Asymptotic decomposition methods in geometry, dynamics and operator algebras Seminar

Monday 27/03/17

PM

Romain Tessera , Paris Sud - Orsay

Title: Proper actions on Lp spaces for relatively hyperbolic groups

Abstract: In a joint work with Erik Guentner, we prove that a relatively hyperbolic group whose peripheral subgroups have polynomial growth admits a proper action on some L^p space. We shall also discuss the general case (without assumptions on the peripheral groups).

Arthur Bartels , Universtät Munster

Title: The Farrell-Jones Conjecture and (coarse) flow spaces

Abstract: The talk will give an introduction to the Farrell-Jones Conjecture and to the part dynamics plays in proofs of instances of the conjecture.

Indira Chatterji , Université de Nice

Title: TBA

Tuesday 28/03/17

AM

Jiawen Zhang, University of Southampton

Title: A new proof of finite asymptotic dimension for CAT(0) cube complexes

Abstract: This is a joint work with G. Arzhantseva, G. Niblo and N. Wright. In 2010, N. Wright proved that the asymptotic dimension of a CAT(0) cube complex is bounded by its dimension. He proved this by constructing a family of Lipschitz cobounded maps to CAT(0) cube complexes. In my talk, I will introduce a new direct (i.e. to construct bounded covers directly) proof of the finiteness part of this result. It consists of a characterization for finite asymptotic dimension and some study on the normal spheres of CAT(0) cube complexes. The work is partially supported by the European Research Council grant of Goulnara ARZHANTSEVA, no. 259527 and the Sino-British Fellowship Trust by Royal Society.

Rufus Willett , University of Hawaii

Title: Dynamic asymptotic dimension and applications

Abstract: This is based on joint work with Guoliang Yu and Erik Guentner. I will introduce a concept called dynamic asymptotic dimension for an action of a discrete group on a compact topological space, motivated by the asymptotic dimension of a metric space in the sense of Gromov.  Over the course of two lectures, I’ll sketch examples, connections to the work of Bartels, Lück and Reich in controlled topology, connections to the work of Winter and Zacharias in C*-algebra theory, and applications to the computation of C*-algebra K-theory.

Damian Sawicki , Polish Academy of Sciences

Title: TBA

PM

Nick Wright , University of Southampton

Title: TBA

Pierre Pansu , Paris Sud-Orsay

Title: $L^p$-cohomology, coarse and conformal embeddings

Abstract: Invariants under coarse embeddings are scarce. We show that $L^p$-cohomology is one. In fact, it is natural for an even larger class of maps, large scale conformal maps. Using it, we show that a real valued invariant of hyperbolic metric spaces, conformal dimension, is non decreasing under coarse (and large scale conformal) embeddings.

Wednesday    29/03/17

AM

Arthur Bartels , Universtät Munster

Title: The Farrell-Jones Conjecture and (coarse) flow spaces

Abstract: The talk will give an introduction to the Farrell-Jones Conjecture and to the part dynamics plays in proofs of instances of the conjecture.

Kang Li , Universtät Munster

Title: Structure and K-theory of uniform Roe algebras

Abstract: I will report on recent developments in the structure and K-theory of uniform Roe algebras associated to metric spaces with bounded geometry. In particular, we show that for uniform Roe algebras, being AF, having stable rank one, and having cancellation, are all equivalent to the underlying metric space having asymptotic dimension zero. A countable group has asymptotic dimension zero if and only if it is locally finite. We also show that uniform Roe algebras of locally finite countable groups can be completely classified by $K_0$ groups together with units. To our best knowledge, this is the first classification result for non-separable and non-simple $C^*$-algebras. As a contrast, if the metric space $X$ is non-amenable and has asymptotic dimension one, then the $K_0$ group of the uniform Roe algebra over $X$ is always zero. Finally, we answer negatively to a question of Elliott and Sierakowski about the vanishing of $K_0$ of the uniform Roe algebras of non-amenable groups.
This is joint work with Hung-Chang Liao and Rufus Willett.

David Rosenthal , St John’s University

Title: Regular Finite Decomposition Complexity

Abstract: We introduce the notion of "regular finite decomposition complexity" of a metric family. This generalizes Gromov's finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the permanence properties that are known for FDC, as well as a new one. We show that for a collection containing all metric families with finite asymptotic dimension all other permanence properties follow from Fibering Permanence. This is joint work with Daniel Kasprowski and Andrew Nicas.

PM

Yves Benoist , Paris Sud - Orsay

Title: TBA

Thursday        30/03/17

AM

Rufus Willett , University of Hawaii

Title: Dynamic asymptotic dimension and applications

Abstract: This is based on joint work with Guoliang Yu and Erik Guentner. I will introduce a concept called dynamic asymptotic dimension for an action of a discrete group on a compact topological space, motivated by the asymptotic dimension of a metric space in the sense of Gromov.  Over the course of two lectures, I’ll sketch examples, connections to the work of Bartels, Lück and Reich in controlled topology, connections to the work of Winter and Zacharias in C*-algebra theory, and applications to the computation of C*-algebra K-theory.

David Hume , Oxford University

Title: Coarse embeddings, and how to avoid them

Abstract: Coarse embeddings occur completely naturally in geometric group theory: every finitely generated subgroup of a finitely generated group is coarsely embedded. Since even very nice classes of groups - hyperbolic groups or right-angled Artin groups for example - are known to have 'wild' collections of subgroups, there are precious few invariants that one may use to prove a statement of the form '$H$ does not coarsely embed into $G$' for two finitely generated groups $G,H$.
The growth function and the asymptotic dimension are two coarse invariants which which have been extensively studied, and a more recent invariant is the separation profile of Benjamini-Schramm-Timar.
In this talk I will describe a new spectrum of coarse invariants, which include both the separation profile and the growth function, and can be used to tackle many interesting problems, for instance: Does there exist a coarse embedding of the Baumslag-Solitar group $BS(1,2)$ into a hyperbolic group?
This is part of an ongoing collaboration with John Mackay and Romain Tessera.

Michael Farber , Queen Mary University of London

Title: Topology of large random spaces

Abstract: I will discuss probabilistic models generating random simplicial complexes. One is able to predict their topological properties with probability tending to one when the spaces are large, i.e. depend on a growing number of independent random variables.

PM

Karen Strung , Polish Academy of Sciences

Title: TBA

Gábor Szabó , University of Aberdeen

Title: TBA

Friday              31/03/17

AM

Daniel Kasprowski , Universität Bonn

Title: On the K-theory of groups with finite decomposition complexity

We will show that for every ring $R$ the assembly map in algebraic $K$-theory
\[H^G_n(\underbar{E}G;\mathbb{K}_R)\to K_n(R[G])\]
is split injective for certain groups with finite decomposition complexity. In particular, for every subgroup $G$ of a linear group which admits a finite-dimensional model for the classifying space $\underbar{E}G$ for proper actions. The concept of finite decomposition complexity was first introduced by Guentner, Tessera, and Yu. It is a coarse invariant of metric spaces and generalizes the notion of finite asymptotic dimension.

Joachim Zacharias , University of Glasgow

Title: TBA

Mark Hagen , Cambridge University

Title: Finite asymptotic dimension of hierarchically hyperbolic spaces

Abstract: The class of "hierarchically hyperbolic spaces/groups" generalises the notion of a hyperbolic metric space and was introduced to abstract some common geometric features of mapping class groups and cubical groups.  It provides a unified set of tools for studying a wide variety of spaces and groups, including Teichmuller space and most fundamental groups of 3--manifolds, in addition to the examples already mentioned.  In this talk, I will explain why hierarchically hyperbolic groups have finite asymptotic dimension.  In the case of the mapping class group, we get and explicit upper bound which is quadratic in the complexity.