79 research outputs found
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
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
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
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
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
- …