Publications

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

We use the language of proper CAT(-1) spaces to study thick, locally compact trees, the real, complex and quaternionic hyperbolic spaces and the hyperbolic plane over the octonions. These are rank 1 Euclidean buildings, respectively rank 1 symmetric spaces of non-compact type. We give a uniform proof that these spaces may be reconstructed using the cross ratio on their visual boundary, bringing together the work of Tits and Bourdon.

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

Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self-maps of spaces related to the given objects. Here we address a question posed by Gonçalves and Wong in the mid 2000s: we construct an infinite series of compact connected solvmanifolds (that are not nilmanifolds) of strictly increasing dimensions and all of whose self-homotopy equivalences have vanishing Nielsen number. To this end, we establish a sufficient condition for a prominent (infinite) family of soluble linear groups to have the so-called property R∞. In particular, we generalize or complement earlier results due to Dekimpe, Gonçalves, Kochloukova, Nasybullov, Taback, Tertooy, Van den Bussche, and Wong, showing that many soluble S-arithmetic groups have R∞ and suggesting a conjecture in this direction.

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

In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups -- a result which is possibly of independent interest.

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

We show that if one of various cycle types occurs in the permutation action of a finite group on the cosets of a given subgroup, then every almost conjugate subgroup is conjugate. As a number theoretic application, corresponding decomposition types of primes effect that a number field is determined by the Dedekind zeta function. As a geometric application, coverings of Riemannian manifolds with certain geodesic lifting behaviors must be isometric.

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

Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and Köhl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple.

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

We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, the Iwasawa decomposition G=KAU provides a weak homotopy equivalence between K and G, implying π1(G)=π1(K). It thus suffices to determine π1(K) which we achieve by investigating the fundamental groups of generalized flag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized flag variety is a CW decomposition − in particular, we cover the complete symmetrizable situation; the result concerning the structure of π1(K) more generally also holds in the non-symmetrizable two-spherical situation.

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

In the 1970s, Williams developed an algorithm that has been used to construct modular links. We introduce the notion of bunches to provide a more efficient algorithm for constructing modular links in the Lorenz template. Using the bunch perspective, we construct parent manifolds for modular link complements and provide the first upper volume bound that is independent of word exponents and quadratic in the braid index. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented.

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

very oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic's continued fraction expansion. Consequently, we give a volume's upper bound for some sequences of Lorenz knots complements, linearly in terms of the corresponding braid index. Also, for any punctured hyperbolic surface we give volume's bounds for the canonical lift complement relative to some sequences of sets of closed geodesics in terms of the geodesics length.

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

We construct a sequence of geodesics on the modular surface such that the complement of the canonical lifts to the unit tangent bundle are arithmetic 3-manifolds.

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

We prove a positive mass theorem for spin initial data sets \((M,g,k)\) that contain an asymptotically flat end and a shield of dominant energy (a subset of M on which the dominant energy scalar \(μ−|J|\) has a positive lower bound). In a similar vein, we show that for an asymptotically flat end \(\mathcal{E}\) that violates the positive mass theorem (i.e. \(\mathrm{E}<|\mathrm{P}|\)), there exists a constant \(R > 0\), depending only on \(\mathcal{E}\), such that any initial data set containing \(\mathcal{E}\) must violate the hypotheses of Witten's proof of the positive mass theorem in an \(R\)-neighborhood of \(\mathcal{E}\). This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten's approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

Let \(M\) be an orientable connected \(n\)-dimensional manifold with \(n\in\{6,7\}\) and let \(Y\subset M\) be a two-sided closed connected incompressible hypersurface which does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of \(M\) and \(Y\) are either both spin or both non-spin. Using Gromov's \(\mu\)-bubbles, we show that \(M\) does not admit a complete metric of psc. We provide an example showing that the spin/non-spin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension \(7\), a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension two. We deduce as special cases that, if \(Y\) does not admit a metric of psc and \(\dim(Y) \neq 4\), then \(M := Y\times\mathbb{R}\) does not carry a complete metric of psc and \(N := Y \times \mathbb{R}^2\) does not carry a complete metric of uniformly psc provided that \(\dim(M) \leq 7\) and \(\dim(N) \leq 7\), respectively. This solves, up to dimension \(7\), a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

In this survey, we give an overview of recent applications of Callias operators to the geometry of scalar curvature. A Callias operator is an operator of the form \(\mathcal{B}_\psi = \mathcal{D} + \mathcal{G}_\psi\), where \(\mathcal{D}\) is a Dirac operator and \(\mathcal{G}_\psi\) is an order zero term depending on a scalar-valued function \(\psi\). The zero order term modifies the Schrödinger–Lichnerowicz formula by a differential expression in the function \(\psi\) that can be related to distance estimates. This fact allows to use the Dirac method to derive sharp quantitative estimates in the presence of lower scalar curvature bounds in the spirit of metric inequalities with scalar curvature as proposed by Gromov.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

We construct pairs of residually finite groups with isomorphic profinite completions such that one has non-vanishing and the other has vanishing real second bounded cohomology. The examples are lattices in different higher rank simple Lie groups. Using Galois cohomology, we actually show that \(\operatorname{SO}^0(n,2)\) for \(n \ge 6\) and the exceptional groups \(E_{6(-14)}\) and \(E_{7(-25)}\) constitute the complete list of higher rank Lie groups admitting such examples.

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