We study the topology of the boundary manifold of a line arrangement in CP^2, with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial Delta(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representation. We give an explicit description of the unit ball in the Alexander norm, and use it to analyze certain Bieri–Neumann–Strebel

We explore the codimension-one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex line arrangements are quasi-projective. We also give

We explore the geometry of the Abel-Jacobi map f from a closed, orientable Riemannian manifold X to its Jacobi torus. Applying Gromov's filling inequality to the typical fiber of f, we prove an interpolating inequality for two flavors of shortest length invariants of loops. The procedure works, provided the lift of the fiber is non-trivial in the homology of the maximal free abelian cover, X~, classified by f. We show that the finite-dimensionality

Bestvina-Brady groups arise as kernels of length homomorphisms from right-angled Artin groups G_Gamma to the integers. Under some connectivity assumptions on the flag complex Delta_Gamma, we compute several algebraic invariants of such a group N_Gamma, directly from the underlying graph Gamma. As an application, we give examples of Bestvina-Brady groups which are not isomorphic to any Artin group or arrangement group.

A finite simple graph G determines a right-angled Artin group G_G, with one generator for each vertex v, and with one commutator relation vw=wv for each pair of vertices joined by an edge. The Bestvina-Brady group N_G is the kernel of the projection G_G o , which sends each generator v to 1. We establish precisely which graphs G give rise to quasi-K"ahler (respectively, K"ahler) groups N_G. This yields examples of quasi-projective groups which are

Associated to every finite simplicial complex K there is a ``moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications

Let A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra H. Suppose H_3 is a free abelian group of minimum possible rank, given the values the M"obius function mu: L_2 -> Z takes on the rank 2 flats of A. Then the associated graded Lie algebra of G decomposes (in degrees 2 and higher) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge-theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology