TeeSeminar der AG KramerZeit und Ort:

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 email an Dr. Sathaye.
Vorträge:
 18.10.2021 Amandine Escalier (Münster) LocaltoGlobal Rigidity of quasibuildings
 08.11.2021 Kevin Schreve (Lousiana State University) Torsion invariants of complexes of groups
 15.11.2021 Justin Lanier (University of Chicago) Mapping class groups and dense conjugacy classes
 22.11.2021 Robert Kropholler (University of Warwick) Foldinglike techniques for CAT(0) cube complexes
 13.12.2021 Oussama Bensaid (University of Paris) Coarse Embeddings of Symmetric Spaces and Euclidean Buildings
 20.12.2021 Mireille Soergel (L'Institut de Mathématiques de Bourgogne) Systolic complexes and group presentations
 10.01.2022 Harry Petyt (University of Bristol) Injective metric spaces and group actions
 24.01.2022 Damian Osajda (University of Wroclaw) Helly groups
 31.01.2022 Nima Hoda (ENS Paris) Asymptotic Cones of Snowflake Groups and the Strong Shortcut Property
Abstract: We say that a graph G is LocaltoGlobal 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 quasiisometric to the building among which are the torsionfree lattices of PSL_{n}(Q_{p}).
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 modp 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 rightangled Artin groups. This is joint work with Boris Okun, and I will also talk about earlier work with Grigori Avramidi and Boris Okun.
Abstract: I’ll start by introducing infinitetype surfacesthose with infinite genus or infinitely many puncturesand the emerging study of their mapping class groups. One difference from the finitetype setting is that these mapping class groups come with natural nondiscrete 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.
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 DaniLevcovitz used Stallingslike methods to study subgroups of rightangled Coxeter groups, which act geometrically on CAT(0) cube complexes. We extend their techniques to fundamental groups of nonpositively curved cube complexes. This is joint work with Rylee Lyman and Michael BenZvi.
Abstract: Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasiisometric embeddings. We will be particularly interested in coarse embeddings between symmetric spaces and euclidean buildings. The quasiisometric case is very well understood thanks to the rigidity results of AndersonSchroeder, KleinerLeeb and EskinFarb in the 90's. In particular, it is well known that the rank of these spaces is monotonous under quasiisometric embeddings. I will start by introducing these spaces and their largescale geometry, and show that in the absence of a euclidean factor in the domain, the rank is still monotonous under coarse embeddings.
Abstract: Systolic complexes were introduced by Januszkiewicz and Swiatkowski as a combinatorial form of nonpositive 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 welldefined 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.
Abstract: Injective metric spaces were introduced in the 50s, and they have some nice metric and fixedpoint 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.
Abstract: A simplicial graph is Helly if each family of pairwise intersecting (combinatorial) balls has nonempty 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 BaumConnes conjecture and the FarrellJones 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.
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.