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 $CAT\left(0\right)$ cubical complex. As a main application, we establish the lack of Kazhdan’s property (T)for Burnside groups.

Suppose that $G$ is a complex, reductive algebraic group. A real form of $G$ is an antiholomorphic involutive automorphism $\sigma$, so $G\left(\mathbb{R}\right)=G(\mathbb{C}{)}^{\sigma }$ is a real Lie group. Write $H^1(\sigma ,G)$ for the Galois cohomology (pointed) set $H^1(Gal(\mathbb{C}/\mathbb{R}),G)$. A Cartan involution for $\sigma$ is an involutive holomorphic automorphism $\theta$ of $G$, commuting with $\sigma$, so that $\theta \sigma$ is a compact real form of $G$. Let $H^1(\theta ,G)$ be the set $H^1({\mathbb{Z}}_2,G)$, where the action of the nontrivial element of ${\mathbb{Z}}_2$ is by $\theta$. By analogy with the Galois group, we refer to $H^1(\theta ,G)$ as the Cartan cohomology of $G$ with respect to $\theta$. Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism $H^1(\sigma ,G_{\mathrm{ad}})\simeq H^1(\theta ,G_{\mathrm{ad}})$, where $G_{\mathrm{ad}}$ is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism $H^1(\sigma ,G)\simeq H^1(\theta ,G)$.

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 $H^1(\sigma ,G)$ 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 $G$ is connected.

Suppose that $\eta$ is a Schramm–Loewner evolution (${SLE}_{\kappa }$) in a smoothly bounded simply connected domain $D\subset \mathbf{C}$ and that $\varphi$ is a conformal map from $\mathbf{D}$ to a connected component of $D\setminus \eta \left(\right[0,t\left]\right)$ for some $t>0$. The multifractal spectrum of $\eta$ is the function $(-1,1)\to [0,\infty )$ which, for each $s\in (-1,1)$, gives the Hausdorff dimension of the set of points $x\in \partial \mathbf{D}$ such that $\left|\varphi \text{'}\right((1-ϵ)x\left)\right|={ϵ}^{-s+o\left(1\right)}$ as $ϵ\to 0$. We rigorously compute the almost sure multifractal spectrum of $SLE$, 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 $SLE$, we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an $SLE$ curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the $SLE$ curve for $\kappa \le 4$. Our results also hold for the ${SLE}_{\kappa }\left(\underline{\rho }\right)$ processes with general vectors of weight $\underline{\rho }$.