Wednesdays 11:00-12:00 in N0.008, coffee from 10:30 in the Hausdorff room.
Organizer: Jonathan Fruchter.
Dario Ascari (University of the Basque Country), 2.5.24 (different day due to holiday)
Lawk Mineh (Southampton), 8.5.24
No talk (dies academicus), 15.5.24
No talk (midsemester break), 22.5.24
Amir Behar (Hebrew University), 29.5.24
Claudio Llosa Isenrich (KIT), 5.6.24
Sam Fisher (Oxford), 12.6.24
Stephen Cantrell (Warwick), 19.6.24
Olga Varghese (Düsseldorf), 26.6.24
Shaked Bader (Oxford), 3.7.24
Stefanie Zbinden (Heriot-Watt), 10.7.24
Wednesdays 11:00-12:00 in 0.011, coffee from 10:30 in the Hausdorff room.
Organizer: Giles Gardam.
Eduardo Reyes (MPIM, Bonn), 18.10.23
Comparing group actions on CAT(0) cube complexes
Given a set of isometric actions of a given group that share common characteristics and a natural notion of equivalence between them, we can obtain a rich deformation space. Classical examples include (higher) Teichmuller spaces, outer spaces for free groups, and negatively curved metric on manifolds. For different points in this deformation space, it is also desirable to have a quantitative notion of how different they are. In the case of negatively curved metrics and Anosov representations, this can be answered from a statistical point of view, by encoding the actions in (analogues of) the geodesic flow. In this talk I will explain a joint work with Stephen Cantrell, in which we extend this philosophy and compare "compatible" geometric actions on CAT(0) cube complexes.
Jonathan Fruchter (Uni Bonn), 25.10.23
Virtual homology, residually free groups and profinite rigidity
The (co)homology of finite index subgroups of a finitely generated and residually finite group G can sometimes be read from the list of finite quotients of G. The goal of this talk is to illustrate how this simple observation can be used to profinitely distinguish some residually free groups from one another. We will use topological methods to compute virtual homological invariants of (fully) residually free groups, and use these calculations along with additional properties of residually free groups to show that direct products of free, surface and free abelian groups are profinitely rigid among finitely presented residually free groups. The talk is based on joint work with Ismael Morales.
No talk (all saints), 1.11.23
Jacques Audibert (MPIM, Bonn), 8.11.23
Surface subgroups in lattices of the symplectic group
Considering a lattice in a Lie group, we say that a subgroup of this lattice is "thin" if it is infinite index yet Zariski-dense. Thin subgroups share a lot of properties with lattices themselves and are thus an active field of research. However, we are lacking examples of thin groups that are not virtually free groups. In this talk, I will present a construction of thin surface subgroups in lattices of symplectic groups. The proof will rely on the theory of maximal representations.
Giles Gardam (Uni Bonn), 15.11.23
Graphs and groups with unique geodesics
A graph is called geodetic if any two vertices are joined by a unique shortest path. We show that a finitely generated group whose Cayley graph is geodetic is virtually a free group. More generally, a locally finite quasi-transitive geodetic graph is a quasi-tree. The proof exploits the interplay between local structure and the structure of a boundary. This is joint work with Murray Elder, Adam Piggott, Davide Spriano and Kane Townsend.
Grigori Avramidi (MPIM, Bonn), 22.11.23
Group rings and hyperbolic geometry
In the 60's Cohn showed that all ideals in the group algebra of a free group are free. Bass and Wall used this result to show that all two-dimensional complexes with free fundamental groups are standard: they are all homotopy equivalent to wedges of circles and 2-spheres. The goal of this talk is to describe recent results of this type for groups acting on hyperbolic spaces. I will explain an algorithm showing that in the group algebra of a group acting on a hyperbolic space, ideals generated by "few" elements are free (where "few" is a function of the minimal displacement of the action) and mention applications to cohomological dimension of "few relator" groups, topology of 2-complexes with hyperbolic fundamental groups, and complexity of cell decompositions of hyperbolic manifolds. Joint work with Thomas Delzant.
Igor Spiridonov (MPIM, Bonn), 29.11.23
On the homology of the Torelli group
The theory of surface mapping class groups is closely related to topology of 3-manifolds, geometry of moduli spaces, and automorphisms of free groups. The Torelli group is defined as the kernel of the mapping class group action on the homology of the surface. Its Eilenberg-MacLane space can be constructed as the moduli space of smooth complex curves equipped with a homology framing. However, unlike the case of mapping class group, we know very little about the homology of the Torelli group. In the 1980s, Johnson computed its abelianization, but no other nontrivial homology groups had been computed explicitly yet. In 2007, Bestvina, Bux and Margalit proved that the cohomological dimension of the genus g Torelli group is 3g-5. In this talk, I will present a complete description of the top homology of the Torelli group in genus 3.
No talk (dies academicus), 6.12.23
Adele Jackson (Oxford), 13.12.23
Algorithms for Seifert fibered spaces
Given two mathematical objects, the most basic question we can ask is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a plan to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces.
Marco Linton (Oxford), 10.1.24
The coherence of one-relator groups
(Joint work with Andrei Jaikin-Zapirain.) A group is said to be coherent if all of its finitely generated subgroups are finitely presented. In this talk I will sketch a proof of Baumslag’s conjecture that all one-relator groups are coherent, discussing connections with the non-positive immersions property and the vanishing of the second L^2 Betti number.
Uri Bader (Weizmann), 17.1.24
Characters of arithmetic groups and related topics
This will be a survey style talk. I will discuss the spaces of traces and characters on a countable group and their relations with the spaces of normal subgroups, invariant random subgroups and other related structures. I will give some examples and then specialize at arithmetic groups, explain the state of the art and discuss some conjectures.
Davide Spriano (Oxford), 24.1.24
The Morse local-to-global property
One of the best understood classes of infinite groups are Gromov hyperbolic groups. This is largely because they present many aspects of negative curvature, which have important algebraic and geometric consequences. Pushing beyond the boundaries of hyperbolic groups, one can consider groups that are “negatively curved” only along some directions, known as the Morse geodesics. The philosophy is that theorems that are true for hyperbolic groups should be true for general groups along Morse direction. Although appealing, this philosophy turned out to be too optimistic, as researcher discovered pathological examples where Morse geodesics do not, in fact, behave as geodesics in hyperbolic groups. However, such examples are very artificial. The Morse local-to-global property is a property satisfied by a very large class of metric spaces that, morally, guarantees that Morse geodesics “behave as they should”. In this talk we will elaborate on the background and survey new developments in the theory of Morse local-to-global groups.