Publications

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 prove that the minimal representation dimension of a direct product G of non-abelian groups G1,…,Gn is bounded below by n+1 and thereby answer a question of Abért. If each Gi is moreover non-solvable, then this lower bound can be improved to be 2n. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups we show that branch groups cannot be boundedly generated.

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

We introduce the notion of a telescope of groups. Very roughly a telescope is a directed system of groups that contains various commuting images of some fixed group B. Telescopes are inspired from the theory of groups acting on rooted trees. Imitating known constructions of branch groups, we obtain a number of examples of B-telescopes and discuss several applications. We give examples of 2-generated infinite amenable simple groups. We show that every finitely generated residually finite (amenable) group embeds into a finitely generated (amenable) LEF simple group. We construct 2-generated frames in products of finite simple groups and show that there are Grothendieck pairs consisting of amenable groups and groups with property (τ). We give examples of automorphisms of finitely generated, residually finite, amenable groups that are not inner, but become inner in the profinite completion. We describe non-elementary amenable examples of finitely generated, residually finite groups all of whose finitely generated subnormal subgroups are direct factors.

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

We show that Ricci flow does not preserve a range of curvature conditions that interpolate between positive sectional and positive scalar curvature. 

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

We prove an analogue of Kostant's convexity theorem for split real and complex Kac-Moody groups associated to free and cofree root data. The result can be seen as a first step towards describing the multiplication map in a Kac-Moody group in terms of Iwasawa coordinates. Our method involves a detailed analysis of the geometry of Weyl group orbits in the Cartan subalgebra of a real Kac-Moody algebra. It provides an alternative proof of Kostant convexity for semisimple Lie groups and also generalizes a linear analogue of Kostant's theorem for Kac-Moody algebras that has been established by Kac and Peterson in 1984.

Related project(s):
61At infinity of symmetric spaces

A singular foliation \(\mathcal{F}\) on a complete Riemannian manifold \(M\) is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of M into orbits of an isometric action. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRF with special properties, e.g. isoparametric foliations, SRF on fibers bundles with Sasaki metric, and orbit-like foliations. More precisely, we prove two results analogous to Palais' Principle of Symmetric Criticality, one is a general principle for \(\mathcal{F}\) symmetric operators on the Hilbert space \(W^{1,2}(M)\), the other one is for \(\mathcal{F}\) symmetric integral operators on the Banach spaces \(W^{1,p}(M)\). These results together with a \(\mathcal{F}\) version of Rellich Kondrachov Hebey Vaugon Embedding Theorem allow us to circumvent difficulties with Sobolev's critical exponents when considering applications of Calculus of Variations to find solutions to PDEs. To exemplify this we prove the existence of weak solutions to a class of variational problems which includes \(p\)-Kirschoff problems.

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

We study obstructions to the existence of Riemannian metrics of positive scalar curvature on closed smooth manifolds arising from torsion classes in the integral homology of their fundamental groups. As an application, we construct new examples of manifolds which do not admit positive scalar curvature metrics, but whose Cartesian products admit such metrics. 

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We consider a geodesic billiard system consisting of a complete Riemannian manifold and an obstacle submanifold with boundary at which the trajectories of the geodesic flow experience specular reflections. We show that if the geodesic billiard system is hyperbolic on its trapped set and the latter is compact and non-grazing the techniques for open hyperbolic systems developed by Dyatlov and Guillarmou can be applied to a smooth model for the discontinuous flow defined by the non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent for the generator of the billiard flow. As an application we prove a meromorphic continuation of weighted zeta functions together with explicit residue formulae. In particular, our results apply to scattering by convex obstacles in the Euclidean plane.

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