Tee-Seminar der AG Kramer

Zeit und Ort:

Semester:		WS 2021
Zeit:		Mo 14:00 - 14:45
Ort:		SRZ 214 (hybrid)

Inhalt:

Mitglieder der Arbeitsgruppe und Gäste tragen über ihre laufenden Forschungsarbeiten vor, oder über Themen, die uns interessieren. Wenn Sie am Seminar über ZOOM teilnehmen möchten, schicken Sie bitte eine e-mail an Dr. Sathaye.

Vorträge:

18.10.2021 Amandine Escalier (Münster) Local-to-Global Rigidity of quasi-buildings

Abstract: We say that a graph G is Local-to-Global rigid if there exists R>0 such that every other graph whose balls of radius R are isometric to the balls of radius R in G is covered by G. Examples include the Euclidean building of PSL_n(Q_p). We prove that the rigidity of the building goes further by proving that a reconstruction is possible from only a partial local information, called “print”. We use this to prove the rigidity of graphs quasi-isometric to the building among which are the torsion-free lattices of PSL_n(Q_p).

08.11.2021 Kevin Schreve (Lousiana State University) Torsion invariants of complexes of groups

Abstract: For a residually finite group G, one can consider various homological growths of a chain of finite index normal subgroups of G. We will be interested in the mod-p homology growth and growth of torsion in integral homology. The groups we consider all act on contractible complexes with strict fundamental domain Q, where stabilizers are either trivial or have vanishing homological growth. We then show the homology of the subcomplex of Q consisting of cells with nontrivial stabilizer completely determines the homological growth of G. For example, we give an exact calculation of these invariants for right-angled Artin groups. This is joint work with Boris Okun, and I will also talk about earlier work with Grigori Avramidi and Boris Okun.

15.11.2021 Justin Lanier (University of Chicago) Mapping class groups and dense conjugacy classes

Abstract: I’ll start by introducing infinite-type surfaces-those with infinite genus or infinitely many punctures-and the emerging study of their mapping class groups. One difference from the finite-type setting is that these mapping class groups come with natural non-discrete topologies. I’ll discuss joint work with Nick Vlamis where we fully characterize which surfaces have mapping class groups with dense conjugacy classes, so that there is a single element that well approximates every mapping class, up to conjugacy.

22.11.2021 Robert Kropholler (University of Warwick) Folding-like techniques for CAT(0) cube complexes

Abstract: In a seminal paper, Stallings introduced folding of morphisms of graphs. Stallings's methods give effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani-Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. We extend their techniques to fundamental groups of non-positively curved cube complexes. This is joint work with Rylee Lyman and Michael Ben-Zvi.

13.12.2021 Oussama Bensaid (University of Paris) Coarse Embeddings of Symmetric Spaces and Euclidean Buildings

Abstract: Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings. We will be particularly interested in coarse embeddings between symmetric spaces and euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results of Anderson--Schroeder, Kleiner--Leeb and Eskin--Farb in the 90's. In particular, it is well known that the rank of these spaces is monotonous under quasi-isometric embeddings. I will start by introducing these spaces and their large-scale geometry, and show that in the absence of a euclidean factor in the domain, the rank is still monotonous under coarse embeddings.

20.12.2021 Mireille Soergel (L'Institut de Mathématiques de Bourgogne) Systolic complexes and group presentations

Abstract: Systolic complexes were introduced by Januszkiewicz and Swiatkowski as a combinatorial form of non-positive curvature. We want to construct systolic complexes using Cayley graphs. We give conditions on a presentation of a group, which imply that the flag complex of the Cayley graph is well-defined and systolic. The main applications of the result concern Garside groups and Artin groups. We will give a classification of the Garside groups whose presentation using the simple elements as generators satisfy our conditions.

10.01.2022 Harry Petyt (University of Bristol) Injective metric spaces and group actions

Abstract: Injective metric spaces were introduced in the 50s, and they have some nice metric and fixed-point properties. It turns out that every metric space embeds in a "smallest" injective space, called its injective hull. In this talk we'll discuss injective hulls and some applications to group theory.

24.01.2022 Damian Osajda (University of Wroclaw) Helly groups

Abstract: A simplicial graph is Helly if each family of pairwise intersecting (combinatorial) balls has non-empty intersection. Groups acting geometrically on such graphs are themselves called Helly. The family of such groups is vast, it contains: Gromov hyperbolic groups, CAT(0) cubical groups, Garside groups, FC type Artin groups, some lattices in buildings, and others. On the other hand, being Helly implies many important algorithmic and geometric features of the group. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a kind of CAT(0)-like structure. One immediate consequence is that Helly groups satisfy the coarse Baum-Connes conjecture and the Farrell-Jones conjecture, allowing us to prove these conjectures for new classes of groups. I will present basic properties and examples of Helly groups. The talk is based on joint works with Jeremie Chalopin, Victor Chepoi, Anthony Genevois, Hiroshi Hirai, Jingyin Huang, Motiejus Valiunas, Thomas Haettel.

31.01.2022 Nima Hoda (ENS Paris) Asymptotic Cones of Snowflake Groups and the Strong Shortcut Property

Abstract: Snowflake groups were introduced by Brady and Bridson in order to prove that the set of degrees of polynomial Dehn functions of finitely presented groups is dense above 2. I will discuss recent work of Cashen, Woodhouse and mine in which we show that an infinite family of snowflake groups has all asymptotic cones simply connected. This is a property shared by strongly shortcut groups, which are groups acting on graphs having strong cycle distortion properties. By contrast, we show that our infinite family of snowflake groups have natural Cayley graphs that have isometrically embedded cycles of arbitrary length. Thus we either exhibit first examples of not strongly shortcut groups with all asymptotic cones simply connected or we exhibit first examples of strongly shortcut groups not all of whose Cayley graphs are strongly shortcut.