Speakers

Gromov conjectured that the L^p-cohomology of simple Lie groups vanishes below the rank. Farb conjectured a fixed point property for actions of lattices in such groups on CAT(0) cell complexes of dimension lower than the rank. In this talk I will give a short introduction to group cohomology and prove that vanishing of \ell^1-cohomology up to degree n implies a finite orbit for every action on an n-dimensional contrcatible complex, thus in particular establishing that Gromov's conjecture implies Farb's conjecture. I will then give some insight to the proof of Gromov's conjecture. This talk is based on joint work with Saar Bader, Uri Bader and Roman Sauer.

In this talk I will present the construction of a suitable cube complex X with a G-action, for G a generalized Baumslag-Solitar group BS(p_1, q_1;...;p_n, q_n) = <a, t_1,..., t_n | t_i a^{p_i} = a^{q_i} t_i, t_i t_j = t_j t_i for all i,j>. Such a complex X is isomorphic to the product of the Bass-Serre trees of the classical BS(p_i, q_i), and by devising an easy-to-check criterion for computing the essential connectivity properties of products of trees (equipped with suitable height functions), the complex can be used to describe the full Σ-theory of the generalized Baumslag-Solitar groups. This is a joint work with Kevin Klinge and José Pedro Quintanilha

Gromov-Thuston manifolds have been introduced in the 80s as new examples of negatively curved Riemannian manifolds that are not homotopy equivalent to symmetric spaces. Topologically, they are constructed as branched coverings of a hyperbolic manifold along a suitable chosen codimension 2 totally geodesic submanifold. In a joint work with M. Incerti-Medici we compute the entropy of general branched coverings with curvature bounded above. I will also discuss an ongoing project where we try to understand if two different branched coverings on the base couple have quasi-isometric fundamental groups: this is already a relevant open question in the case of Gromov-Thurston manifolds.

The study of hyperbolic geometry in dimensions two and three has long been intertwined with holomorphic quadratic differentials on Riemann surfaces. We will focus on three settings in which this connection appears: the Bers embedding of Teichmüller space, Wolf’s parametrization via harmonic maps, and the description of almost Fuchsian manifolds through the second fundamental forms of their minimal surfaces. In this talk, I will present work in progress with Nathaniel Sagman showing that these three topics, traditionally regarded as unrelated, can instead be understood as different restrictions of a more general "holomorphic" framework.

A group is coherent if all its finitely generated subgroups are finitely presented. Wise proposes a bold classification of coherent two dimensional groups: he conjectures that G is coherent if and only if G is virtually the fundamental group of a 2-complex with nonpositive immersions if and only if the second L²-Betti number of G vanishes. In this talk we will define the concepts appearing in Wise's Conjecture, and we will see that the conjecture holds in the class of virtually special groups. We will also discuss applications to coherence of Coxeter groups and random groups, as well similar results for higher-dimensional coherent groups. This is based on joint work with Pablo Sánchez-Peralta.

A strongly invertible knot is a knot embedded in the 3-sphere together with a suitable involution of the ambient space that preserves the knot. We introduce a variant of grid homology that incorporates these symmetry properties. Using this framework, we define a pair of invariants that provide a lower bound for the slice genus of equivariant cobordisms between strongly invertible knots.

In the seminal work of Culler and Shalen (1983) a method is outlined to detect essential surfaces in a three-manifold by studying their SL(2,C)-character variety. The method underscores connections between the theory of incompressible surfaces in three-manifolds, splittings of fundamental groups, group actions on trees, and the geometry of representation varieties. In this talk, we will motivate and then lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over F, an algebraically closed field of positive characteristic. We then apply the extended theory to a variety of settings. This is joint work with Stephan Tillmann.

For negatively curved manifolds, a way of building classes in bounded cohomology is to consider cocycles defined by integrating differential forms over straight simplices. A classical result of Barge and Ghys shows that for closed, negatively curved surfaces this procedure defines an injective map from the space of differential 2-forms to the second bounded cohomology of the surface. Recently, Battista, Francaviglia, Moraschini, Sartini and Savini showed that a similar result holds for 2-forms on closed negatively curved manifolds in any dimension. In this talk, I will present a generalisation of these results for non-compact hyperbolic manifolds, characterising the injectivity of the Barge-Ghys map in terms of the action of the fundamental group on hyperbolic space. This is (ongoing) joint work with Gian Maria dall'Ara and Roberto Frigerio.

