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For any right-angled Artin group , we construct a finite-dimensional space on which the group of outer automorphisms of acts with finite point stabilizers. We prove that is contractible, so that the quotient is a rational classifying space for . The space blends features of the symmetric space of lattices in with those of outer space for the free group . Points in are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cube complexes, marked by an isomorphism of their fundamental group with .
We give a lower bound of the δ-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.
Let , , be three cuspidal automorphic representations for the group , where and are fixed and has large analytic conductor. We prove a subconvex bound for of Weyl-type quality. Allowing to be an Eisenstein series, we also obtain a Weyl-type subconvex bound for .