Publications

For vector-valued Maass cusp forms for SL(2,Z) with real weight k∈R and spectral parameter s∈C, Res∈(0,1), s≢±k/2 mod 1, we propose a notion of vector-valued period functions, and we establish a linear isomorphism between the spaces of Maass cusp forms and period functions by means of a cohomological approach. The period functions are a generalization of those for the classical Maass cusp forms, being solutions of a finite-term functional equation or, equivalently, eigenfunctions with eigenvalue 1 of a transfer operator deduced from the geodesic flow on the modular surface. We apply this result to deduce a notion of period functions and related linear isomorphism for Jacobi Maass forms of weight k+1/2 for the semi-direct product of SL2(Z) with the integer points Hei(Z) of the Heisenberg group.

Related project(s):
70Spectral theory with non-unitary twists

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler-Einstein manifolds, quaternionic Kähler manifolds and manifolds with a harmonic 1-form of constant length.

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

We investigate groups that act amenably on their Higson corona (also known as bi-exact groups) and we provide reformulations of this in relation to the stable Higson corona, nuclearity of crossed products and to positive type kernels. We further investigate implications of this in relation to the Baum-Connes conjecture, and prove that Gromov hyperbolic groups have isomorphic equivariant K-theories of their Gromov boundary and their stable Higson corona.

Related project(s):
45Macroscopic invariants of manifolds

We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first ℓ2-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

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

We prove that every finitely generated, residually finite group G embeds into a finitely generated perfect branch group such that many properties of G are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non-abelian free group. As an application, we construct a finitely generated, non-amenable torsion branch group.

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

We study \(\mathsf{RCD}\)-spaces \((X,d,\mathfrak{m})\) with group actions by isometries preserving the reference measure \(\mathfrak{m}\) and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that each slice at a point is homeomorphic to a non-negatively curved \(\mathsf{RCD}\)-space. Under the assumption that \(X\) is non-collapsed we further show that the slices are homeomorphic to metric cones over homogeneous spaces with \(\mathrm{Ric}\geq 0\). As a consequence we obtain complete topological structural results and a principal orbit representation theorem. Conversely, we show how to construct new \(\mathsf{RCD}\)-spaces from a cohomogeneity one group diagram, giving a complete description of \(\mathsf{RCD}\)-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed \(\mathsf{RCD}\)-spaces of essential dimension at most 4.

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

We show that the sign of the Euler characteristic of an S-arithmetic subgroup of a simple k-group over a number field k depends on the S-congruence completion only. Consequently, the sign is a profinite invariant for such S-arithmetic groups with the congruence subgroup property. This generalizes previous work of the first author with Kionke-Raimbault-Sauer.

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

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped  with   strictly log-concave warping functions. This generalizes earlier results of  Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least $3$ minus two antipodal points, thus resolving a problem in Gromov's  ``Four Lectures'' in all dimensions. Our arguments are based on spin geometry.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We prove that there exist ????????(3)-invariant metrics on Aloff-Wallach spaces W^7_{k1,k2}, as well as ????????(5)-invariant metrics on the Berger space B^{13}, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.

Related project(s):
79Alexandrov geometry in the light of symmetry and topology

By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Guéritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Related project(s):
65Resonances for non-compact locally symmetric spaces

For negatively curved symmetric spaces it is known that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.

Related project(s):
65Resonances for non-compact locally symmetric spaces

We present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension on a compact manifold, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that a closed flat regular Riemannian foliation with leaves of positive dimension on a compact simply-connected manifold is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions.

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

The fine curve graph was introduced as a geometric tool to study the homeomorphisms of surfaces. In this paper we study the Gromov boundary of this space and the local topology near points associated with minimal measurable foliations. We then give several applications including finding explicit elements with positive stable commutator length, and proving a Tits alternative for subgroups of the homemorphism group of a closed surface containing a pseudo-Anosov map, generalizing a result of Hurtado-Xue.

Related project(s):
38Geometry of surface homeomorphism groups

In the framework of infinite ergodic theory, we derive equidistribution results for suitable weighted sequences of cusp points of Hecke triangle groups encoded by group elements of constant word length with respect to a set of natural generators. This is a generalization of the corresponding results for the modular group, for which we rely on advanced results from infinite ergodic theory and transfer operator techniques developed for AFN-maps.

Related project(s):
70Spectral theory with non-unitary twists

We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the maximal domain. Under additional geometric assumptions, and assumptions on an adapted boundary operator, we obtain a trace theorem on the maximal domain. This allows us to systematically study both local and nonlocal boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition occurs as a special case. Furthermore, we study contexts which induce semi-Fredholm and Fredholm extensions.

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

Every finite collection of oriented closed geodesics in the modular surface has a canonically associated link in its unit tangent bundle coming from the periodic orbits of the geodesic flow. We study the volume of the associated link complement with respect to its unique complete hyperbolic metric. We provide the first lower volume bound that is linear in terms of the number of distinct exponents in the code words corresponding to the collection of closed geodesics.

Related project(s):
38Geometry of surface homeomorphism groups

We show that locally homogeneous C^0-Riemannian manifolds are smooth.

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

We conduct a review of the basic definitions and the principal results in the study of wavelike spacetimes, that is spacetimes whose metric models massless radiation moving at the speed of light, focusing in particular on those geometries with parallel rays. In particular, we motivate and connect their various definitions, outline their coordinate descriptions and present some classical results in their study in a language more accessible to modern readers, including the existence of "null coordinates" and the construction of Penrose limits. We also present a thorough summary of recent work on causality in pp-waves, and describe progress in addressing an open question in the field - the Ehlers-Kundt conjecture.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We study timelike, totally umbilic hypersurfaces -- called photon surfaces -- in n+1-dimensional static, asymptotically flat spacetimes, for n+1≥4. First, we give a complete characterization of photon surfaces in a class of spherically symmetric spacetimes containing the (exterior) subextremal Reissner--Nordström spacetimes, and hence in particular the (exterior) positive mass Schwarzschild spacetimes. Next, we give new insights into the spacetime geometry near equipotential photon surfaces and provide a new characterization of photon spheres (not appealing to any field equations).

We furthermore show that any asymptotically flat electrostatic electro-vacuum spacetime with inner boundary consisting of equipotential, (quasi-locally) subextremal photon surfaces and/or non-degenerate black hole horizons must be isometric to a suitable piece of the necessarily subextremal Reissner--Norström spacetime of the same mass and charge. Our uniqueness result applies work by Jahns and extends and complements several existing uniqueness theorems. Its proof fundamentally relies on the lower regularity rigidity case of the Riemannian Positive Mass Theorem.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

The coordinate freedom of General Relativity makes it challenging to find mathematically rigorous and physically sound definitions for physical quantities such as the center of mass of an isolated gravitating system. We will argue that a similar phenomenon occurs in Newtonian Gravity once one ahistorically drops the restriction that one should only work in Cartesian coordinates when studying Newtonian Gravity. This will also shed light on the nature of the challenge of defining the center of mass in General Relativity. Relatedly, we will give explicit examples of asymptotically Euclidean relativistic initial data sets which do not satisfy the Regge--Teitelboim parity conditions often used to achieve a satisfactory definition of center of mass. These originate in our joint work with Jan Metzger. This will require appealing to Bartnik's asymptotic harmonic coordinates.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity