## 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“.

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.

Journal | Adv. Math |

Volume | 408 Part B |

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

Journal | Invent. math. |

Publisher | Springer |

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

Given a Chevalley group $\mathcal{G}$ of classical type and a Borel subgroup $\mathcal{B} \subseteq \mathcal{G}$, we compute the $\Sigma$-invariants of the $S$-arithmetic groups $\mathcal{B}(\Z[1/N])$, where $N$ is a product of large enough primes. To this end, we let $\mathcal{B}(\Z[1/N])$ act on a Euclidean building $X$ that is given by the product of Bruhat--Tits buildings $X_p$ associated to $\mathcal{G}$, where $p$ is a prime dividing $N$. In the course of the proof we introduce necessary and sufficient conditions for convex functions on $\CAT(0)$-spaces to be continuous. We apply these conditions to associate to each simplex at infinity $\tau \subset \pinf X$ its so-called parabolic building $X^{\tau}$ and to study it from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential $n$-connectivity rather than actual $n$-connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building $\Delta$ contains an apartment, provided $\Delta$ is thick enough and $\Aut(\Delta)$ acts chamber transitively on $\Delta$.

Journal | Transactions of the American Mathematical Society |

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

The main theme of this paper is higher virtual algebraic fibering properties of right-angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building.

We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected.

From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to Z whose kernel has strong topological finiteness properties.

We additionally use our techniques to present examples where the kernel is of type F_2 but not FP_3, and examples where the RACG is hyperbolic and the kernel is finitely generated and non-hyperbolic.

The key tool we use is a generalization of an approach due to Jankiewicz--Norin--Wise involving Bestvina--Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.

Journal | Journal of Topology |

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

Given a finitely generated group G that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup H of G that is bounded in the relative Cayley graph of G is conjugate into a peripheral subgroup. As an application, we obtain a trichotomy for subgroups of relatively hyperbolic groups. Moreover we prove the existence of the relative exponential growth rate for all subgroups of limit groups.

Journal | Algebraic and Geometric Topology |

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

We give a topological classification of Lawson's bipolar minimal surfaces corresponding to his ξ- and η-family. Therefrom we deduce upper as well as lower bounds on the area of these surfaces, and find that they are not embedded.

**Related project(s):****25**The Willmore energy of degenerating surfaces and singularities of geometric flows**67**Asymptotics of singularities and deformations

We obtain estimates on nonlocal quantities appearing in the Volume Preserving Mean Curvature Flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time singularities of VPMCF are ancient solutions to Mean Curvature Flow (MCF), prove that monotonicity methods may always be applied at finite times and obtain information on the asymptotics of the flow.

**Related project(s):****25**The Willmore energy of degenerating surfaces and singularities of geometric flows**67**Asymptotics of singularities and deformations

We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle-Huisken to allow for mean curvature flow with surgery for the Alexandrov immersed, 2-dimensional setting.

**Related project(s):****25**The Willmore energy of degenerating surfaces and singularities of geometric flows**67**Asymptotics of singularities and deformations

This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by c⋅t−1 converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott.

**Related project(s):****75**Solutions to Ricci flow whose scalar curvature is bounded in L^p II

Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the \(C^{2,\alpha}\)-topology. In dimension 3 the original manifold only needs to have finite volume, and the volume can be arbitrarily large. Applications include a new proof of the hyperbolization of 3-manifolds of large Hempel distance yielding some new geometric control on the hyperbolic metric, and an analytic proof of Dehn filling and drilling that allows the filling and drilling of arbitrary many cusps and tubes.

**Related project(s):****51**The geometry of locally symmetric manifolds via natural maps

We describe the Gromov-Hausdorff closure of the class of length spaces being homeomorphic to a fixed closed surface.

**Related project(s):****66**Minimal surfaces in metric spaces II

If a group Γ acts geometrically on a CAT(0) space *X* without 3-flats, then either *X* contains a Γ-periodic geodesic which does not bound a flat half-plane, or else *X* is a rank 2 Riemannian symmetric space, a 2-dimensional Euclidean building or non-trivially splits as a metric product. Consequently all such groups satisfy a strong form of the Tits Alternative.

**Related project(s):****66**Minimal surfaces in metric spaces II

This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let *X* be a CAT(0) space with a geometric group action. Suppose that every geodesic in *X* lies in an *n*-flat, *n*≥2. If *X* contains a periodic *n*-flat which does not bound a flat (*n*+1)-half-space, then *X* is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.

**Related project(s):****66**Minimal surfaces in metric spaces II

A CAT(0) space has rank at least *n* if every geodesic lies in an *n*-flat. Ballmann's Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann's conjecture. Here we prove that a CAT(0) space of rank at least *n*≥2 is rigid if it contains a periodic *n*-flat and its Tits boundary has dimension (*n*−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces -- so-called Morse flats. We show that the Tits boundary ∂*T**F* of a periodic Morse *n*-flat *F* contains a regular point -- a point with a Tits-neighborhood entirely contained in ∂*T**F*. More precisely, we show that the set of singular points in ∂*T**F* can be covered by finitely many round spheres of positive codimension.

**Related project(s):****66**Minimal surfaces in metric spaces II

We describe the Gromov-Hausdorff closure of the class of length spaces being homeomorphic to a fixed closed surface.

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

We discuss solutions of several questions concerning the geometry of conformal planes.

**Related project(s):****66**Minimal surfaces in metric spaces II

H-type foliations (????,H,g) are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping ???? with the Bott connection we consider the scalar horizontal curvature κ as well as a new local invariant τ induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannanian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of κ and τ. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of H-type foliations allows us to consider the pull-back of Korányi balls to ????. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when ???? is locally isometric as a sub-Riemannian manifold to its H-type tangent group.

**Related project(s):****6**Spectral Analysis of Sub-Riemannian Structures

On the seven dimensional Euclidean sphere S^7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution T is induced by a Clifford module structure of R^8. More precisely, T is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution Q of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration ????3↪????7→????4. Answering a question in arXiv:0901.1406 we first show that Q does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures, the intrinsic sublaplacians ΔTsub and ΔQsub and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

Journal | J. Geom. Anal. |

Volume | 32 |

Pages | 37pp |

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**Related project(s):****6**Spectral Analysis of Sub-Riemannian Structures

We prove regularity estimates for the eigenfunctions of Schrödinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body operators are covered by our result; in that case, the weight is in terms of the (euclidean) distance to the collision planes. The technique of proof is based on blow-ups and Lie manifolds. More precisely, we first blow-up the spheres at infinity of the collision planes to obtain the Georgescu-Vasy compactification and then we blow-up the collision planes. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher order operators and matrices of scalar operators.

Journal | to appear in Letters Math. Phys. |

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**Related project(s):****35**Geometric operators on singular domains

The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.

**Related project(s):****60**Property (T)