conformal map from to a connected component of for some . The multifractal spectrum of is the function which, for each , gives the Hausdorff dimension of the set of points such that as . We rigorously compute the almost sure multifractal spectrum of , confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure bulk integral means spectrum of , we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the curve for . Our results also hold for the processes with general vectors of weight .