Publications

This article presents a method for proving upper bounds for the first \(\ell^2\)-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first \(\ell^2\)-Betti number.

Our approach extends to generalizations of \(\ell^2\)-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

Related project(s):
58Profinite perspectives on l2-cohomology

We provide a posteriori error estimates in the energy norm for temporal semi-discretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak-strong stability estimates which we combine with suitable reconstructions of the numerical solution. We present time-adaptive numerical simulations based on the a posteriori error estimators for solutions involving blow-up.

Related project(s):
67Asymptotics of singularities and deformations

This is the second in a two part series of papers concerning Morse quasiflats - higher dimensional analogs of Morse quasigeodesics. Our focus here is on their asymptotic structure. In metric spaces with convex geodesic bicombings, we prove asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity for Morse quasiflats. Moreover, we provide some immediate consequences.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We analyze weak convergence on CAT(0) spaces and the existence and properties of corresponding weak topologies.

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

We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.

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

We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.

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

We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

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

We prove that a topological 4-manifold of globally non-positive curvature is homeomorphic to Euclidean space.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.

Related project(s):
24Minimal surfaces in metric spaces66Minimal surfaces in metric spaces II

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-diffeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-diffeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

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

Let (N,h) be a time- and space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of N with induced Riemannian metric g and second fundamental form K. If (N,h) satisfies the dominant energy condition in a strict sense, then the Dirac-Witten operator of M⊂ N is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial data pairs on $M$ satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's α-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac-Witten operator may be non-invertible, and we will study the kernel of this operator in this case.

We will show that the kernel may only be non-trivial if π1(M) is virtually solvable of derived length at most 2. This allows to extend the index theoretical methods to spaces of initial data pairs, satisfying the dominant energy condition in the weak sense.

We will show further that a spinor φ is in the kernel of the Dirac--Witten operator on (M,g,K) if and only if (M,g,K,φ) admits an extension to a Lorentzian manifold (N,h) with parallel spinor ψ such that M is a Cauchy hypersurface of (N,h), such that g and K are the induced metric and second fundamental form of M, respectively, and φ is the restriction of ψ to M.

Back to my Homepage

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

We provide a direct connection between the Zmax (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank 2.

We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the Zmax-JSJ decomposition of such groups.

Related project(s):
60Property (T)

We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise, and are intricately related to, groups with no Baumslag-Solitar subgroups. Indeed, for groups of cohomological dimension 2 we prove that algebraic hyperbolicity is equivalent to containing no Baumslag-Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension 2.

We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.

Related project(s):
60Property (T)

We prove that twisted ℓ2-Betti numbers of locally indicable groups are equal to the usual ℓ2-Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration E with base space B having locally indicable fundamental group, and with a simply-connected fibre F, the first formula bounds ℓ2-Betti numbers b(2)i(E) of E in terms of ℓ2-Betti numbers of B and usual Betti numbers of F; the second formula computes b(2)i(E) exactly in terms of the same data, provided that F is a high-dimensional sphere.

We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and 3-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.

Related project(s):
60Property (T)

We construct a new type of expanders, from measure-preserving affine actions with spectral gap on origami surfaces, in each genus g⩾1. These actions are the first examples of actions with spectral gap on surfaces of genus g>1. We prove that the new expanders are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. In genus g=1, this implies that the Margulis expander, and hence the Gabber--Galil expander, is coarsely distinct from the Selberg expander. For the proof, we use the concept of piecewise asymptotic dimension and show a coarse rigidity result: A coarse embedding of either R2 or H2 into either R2 or H2 is a quasi-isometry.

Related project(s):
60Property (T)

The present book deals with the spectral geometry of infinite graphs. This topic involves the interplay of three different subjects: geometry, the spectral theory of Laplacians and the heat flow of the underlying graph. These three subjects are brought together under the unifying perspective of Dirichlet forms. The spectral geometry of manifolds is a well-established field of mathematics. On manifolds, the focus is on how Riemannian geometry, the spectral theory of the Laplace–Beltrami operator, Brownian motion and heat evolution interact. In the last twenty years large parts of this theory have been subsumed within the framework of strongly local Dirichlet forms. Indeed, this point of view has proven extremely fruitful.

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

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schrödinger operators on these graphs. We obtain estimates on the first and second order term of the eigenvalue asymptotics. Moreover, we prove a unique continuation result for eigenfunctions and decay properties of general eigenfunctions. The proofs rely on a detailed analysis of the geometry which employs a Copy-and-Paste procedure based on the Gauß-Bonnet theorem.

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

In this article we prove upper bounds for the k-th Laplace eigenvalues below the essential spectrum for strictly negatively curved Cartan–Hadamard manifolds. Our bound is given in terms of k^2 and specific geometric data of the manifold. This applies also to the particular case of non‐compact manifolds whose sectional curvature tends to minus infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.

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

We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts.

On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse $\mathrm{K}$-theory and -homology, the secondary products correspond to canonical primary products between the $\mathrm{K}$-theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.

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