56 research outputs found
Forgetful maps between Deligne-Mostow ball quotients
We study forgetful maps between Deligne-Mostow moduli spaces of weighted
points on P^1, and classify the forgetful maps that extend to a map of
orbifolds between the stable completions. The cases where this happens include
the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic
surfaces. They also include a retraction of a 3-dimensional ball quotient onto
one of its 1-dimensional totally geodesic complex submanifolds
Manifolds with small Dirac eigenvalues are nilmanifolds
Consider the class of n-dimensional Riemannian spin manifolds with bounded
sectional curvatures and diameter, and almost non-negative scalar curvature.
Let r=1 if n=2,3 and r=2^{[n/2]-1}+1 if n\geq 4. We show that if the square of
the Dirac operator on such a manifold has small eigenvalues, then the
manifold is diffeomorphic to a nilmanifold and has trivial spin structure.
Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a
non-trivial spin structure, then there exists a uniform lower bound on the r-th
eigenvalue of the square of the Dirac operator. If a manifold with almost
nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume
is not too small, then we show that the metric is close to a Ricci-flat metric
on M with a parallel spinor. In dimension 4 this implies that M is either a
torus or a K3-surface
On semistable principal bundles over a complex projective manifold, II
Let (X, \omega) be a compact connected Kaehler manifold of complex dimension
d and E_G a holomorphic principal G-bundle on X, where G is a connected
reductive linear algebraic group defined over C. Let Z (G) denote the center of
G. We prove that the following three statements are equivalent: (1) There is a
parabolic subgroup P of G and a holomorphic reduction of the structure group of
E_G to P (say, E_P) such that the bundle obtained by extending the structure
group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat
connection; (2) The adjoint vector bundle ad(E_G) is numerically flat; (3) The
principal G-bundle E_G is pseudostable, and the degree of the charateristic
class c_2(ad(E_G) is zero.Comment: 15 page
B^F Theory and Flat Spacetimes
We propose a reduced constrained Hamiltonian formalism for the exactly
soluble theory of flat connections and closed two-forms over
manifolds with topology . The reduced phase space
variables are the holonomies of a flat connection for loops which form a basis
of the first homotopy group , and elements of the second
cohomology group of with value in the Lie algebra . When
, and if the two-form can be expressed as , for some
vierbein field , then the variables represent a flat spacetime. This is not
always possible: We show that the solutions of the theory generally represent
spacetimes with ``global torsion''. We describe the dynamical evolution of
spacetimes with and without global torsion, and classify the flat spacetimes
which admit a locally homogeneous foliation, following Thurston's
classification of geometric structures.Comment: 21 pp., Mexico Preprint ICN-UNAM-93-1
Automorphism groups of polycyclic-by-finite groups and arithmetic groups
We show that the outer automorphism group of a polycyclic-by-finite group is
an arithmetic group. This result follows from a detailed structural analysis of
the automorphism groups of such groups. We use an extended version of the
theory of the algebraic hull functor initiated by Mostow. We thus make
applicable refined methods from the theory of algebraic and arithmetic groups.
We also construct examples of polycyclic-by-finite groups which have an
automorphism group which does not contain an arithmetic group of finite index.
Finally we discuss applications of our results to the groups of homotopy
self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of
arithmeticity results of Sullivan in rational homotopy theory
Monodromy of Cyclic Coverings of the Projective Line
We show that the image of the pure braid group under the monodromy action on
the homology of a cyclic covering of degree d of the projective line is an
arithmetic group provided the number of branch points is sufficiently large
compared to the degree.Comment: 47 pages (to appear in Inventiones Mathematicae
Arithmeticity vs. non-linearity for irreducible lattices
We establish an arithmeticity vs. non-linearity alternative for irreducible
lattices in suitable product groups, such as for instance products of
topologically simple groups. This applies notably to a (large class of)
Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as
we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page
Algebraic volume density property of affine algebraic manifolds
We introduce the notion of algebraic volume density property for affine
algebraic manifolds and prove some important basic facts about it, in
particular that it implies the volume density property. The main results of the
paper are producing two big classes of examples of Stein manifolds with volume
density property. One class consists of certain affine modifications of \C^n
equipped with a canonical volume form, the other is the class of all Linear
Algebraic Groups equipped with the left invariant volume form.Comment: 35 page
Equivariant geometric K-homology for compact Lie group actions
Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant
geometric K-homology groups K^G_*(X), using an obvious equivariant version of
the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural
transformations to and from equivariant K-homology defined via KK-theory (the
"official" equivariant K-homology groups) and show that these are isomorphism.Comment: 25 pages. v2: some mistakes corrected, more detail added, Michael
Walter as author added. To appear in Abhandlungen aus dem Mathematischen
Seminar der Universit\"at Hambur
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