Publications

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 show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian (????+2)-manifold, with regular leaves homeomorphic to the n-torus, is given by a smooth effective n-torus action. This solves in the negative for the codimension 2 case a question about the existence of foliations by exotic tori on simply-connected manifolds.

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

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

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

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

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.

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

We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.

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

For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. Assuming Grothendieck rigidity, we essentially solve the problem by Galois cohomological means.

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

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

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

Building on work of Bowden-Hensel-Webb, we study the action of the homeomorphism group of a surface $S$ on the fine curve graph $\mathcal{C}^\dagger(S)$.  While the definition of $\mathcal{C}^\dagger(S)$ parallels the classical curve graph for mapping class groups, we show that the dynamics of the action of $\mathrm{Homeo}(S)$ on $\mathcal{C}^\dagger(S)$ is much richer:  homeomorphisms induce parabolic isometries in addition to elliptics and hyperbolics, and all positive reals are realized as asymptotic translation lengths. 

When the surface $S$ is a torus, we relate the dynamics of the action of a homeomorphism on $\mathcal{C}^\dagger(S)$ to the dynamics of its action on the torus via the classical theory of {\em rotation sets}.  We characterize homeomorphisms acting hyperbolically, show asymptotic translation length provides a lower bound for the area of the rotation set, and, while no characterisation purely in terms of rotation sets is possible, we give sufficient conditions for elements to be elliptic or parabolic.  

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