Publications

If a group Γ acts geometrically on a CAT(0) space X without 3-flats, then either X contains a Γ-periodic geodesic which does not bound a flat half-plane, or else X is a rank 2 Riemannian symmetric space, a 2-dimensional Euclidean building or non-trivially splits as a metric product. Consequently all such groups satisfy a strong form of the Tits Alternative.

Related project(s):
66Minimal surfaces in metric spaces II

This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let X be a CAT(0) space with a geometric group action. Suppose that every geodesic in X lies in an n-flat, n≥2. If X contains a periodic n-flat which does not bound a flat (n+1)-half-space, then X is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.

Related project(s):
66Minimal surfaces in metric spaces II

A CAT(0) space has rank at least n if every geodesic lies in an n-flat. Ballmann's Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann's conjecture. Here we prove that a CAT(0) space of rank at least n≥2 is rigid if it contains a periodic n-flat and its Tits boundary has dimension (n−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces -- so-called Morse flats. We show that the Tits boundary ∂TF of a periodic Morse n-flat F contains a regular point -- a point with a Tits-neighborhood entirely contained in ∂TF. More precisely, we show that the set of singular points in ∂TF can be covered by finitely many round spheres of positive codimension.

Related project(s):
66Minimal surfaces in metric spaces II

We describe the Gromov-Hausdorff closure of the class of length spaces being homeomorphic to a fixed closed surface.

Related project(s):
24Minimal surfaces in metric spaces

We discuss solutions of several questions concerning the geometry of conformal planes.

Related project(s):
66Minimal surfaces in metric spaces II

H-type foliations (????,H,g) are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping ???? with the Bott connection we consider the scalar horizontal curvature κ as well as a new local invariant τ induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannanian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of κ and τ. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of H-type foliations allows us to consider the pull-back of Korányi balls to ????. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when ???? is locally isometric as a sub-Riemannian manifold to its H-type tangent group.

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

On the seven dimensional Euclidean sphere S^7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution T  is induced by a Clifford module structure of R^8. More precisely, T is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution Q of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration ????3↪????7→????4. Answering a question in arXiv:0901.1406 we first show that Q does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures, the intrinsic sublaplacians ΔTsub and ΔQsub and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

We prove regularity estimates for the eigenfunctions of Schrödinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body operators are covered by our result; in that case, the weight is in terms of the (euclidean) distance to the collision planes. The technique of proof is based on blow-ups and Lie manifolds. More precisely, we first blow-up the spheres at infinity of the collision planes to obtain the Georgescu-Vasy compactification and then we blow-up the collision planes. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher order operators and matrices of scalar operators.

Related project(s):
35Geometric operators on singular domains

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.

Related project(s):
60Property (T)

We introduce the classes of TAP groups, in which various types of algebraic fibring are detected by the non-vanishing of twisted Alexander polynomials. We show that finitely presented LERF groups lie in the class TAP1(R) for every integral domain R, and deduce that algebraic fibring is a profinite property for such groups. We offer stronger results for algebraic fibring of products of limit groups, as well as applications to profinite rigidity of Poincaré duality groups in dimension 3 and RFRS groups.

Related project(s):
60Property (T)

We introduce the notion of quasi-BNS invariants, where we replace homomorphism to R by homogenous quasimorphisms to R in the theory of Bieri-Neumann-Strebel invariants. We prove that the quasi-BNS invariant QΣ(G) of a finitely generated group G is open; we connect it to approximate finite generation of almost kernels of homogenous quasimorphisms; finally we prove a Sikorav-style theorem connecting QΣ(G) to the vanishing of the suitably defined Novikov homology.

Related project(s):
60Property (T)

In order to obtain a closed orientable convex projective 4-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic 4-manifold through a continuous path of projective cone-manifolds.

Related project(s):
1Hitchin components for orbifolds

In previous definition of $\mathrm{E}$-theory, separability of the $\mathrm{C}^*$-algebras is needed either to construct the composition product or to prove the long exact sequences.

Considering the latter, the potential failure of the long exact sequences can be traced back to the fact that these $\mathrm{E}$-theory groups accommodate information about asymptotic processes in which one real parameter goes to infinity, but not about more complicated asymptotics parametrized by directed sets.

We propose a definition for $\mathrm{E}$-theory which also incorporates this additional information by generalizing the notion of asymptotic algebras. As a consequence, it not only has all desirable products but also all long exact sequences, even for non-separable $\mathrm{C}^*$-algebras.

More precisely, our construction yields equivariant $\mathrm{E}$-theory for $\mathbb{Z}_2$-graded $G$-$\mathrm{C}^*$-algebras for arbitrary discrete groups $G$.

We suspect that our model for $\mathrm{E}$-theory could be the right entity to investigate index theory on infinite dimensional manifolds.

Related project(s):
10Duality and the coarse assembly map78Duality and the coarse assembly map II

We expand upon the notion of a pre-section for a singular Riemannian foliation \((M,\mathcal{F})\), i.e. a proper submanifold \(N\subset M\) retaining all the transverse geometry of the foliation. This generalization of a polar foliation provides a similar reduction, allowing one to recognize certain geometric or topological properties of \((M,\mathcal{F})\)  and the leaf space \(M/\mathcal{F}\). In particular, we show that if a foliated manifold  \(M\) has positive sectional curvature and contains a non-trivial pre-section, then the leaf space \(M/\mathcal{F}\) has nonempty boundary. We recover as corollaries the known result for the special case of polar foliations as well as the well-known analogue for isometric group actions.

Related project(s):
43Singular Riemannian foliations and collapse

Beschreibung

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima’s ergodic theorem for the harmonic functions in the domain of the Lp generator. Secondly we prove analogues of Yau’s and Karp’s Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain growth criteria must be constant. As consequence we give an integral criterion for recurrence.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

Let K be a number field with ring of integers D and let G be a Chevalley group scheme not of type E8, F4 or G2. We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that H^vcd(G(D);Q)=0 if D is Euclidean.

Related project(s):
62A unified approach to Euclidean buildings and symmetric spaces of noncompact type

Non-local boundary conditions, such as the Atiyah-Patodi-Singer (APS) conditions, for Dirac operators on Riemannian manifolds are well under\-stood while not much is known for such operators on spacetimes with timelike boundary. We define a class of Lorentzian boundary conditions that are local in time and non-local in the spatial directions and show that they lead to a well-posed Cauchy problem for the Dirac operator. This applies in particular to the APS conditions imposed on each level set of a given Cauchy temporal function.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds

We establish a one-to-one correspondence between Finsler structures on the \(2\)-sphere with constant curvature \(1\) and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $$\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$$ of weighted projective spaces provide examples of Finsler \(2\)-spheres of constant curvature and all geodesics closed.

Related project(s):
68Minimal Lagrangian connections and related structures

We associate a flow \(\phi\) to a solution of the vortex equations on a closed oriented Riemannian 2-manifold \((M,g)\) of negative Euler characteristic and investigate its properties. We show that \(\phi\) always admits a dominated splitting and identify special cases in which \(\phi\) is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of \((M,g)\).

Related project(s):
68Minimal Lagrangian connections and related structures

Given a parabolic geometry on a smooth manifold \(M\), we study a natural affine bundle \(A \to M\), whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on \(A\), which induces an almost bi-Lagrangian structure on \(A\) and a compatible linear connection on \(TA\). We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with non-zero scalar curvature, provided the parabolic geometry is torsion-free and \(|1|\)-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in \(A\). For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampere equation and thus to properly convex projective structures.

Related project(s):
68Minimal Lagrangian connections and related structures