For an orientable surface S of finite topological type with genus g ≥ 3, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph $\mathcal{C}$(S). The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set in Aramayona and Leininger, J. Topology Anal.5(2) (2013), 183–203 and Aramayona and Leininger, Pac. J. Math.282(2) (2016), 257–283, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

the cohomology of a right-angled artin group with group ring coefficients is explicitly presented in terms of the cohomology of its defining flag complex.

Acyclic groups of low dimension are considered. To indicate the results simply, let $G^{\prime}$ be the nontrivial perfect commutator subgroup of a finitely presentable group $G$. Then $\mathrm{def}(G)\leq1$. When $\mathrm{def}(G)=1$, $G^{\prime}$ is acyclic provided that it has no integral homology in dimensions above 2 (a sufficient condition for this is that $G^{\prime}$ be finitely generated); moreover, $G/G^{\prime}$ is then $Z$ or $Z^{2}$. Natural examples are the groups of knots and links with Alexander polynomial 1. A further construction is given, based on knots in $S^{2}\times S^{1}$. In these geometric examples, $G^{\prime}$ cannot be finitely generated; in general, it cannot be finitely presentable. When $G$ is a 3-manifold group it fails to be acyclic; on the other hand, if $G^{\prime}$ is finitely generated it has finite index in the group of a $\mathbb{Q}$-homology 3-sphere.