In this talk, I will introduce a class of groups called generalised triangle groups. These groups were originally defined by Stallings, and later studied by Caprace, Conder, Kaluba, and Witzel as candidates for non-residually finite hyperbolic groups. Generalised triangle groups have corresponding complexes which allow us to study them using combinatorial and geometric techniques. I will demonstrate this through the automaticity of non-positively curved k-fold triangle groups and briefly talk about applications.

Stable subgroups of finitely generated groups were introduced by Durham and Taylor as a generalisation of both the quasiconvex subgroups of hyperbolic groups, and the convex cocompact subgroups of mapping class groups. They also nicely coincide with many other classes of subgroups; for example in the general case they are the hyperbolic Morse subgroups, and in right-angled Artin groups they are the purely loxodromic subgroups. In this talk we will extend this to a characterisation of the stable subgroups of graph products with infinite vertex groups. This is joint work with Sahana Balasubramanya, Marissa Chesser, Johanna Mangahas, and Marie Trin.

θ-positivity, introduced by Guichard-Wienhard, provides a unifying framework that extends several known phenomena of positive representations of surface groups into Lie groups. In this talk, we take a geometrical point of view on θ-positivity, based on special subsets of flag varieties, so-called diamonds. We then explore the dynamics that θ-positive representations of finitely generated free groups induce on certain flag varieties.

Maximal representations from the fundamental group of a surface group into a hermitian Lie group G form connected components of the character variety generalising the Teichmüller space. When G=SO(2,3), those representations act cocompactly on some convex in the pseudo-hyperbolic space (a pseudo-riemannian analogue of the hyperbolic space), and one can define a volume functional on the space of maximal representations. In this talk I will explain a surprising result concerning the boundedness of this volume functional.

To provide a unified framework for median graphs, mapping class groups, and $\mathrm{CAT}(0)$ cube complexes, Bowditch introduced the notion of a coarse median on a metric space. Answering a question of Bowditch, Haettel showed that a higher-rank symmetric space of non-compact type admits a coarse median if and only if it is a product of rank-one spaces. In this talk, we will focus on the family of higher-rank symmetric spaces of noncompact type admitting a convex projective model. While these metric spaces do not admit a coarse median in Bowditch's sense, we will construct a generalized coarse median for them. We will conclude by discussing some properties of our ''coarse higher medians''. This is joint work with Mitul Islam.

In general relativity, the universe is modeled by a four-dimensional Lorentzian manifold satisfying Einstein’s equations. A fundamental result by Choquet-Bruhat and Geroch (1969) establishes the existence and uniqueness of a maximal development associated with given initial data. These solutions fall within the framework of globally hyperbolic spacetimes, which are naturally endowed with a partial order relation, leading to the notion of maximal extension. In this talk, I will focus on these questions in the context of conformally flat globally hyperbolic spacetimes. In 2013, C. Rossi proved the existence and uniqueness of a maximal extension in this setting. However, her proof does not provide an explicit description of this extension. I will present an alternative and constructive approach, based on the notion of an enveloping space, within which the maximal extension can be realized explicitly. I will illustrate this construction with several examples.

Besides being natural objects to study, branched covers of closed hyperbolic manifolds M along codimension 2 totally geodesic submanifolds B provide examples of interesting geometric and topological phenomena. For instance, Gromov and Thurston showed that most of them admit a Riemannian metric of pinched negative curvature but no locally symmetric one. As such natural geometric invariants of the covering like its volume are not necessarily topological invariants and key topological analogs such as the simplicial volume are not so simple to compute exactly. This often makes it difficult to answer simple questions such as: Can cyclic branched covers of M along B of different degrees be homotopy equivalent? In joint work with Alessandro Sisto we develop some criteria and tools to answer this question in terms of the arithmetic of the degrees.