We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of ${\mathrm{SL}}_n(\mathbf{R})$. Our main result is inspired by the Hörmander–Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity. The asymptotics also match the Mikhlin formula for an exponentially growing weight with respect to the word length. Additional decay comes imposed by this growth and the Mikhlin condition for high-order terms. Lafforgue and de la Salle’s rigidity theorem fits here. The proof includes a new relation between Fourier and Schur $L_p$-multipliers for nonamenable groups. By transference, matters are reduced to a rather nontrivial $R{C}_p$-inequality for ${\mathrm{SL}}_n(\mathbf{R})$-twisted forms of Riesz transforms associated to fractional Laplacians.

Our second result gives a new and much stronger rigidity theorem for radial multipliers in ${\mathrm{SL}}_n(\mathbf{R})$. More precisely, additional regularity and Mikhlin-type conditions are proved to be necessary up to an order $\sim |\frac{1}{2}-\frac{1}{p}|(n-1)$ for large enough n in terms of p. Locally, necessary and sufficient growth rates match up to that order. Asymptotically, extra decay for the symbol and its derivatives imposes more accurate and additional rigidity in a wider range of $L_p$-spaces. This rigidity increases with the rank, so we can construct radial generating functions satisfying our Hörmander–Mikhlin sufficient conditions in a given rank n and failing the rigidity conditions for ranks $m\gg n$. We also prove automatic regularity and rigidity estimates for first- and higher-order derivatives of K-bi-invariant multipliers in the rank 1 groups $\mathrm{SO}(n,1)$.

We present a method for calculating the Brauer group of a surface given by a diagonal equation in projective space. For diagonal quartic surfaces with coefficients in $\mathbb{Q}$, we determine the Brauer groups over $\mathbb{Q}$ and $\mathbb{Q}(i)$.

An improvement of the Liouville theorem for discrete harmonic functions on ${\mathbb{Z}}^2$ is obtained. More precisely, we prove that there exists a positive constant ε such that if u is discrete harmonic on ${\mathbb{Z}}^2$ and for each sufficiently large square Q centered at the origin $|u|\le 1$ on a $(1-\mathit{\epsilon })$ portion of Q, then u is constant.

We show, under mild hypotheses, that if each element of a finitely generated group acting on a 2-dimensional $\mathrm{CAT}(0)$ complex has a fixed point, then there is a global fixed point. In particular, all actions of finitely generated torsion groups on such complexes have global fixed points. The proofs rely on Masur’s theorem on periodic trajectories in rational billiards, and Ballmann–Brin’s methods for finding closed geodesics in 2-dimensional locally $\mathrm{CAT}(0)$ complexes. As another ingredient, we prove that the image of an immersed loop in a graph of girth $2\mathrm{\pi }$ with length not commensurable with π has diameter $>\mathrm{\pi }$. This is closely related to a theorem of Dehn on tiling rectangles by squares.