282 research outputs found
When does the associated graded Lie algebra of an arrangement group decompose?
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\to \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 by \phi_r(G)=\sum_{X\in \L_2} \phi_r(F_{\mu(X)}), for r\ge 2. We
illustrate this new Lower Central Series formula with several families of
examples.Comment: 14 pages, accepted for publication by Commentarii Mathematici
Helvetic
The spectral sequence of an equivariant chain complex and homology with local coefficients
We study the spectral sequence associated to the filtration by powers of the
augmentation ideal on the (twisted) equivariant chain complex of the universal
cover of a connected CW-complex X. In the process, we identify the d^1
differential in terms of the coalgebra structure of H_*(X,\k), and the
\k\pi_1(X)-module structure on the twisting coefficients. In particular, this
recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic
p-covers of aspherical complexes. This approach provides information on the
homology of all Galois covers of X. It also yields computable upper bounds on
the ranks of the cohomology groups of X, with coefficients in a prime-power
order, rank one local system. When X admits a minimal cell decomposition, we
relate the linearization of the equivariant cochain complex of the universal
abelian cover to the Aomoto complex, arising from the cup-product structure of
H^*(X,\k), thereby generalizing a result of Cohen and Orlik.Comment: 38 pages, 1 figure (section 10 of version 1 has been significantly
expanded into a separate paper, available at arXiv:0901.0105); accepted for
publication in the Transactions of the American Mathematical Societ
Chen Lie algebras
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 the second derived
subalgebra. Following an idea of W.S. Massey, we point out a connection between
the Alexander invariant of a group G defined by commutator-relators, and its
integral holonomy Lie algebra.
As an application, we determine the Chen Lie algebras of several classes of
geometrically defined groups, including surface-like groups, fundamental groups
of certain classical link complements, and fundamental groups of complements of
complex hyperplane arrangements. For link groups, we sharpen Massey and
Traldi's solution of the Murasugi conjecture. For arrangement groups, we prove
that the rational Chen Lie algebra is combinatorially determined.Comment: 23 page
Algebraic invariants for Bestvina-Brady groups
Bestvina-Brady groups arise as kernels of length homomorphisms from
right-angled Artin groups G_\G to the integers. Under some connectivity
assumptions on the flag complex \Delta_\G, we compute several algebraic
invariants of such a group N_\G, directly from the underlying graph \G. As an
application, we give examples of Bestvina-Brady groups which are not isomorphic
to any Artin group or arrangement group.Comment: 22 pages, accepted for publication in the Journal of the London
Mathematical Societ
Vanishing resonance and representations of Lie algebras
We explore a relationship between the classical representation theory of a
complex, semisimple Lie algebra \g and the resonance varieties R(V,K)\subset
V^* attached to irreducible \g-modules V and submodules K\subset V\wedge V. In
the process, we give a precise roots-and-weights criterion insuring the
vanishing of these varieties, or, equivalently, the finiteness of certain
modules W(V,K) over the symmetric algebra on V. In the case when \g=sl_2(C),
our approach sheds new light on the modules studied by Weyman and Eisenbud in
the context of Green's conjecture on free resolutions of canonical curves. In
the case when \g=sl_n(C) or sp_{2g}(C), our approach yields a unified proof of
two vanishing results for the resonance varieties of the (outer) Torelli groups
of surface groups, results which arose in recent work by Dimca, Hain, and the
authors on homological finiteness in the Johnson filtration of mapping class
groups and automorphism groups of free groups.Comment: 17 pages; Corollary 1.3 stated in stronger form, with a shorter proo
Quasi-K\"ahler Bestvina-Brady groups
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 \to \Z, 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 not
commensurable (up to finite kernels) to the fundamental group of any
aspherical, quasi-projective variety.Comment: 11 pages, accepted for publication by the Journal of Algebraic
Geometr
Non-finiteness properties of fundamental groups of smooth projective varieties
For each integer n\ge 2, we construct an irreducible, smooth, complex
projective variety M of dimension n, whose fundamental group has infinitely
generated homology in degree n+1 and whose universal cover is a Stein manifold,
homotopy equivalent to an infinite bouquet of n-dimensional spheres. This
non-finiteness phenomenon is also reflected in the fact that the homotopy group
\pi_n(M), viewed as a module over Z\pi_1(M), is free of infinite rank. As a
result, we give a negative answer to a question of Koll'ar on the existence of
quasi-projective classifying spaces (up to commensurability) for the
fundamental groups of smooth projective varieties. To obtain our examples, we
develop a complex analog of a method in geometric group theory due to Bestvina
and Brady.Comment: 16 page
Alexander polynomials: Essential variables and multiplicities
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 sharp
upper bounds for the twisted Betti ranks of a group, in terms of multiplicities
constructed from the Alexander polynomial. For Seifert links in homology
3-spheres, these bounds become equalities, and our formula shows explicitly how
the Alexander polynomial determines all the characteristic varieties.Comment: 27 page
Long-Range Response to Transmission Line Disturbances in DC Electricity Grids
We consider a DC electricity grid composed of transmission lines connecting
power generators and consumers at its nodes. The DC grid is described by
nonlinear equations derived from Kirchhoff's law. For an initial distribution
of consumed and generated power, and given transmission line conductances, we
determine the geographical distribution of voltages at the nodes. Adjusting the
generated power for the Joule heating losses, we then calculate the electrical
power flow through the transmission lines. Next, we study the response of the
grid to an additional transmission line between two sites of the grid and
calculate the resulting change in the power flow distribution. This change is
found to decay slowly in space, with a power of the distance from the
additional line. We find the geographical distribution of the power
transmission, when a link is added. With a finite probability the maximal load
in the grid becomes larger when a transmission line is added, a phenomenon that
is known as Braess' paradox. We find that this phenomenon is more pronounced in
a DC grid described by the nonlinear equations derived from Kirchhoff's law
than in a linearised flow model studied previously in Ref. \cite{witthaut2013}.
We observe furthermore that the increase in the load of the transmission lines
due to an added line is of the same order of magnitude as Joule heating.
Interestingly, for a fixed system size the load of the lines increases with the
degree of disorder in the geographical distribution of consumers and producers.Comment: 10 pages, 13 figure
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