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The Alexander polynomial of a 3manifold and the Thurston norm on cohomology
 Ann. Sci. École Norm. Sup
, 2001
"... Let M be a connected, compact, orientable 3manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kkA kkT between the Alexander norm on H 1 (M;Z), dened in terms of the Alexander polynomial, and the Thurston norm, dened in terms of the Euler ..."
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Cited by 70 (3 self)
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Let M be a connected, compact, orientable 3manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. Our main result is the inequality kkA kkT between the Alexander norm on H 1 (M;Z), dened in terms of the Alexander polynomial, and the Thurston norm, dened in terms of the Euler characteristic of embedded surfaces. (A similar result holds when b1 (M) = 1.) Using this inequality we determine the Thurston norm for most links with 9 or fewer crossings. Contents 1
Characteristic varieties and Betti numbers of free abelian covers
 INTERNATIONAL MATHEMATICS RESEARCH NOTICES
, 2014
"... The regular Zrcovers of a finite cell complex X are parameterized by the Grassmannian of rplanes in H1 (X,Q). Moving about this variety, and recording when the Betti numbers b1,...,bi of the corresponding covers are finite carves out certain subsets Ωi r (X) of the Grassmannian. We present here a ..."
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Cited by 15 (11 self)
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The regular Zrcovers of a finite cell complex X are parameterized by the Grassmannian of rplanes in H1 (X,Q). Moving about this variety, and recording when the Betti numbers b1,...,bi of the corresponding covers are finite carves out certain subsets Ωi r (X) of the Grassmannian. We present here a method, essentially going back to Dwyer and Fried, for computing these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. Using the exponential tangent cones to these jump loci, we show that each Ωinvariant is contained in the complement of a union of Schubert varieties associated to an arrangement of linear subspaces in H1 (X,Q). The theory can be made very explicit in the case when the characteristic varieties of X are unions of translated tori. But even in this setting, the Ωinvariants are not necessarily open, not even when X is a smooth complex projective variety. As an application, we discuss the geometric finiteness properties of some classes of groups.
Intersections of translated algebraic subtori
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 2013
"... We exploit the classical correspondence between finitely generated abelian groups and abelian complex algebraic reductive groups to study the intersection theory of translated subgroups in an abelian complex algebraic reductive group, with special emphasis on intersections of (torsion) translated su ..."
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Cited by 6 (6 self)
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We exploit the classical correspondence between finitely generated abelian groups and abelian complex algebraic reductive groups to study the intersection theory of translated subgroups in an abelian complex algebraic reductive group, with special emphasis on intersections of (torsion) translated subtori in an algebraic torus.
HOMOLOGICAL FINITENESS OF ABELIAN COVERS
 ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA
, 2015
"... We present a method for deciding when a regular abelian cover of a finite CWcomplex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CWcomplex X, with group of deck transformations a fixed abelian group A, which in the case of free ab ..."
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Cited by 4 (4 self)
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We present a method for deciding when a regular abelian cover of a finite CWcomplex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CWcomplex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of rank r coincides with the Grassmanian of rplanes in H1 (X,Q). Inside this parameter space, there is a subsetΩ i A (X) consisting of all the covers with finite Betti numbers up to degree i. Building up on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank 1 local systems on X. For certain spaces, such as smooth, quasiprojective varieties, the generalized Dwyer–Fried invariants that we introduce here can be computed in terms of intersections of algebraic subtori in the character group. For many spaces of interest, the homological finiteness of abelian covers can be tested through the corresponding free abelian covers. Yet in general, abelian covers exhibit different homological finiteness properties than their free abelian counterparts.
KÄHLER GROUPS, QUASIPROJECTIVE GROUPS, AND 3MANIFOLD GROUPS
 JOURNAL OF THE LONDON MATHEMATICAL SOCIETY
, 2014
"... We prove two results relating 3manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3manifold. If N has nonempty, toroidal boundary, and π1(N) is a Kähler group, then N is the product of a torus with an interval. On the other hand, if N ..."
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Cited by 4 (2 self)
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We prove two results relating 3manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3manifold. If N has nonempty, toroidal boundary, and π1(N) is a Kähler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and π1(N) is a quasiprojective group, then all the prime components of N are graph manifolds.