473 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
Arithmetic group symmetry and finiteness properties of Torelli groups
We examine groups whose resonance varieties, characteristic varieties and
Sigma-invariants have a natural arithmetic group symmetry, and we explore
implications on various finiteness properties of subgroups. We compute
resonance varieties, characteristic varieties and Alexander polynomials of
Torelli groups, and we show that all subgroups containing the Johnson kernel
have finite first Betti number, when the genus is at least four. We also prove
that, in this range, the -adic completion of the Alexander invariant is
finite-dimensional, and the Kahler property for the Torelli group implies the
finite generation of the Johnson kernel.Comment: Updated references, to appear in Ann. of Mat
Universal representations of braid and braid-permutation groups
Drinfel'd used associators to construct families of universal representations
of braid groups. We consider semi-associators (i.e., we drop the pentagonal
axiom and impose a normalization in degree one). We show that the process may
be reversed, to obtain semi-associators from universal representations of
3-braids. We view braid groups as subgroups of braid-permutation groups. We
construct a family of universal representations of braid-permutation groups,
without using associators. All representations in the family are faithful,
defined over \bbQ by simple explicit formulae. We show that they give
universal Vassiliev-type invariants for braid-permutation groups.Comment: 19 pages, references adde
Equivariant chain complexes, twisted homology and relative minimality of arrangements
We show that the equivariant chain complex associated to a minimal
CW-structure X on the complement M(A) of a hyperplane arrangement A, is
independent of X.
When A is a sufficiently general linear section of an aspheric arrangement,
we explain a new way for computing the twisted homology of M(A).Comment: 22 page
Nonabelian cohomology jump loci from an analytic viewpoint
For a topological space, we investigate its cohomology support loci, sitting
inside varieties of (nonabelian) representations of the fundamental group. To
do this, for a CDG (commutative differential graded) algebra, we define its
cohomology jump loci, sitting inside varieties of (algebraic) flat connections.
We prove that the analytic germs at the origin 1 of representation varieties
are determined by the Sullivan 1-minimal model of the space. Under mild
finiteness assumptions, we show that, up to a degree , the two types of jump
loci have the same analytic germs at the origins, when the space and the
algebra have the same -minimal model. We apply this general approach to
formal spaces (for which we establish the degeneration of the Farber-Novikov
spectral sequence), quasi-projective manifolds, and finitely generated
nilpotent groups. When the CDG algebra has positive weights, we elucidate some
of the structure of (rank one complex) topological and algebraic jump loci: up
to degree , all their irreducible components passing through the origin are
connected affine subtori, respectively rational linear subspaces. Furthermore,
the global exponential map sends all algebraic cohomology jump loci, up to
degree , into their topological counterpart.Comment: New Corollary 1.7 added and Theorem D. strengthened. Final version,
to appear in Communications in Contemporary Mathematic
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
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