## Publications of SPP2026

On this site you find preprints and publications produced within the projects and with the support of the DFG priority programme „Geometry at Infinity“.

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.

Journal | Acta Math. |

Volume | 221 |

Pages | 159-202 |

Link to preprint version | |

Link to published version |

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

We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.

Journal | Math. Ann. |

Volume | Online First |

Link to preprint version |

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

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

Journal | Geom. Funct. Anal. |

Volume | To appear |

Link to preprint version |

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

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.

**Related project(s):****27**Invariants and boundaries of spaces

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

**Related project(s):****5**Index theory on Lorentzian manifolds

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of Higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball, and we compute its dimension explicitly. For example, the Hitchin component of the right-angled hyperbolic \(\ell\)-polygon reflection group into \(\mathrm{PGL}(2m,\mathbb{R})\), resp. \(\mathrm{PGL}(2m+1,\mathbb{R})\), is homeomorphic to an open ball of dimension \((\ell-4)m^2+1\), resp. \((\ell-4)(m^2+m)\). We also give applications to the study of the pressure metric and the deformation theory of real projective structures on 3-manifolds.

**Related project(s):****1**Hitchin components for orbifolds

Geometric structures on manifolds became popular when Thurston used them in his work on the Geometrization Conjecture. They were studied by many people and they play an important role in Higher Teichmüller Theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Thesis that this technique can be used to construct geometric structures in interesting cases. Here we will survey some recent results in this direction, which are joint work with Qiongling Li.

**Related project(s):****1**Hitchin components for orbifolds

Let X be a compact manifold, D a real elliptic operator on X, G a Lie group, P a principal G-bundle on X, and B_P the infinite-dimensional moduli space of all connections on P modulo gauge, as a topological stack. For each connection \nabla_P, we can consider the twisted elliptic operator on X. This is a continuous family of elliptic operators over the base B_P, and so has an orientation bundle O^D_P over B_P, a principal Z_2-bundle parametrizing orientations of KerD^\nabla_Ad(P) + CokerD^\nabla_Ad(P) at each \nabla_P. An orientation on (B_P,D) is a trivialization of O^D_P.

In gauge theory one studies moduli spaces M of connections \nabla_P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X, g). Under good conditions M is a smooth manifold, and orientations on (B_P,D) pull back to

orientations on M in the usual sense of differential geometry.

This is important in areas such as Donaldson theory, where one needs an orientation on M

to define enumerative invariants.

We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (B_P,D), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds,

instantons, the Kapustin-Witten equations, and the Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds.

**Related project(s):****33**Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

Suppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator D. Let G be SU(m) or U(m), and E be a rank m complex bundle with G-structure on X. Write B_E for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z_2-bundle O^D_E on B_E parametrizing orientations of det D_Ad A for twisted elliptic operators D_Ad A at each [A] in B_E. A theorem of Walpuski shows O^D_E is trivializable.

We prove that if we choose an orientation for det D, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of O^D_E for all such bundles E on X, satisfying natural compatibilities.

Now let (X,\varphi,g) be a compact G_2-manifold, with d(*\varphi)=0. Then we can consider moduli spaces M_E^G_2 of G_2-instantons on E over X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general. The restriction of O^D_E to M_E^G_2 is the Z_2-bundle of orientations on M_E^G_2. Thus, our theorem induces canonical orientations on all such G_2-instanton moduli spaces M_E^G_2.

This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of G_2-manifolds (X,\varphi,g) by counting moduli spaces M_E^G_2, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6.

**Related project(s):****33**Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n-2)-connected. In addition, we show that for every non-trivial free factor system of a free group, the corresponding relative free splitting complex is contractible.

**Related project(s):****8**Parabolics and invariants

We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or, more general, groups with finite decomposition complexity.

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

We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic K-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to prove injectivity results about assembly maps.

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

We show that a finitely generated residually finite rationally solvable (or RFRS) group *G* is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb Z$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of *G* vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of 3-manifolds.

**Related project(s):****8**Parabolics and invariants

We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope.

We use the agrarian invariants to solve the torsion-free case of a conjecture of Friedl--Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, for such groups, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl--Tillmann and Friedl--Lück--Tillmann. Finally, we prove that for agrarian groups of deficiency $1$, the agrarian polytope admits a marking of its vertices which controls the Bieri--Neumann--Strebel invariant of the group, improving a result of the second author and partially answering a question of Friedl--Tillmann.

**Related project(s):****8**Parabolics and invariants

We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers.

Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone and improve bounds by Szabo, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups.

In the appendix, we justify optimality of our result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasi-isometric groups, contrary to the case of box spaces.

Journal | Proc. Lond. Math. Soc. |

Publisher | Wiley |

Link to preprint version | |

Link to published version |

**Related project(s):****8**Parabolics and invariants

We construct a Baum--Connes assembly map localised at the unit element of a discrete group $\Gamma$.

This morphism, called $\mu_\tau$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[\tau]\in KK_\mathbb{R}^\Gamma(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $\Gamma$.

We show that the corresponding $\tau$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture.

**Related project(s):****4**Secondary invariants for foliations

In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the *i*-th ℓ²-Betti number is not a profinite invariant for all *i*≥2.

**Related project(s):****18**Analytic L2-invariants of non-positively curved spaces

Given an S-arithmetic group, we ask how much information on the ambient algebraic group, number field of definition, and set of places S is encoded in the commensurability class of the profinite completion. As a first step, we show that the profinite commensurability class of an S-arithmetic group with CSP determines the number field up to arithmetical equivalence and the places in S above unramified primes. We include some applications to profiniteness questions of group invariants.

**Related project(s):****18**Analytic L2-invariants of non-positively curved spaces

We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators.

Publisher | de Gruyter |

Book | J. Brüning, M. Staudacher (Eds.): Space - Time - Matter |

Pages | 324-348 |

Link to preprint version | |

Link to published version |

**Related project(s):****5**Index theory on Lorentzian manifolds

We show that a properly convex projective structure p on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if p is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that p admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable L2-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

**Related project(s):****26**Projective surfaces, Segre structures and the Hitchin component for PSL(n,R)