## 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“.

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 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

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 |

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Link to published version |

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

We show that in each dimension 4n+3, n>1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension 7, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, inconjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.

Journal | Bulletin of the London Math. Society |

Volume | 50 |

Pages | 96-107 |

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Link to published version |

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible \(\sigma\)-compact space in which the corona sits as a \(\mathbb{Z}\)-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space \(BG\), then our constructions yield a \(\mathbb{Z}\)-structure for the group.

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

We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.

Journal | Journal of Geometry and Physics |

Publisher | Elsevier |

Volume | 112 |

Pages | 59-73 |

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

The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E\times\mathbb{R}^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact rank one symmetric spaces of dimensions multiples of four, and for certain products of spheres. Moreover, we observe that the SCSQ is "stable under tangential homotopy equivalence": if it has an affirmative answer for all vector bundles over a certain manifold \(M\), then the same is true for any manifold tangentially homotopy equivalent to~\(M\). Our main tool is topological K-theory. Over \(B^{13}\), there is essentially one stable class of real vector bundles for which our method fails.

Journal | Journal of Differential Geometry |

Link to preprint version |

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and $7$. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions at most $7$.

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds

We prove that sufficiently collapsed, closed and irreducible three-dimensional Alexandrov spaces are modeled on one of the eight three-dimensional Thurston geometries. This extends a result of Shioya and Yamaguchi, originally formulated for Riemannian manifolds, to the Alexandrov setting.

Journal | Indiana Univ. Math. J. |

Volume | In press. |

Link to preprint version |

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $\zeta$-invariants using lifted defect formulae which express discrepancies of $\zeta$-regularised traces in terms of Wodzicki residues. We derive Atiyah's $L^2$-index theorem as an instance of the $\mathbb{Z}_2$-graded generalisation of the canonical lift of spectral $\zeta$-invariants and we show that certain lifted spectral $\zeta$-invariants for geometric operators are integrals of Pontryagin and Chern forms.

Journal | Trans. Amer. Math. Soc |

Volume | to appear |

Link to preprint version |

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

We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.

Journal | to appear in Journal of Spectral Theory |

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**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We study magnetic Schrödinger operators on graphs. We extend the notion of sparseness of graphs by including a magnetic quantity called the frustration index. This notion of magnetic sparse turn out to be equivalent to the fact that the form domain is an \(\ell^2\) space. As a consequence, we get criteria of discreteness for the spectrum and eigenvalue asymptotics.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this paper we give an algebraic construction of the (active) reflected Dirichlet form. We prove that it is the maximal Silverstein extension whenever the given form does not possess a killing part and we prove that Dirichlet forms need not have a maximal Silverstein extension if a killing is present. For regular Dirichlet forms we provide an alternative construction of the reflected process on a compactification (minus one point) of the underlying space.

Journal | to appear in Potential Analysis |

Link to preprint version | |

Link to published version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

Journal | to appear in Mathematische Zeitschrift |

Link to preprint version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

Given a three-dimensional Riemannian manifold containing a ball with an explicit lower bound on its Ricci curvature and positive lower bound on its volume, we use Ricci flow to perturb the Riemannian metric on the interior to a nearby Riemannian metric still with such lower bounds on its Ricci curvature and volume, but additionally with uniform bounds on its full curvature tensor and all its derivatives. The new manifold is near to the old one not just in the Gromov-Hausdorff sense, but also in the sense that the distance function is uniformly close to what it was before, and additionally we have Hölder/Lipschitz equivalence of the old and new manifolds.

One consequence is that we obtain a local bi-Hölder correspondence between Ricci limit spaces in three dimensions and smooth manifolds.

This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must behomeomorphicto manifolds, and we obtain this in the most general, locally non-collapsed case.

The proofs build on results and ideas from recent papers of Hochard and the current authors.

**Related project(s):****31**Solutions to Ricci flow whose scalar curvature is bounded in Lp.