We associate with each graph (S, E) a 2-step simply connected nilpotent Lie group N and a lattice Γ in N . We determine the group of Lie automorphisms of N and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold N/Γ to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anoso...

We study the Ramond twisted representations of the affine W algebra W(ḡ, f) in the case that f is Richardson. We establish the vanishing and the almost irreducibility of the corresponding BRST cohomology. This confirms some of the recent conjectures of Kac and Wakimoto [KW5]. In type A, our results give the characters of all irreducible ordinary Ramond twisted representations of W(sln, f) for a...

Let G be a group. The orbits of the natural action \({{\,\mathrm{Aut}\,}}(G)\) on are called automorphism G, and number is denoted by \(\omega (G)\). virtually nilpotent group such that (G)< \infty \). We prove \(G = K \rtimes H\) where H torsion subgroup torsion-free radicable characteristic G. Moreover, we \(G^{'}= D \times {{\,\mathrm{Tor}\,}}(G^{'})\) subgroup. In particular, if maximum nor...

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a no...