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 $r$ 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 $B \wedge F$ theory of flat connections and closed two-forms over manifolds with topology $\Sigma^3 \times (0,1)$. The reduced phase space variables are the holonomies of a flat connection for loops which form a basis of the first homotopy group $\pi_1(\Sigma^3)$, and elements of the second cohomology group of $\Sigma^3$ with value in the Lie algebra $L(G)$. When $G=SO(3,1)$, and if the two-form can be expressed as $B= e\wedge e$, for some vierbein field $e$, 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