A finite simplicial graph Gamma determines a right-angled Artin group G_Gamma, with generators corresponding to the vertices of Gamma, and with a relation vw=wv for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Gamma, directly from the graph Gamma.

If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,k), viewed as a module over the exterior algebra E on A: heta_k(G) = dim_k Tor^E_{k-1}(A,k)_k, where k is a field of characteristic 0, and kge 2. The Chen ranks conjecture asserts that, for k sufficiently large, heta_k(G) =(k-1)

We present a method for computing the number of epimorphisms from a finitely presented group G to a finite solvable group Gamma , which generalizes a formula of Gaschütz. Key to this approach are the degree 1 and 2 cohomology groups of G, with certain twisted coefficients. As an application, we count low-index subgroups of G. We also investigate the finite solvable quotients of the Baumslag-Solitar groups, the Baumslag parafree groups, and the Artin

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie algebra associated to the lower central series of pi_1(X). If Y is a formal space, this equality is actually equivalent to the Koszulness

The Chen groups of a finitely-presented group G are the lower central series quotients of its maximal metabelian quotient, G/G''. The direct sum of the Chen groups is a graded Lie algebra, with bracket induced by the group commutator. If G is the fundamental group of a formal space, we give an analog of a basic result of D. Sullivan, by showing that the rational Chen Lie algebra of G is isomorphic to the rational holonomy Lie algebra of G modulo

In a recent paper, Dimca and Nemethi pose the problem of finding a homogeneous polynomial f such that the homology of the complement of the hypersurface defined by f is torsion-free, but the homology of the Milnor fiber of f has torsion. We prove that this is indeed possible, and show by construction that, for each prime p, there is a polynomial with p-torsion in the homology of the Milnor fiber. The techniques make use of properties of characteristic

Let $X$ and $Y$ be finite-type CW-spaces ($X$ connected, $Y$ simply connected), such that the ring $H^*(Y,QQ)$ is a $k$-rescaling of $H^*(X,QQ)$. If $H^*(X,QQ)$ is a Koszul algebra, then the graded Lie algebra $pi_*(Omega Y)otimes QQ$ is the $k$-rescaling of $gr_*(pi_1 X)otimes QQ$. If $Y$ is a formal space, then the converse holds, and $Y$ is coformal. Furthermore, if $X$ is formal, with Koszul cohomology algebra, there exist filtered group isomorphisms

If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $phi_k$, of the lower central series quotients of $pi_1(M)$ can be computed from the Betti numbers, $b_{ii}=dim_{k} Tor^A_i(k,k)_i$, of the linear strand in a (minimal) free resolution of $k$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded

Given a finitely-generated group G, and a finite group Gamma, Philip Hall defined delta_Gamma to be the number of factor groups of G that are isomorphic to Gamma. We show how to compute the Hall invariants by cohomological and combinatorial methods, when G is finitely-presented, and Gamma belongs to a certain class of metabelian groups. Key to this approach is the stratification of the character variety by the jumping loci of the cohomology of G,

We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Z pi_1-module presentation of pi_p, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the