Seminarium z Geometrycznej Teorii Grup
prof. dr hab. Tadeusz Januszkiewicz, dr Piotr Nowak
Thursdays 10:1512:00, room 403 at IMPAN
Date 
Speaker 
Title and abstract 


19 IV
5 IV

Michele Triestino (Université de Bourgogne) de BourgogneUn
Ignacio Vergara (ENS Lyon)

Smoothening singular group actions on manifolds
Abstract: Motivated by the recent results around Zimmer’s program, we study group actions on manifolds, with singular regularity (we require that every element is differentiable at all but countably many points). The groups under considerations have a fixed point property, called FW, which generalizes Kazhdan’s property (T) (in particular we can consider actions of lattices in higherrank simple Lie groups).
The main result is that if a group G has property FW, any singular action of G on a closed manifold 1) either has a finite orbit, 2) or is conjugate to a differentiable action, up to changing the differentiable structure of the manifold. This is a joint work with Yash Lodha and Nicolas Matte Bon.
Radial Schur multipliers


8 III

Damian Orlef (IMPAN)

Title: FL^p properties of random groups, using random graphs Abstract: We will discuss the proof, due to C. Druţu and J. M. Mackay, of the fact that random triangular groups have fixed point properties for affine isometric actions on L^p spaces (FL^p), where the range of p tends to infinity with the number of generators. The argument uses a spectral criterion for FL^p, given by Bourdon, which is a generalisation of the criterion of Żuk and BallmannŚwiątkowski for property (T), and relies on eigenvalues of nonlinear pLaplacians. Central to the proof is the authors' development of new bounds on spectral gaps of pLaplacians of random graphs, which we will follow. We will also mention connections to conformal dimension of the group's boundary, results for the Gromov density model and generalisations due to T. de Laat and M. de la Salle.


22 II

Eric Reckwerdt (IMPAN) 
Title: Relative hyperbolicity and weak amenability


1 II

Yongle Jiang (IMPAN)

Title: Lower degree cohomology groups of algebraic group actions Abstract: We study the 1st and 2nd cohomology groups of an algebraic action of a group G. Under natural assumptions, we could show that these cohomology groups "remember" the "algebraic data" of this action. Then we discuss two applications: (1) the second cohomology group H^2(G, ZG) is torsion free as an abelian group when G has property (T) as a direct corollary of Sorin Popa's celebrated cocycle superrigidity theorem; (2) we answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups using known facts from geometric group theory.


18 I

Piotr Nowak (IMPAN)

Aut(F_5) has property (T) I will present the recent joint work with Marek Kaluba (UAM) and Narutaka Ozawa (RIMS) in which we prove that Aut(F_5), the automorphism group of the free group on 5 generators, has Kazhdan's property (T).


11 I

Ján Špakula (University of Southampton)

Title: Recognizing operators in Roe algebras Abstract: Roe algebras are C*algebras which encode largescale structure (a.k.a. coarse geometry) of metric spaces. They are defined as norm closures of certain algebras of operators (with finite propagation) on some Hilbert spaces. Under a mild assumption about the metric space (namely finite decomposition complexity), we provide a few alternative pictures of Roe algebras, e.g. as operators with finite epsilonpropagation, or via commutation relations with slowly varying functions on the space. This is joint work with A. Tikuisis.


14 XII

Tomasz Prytuła (University of Southampton)

Classifying space for proper actions for groups admitting a strict fundamental domain. For an infinite discrete group G, the classifying space for proper actions EG is a proper GCW complex X, such that for every finite subgroup F ⊂ G the fixed point set XF is contractible. In a joint work with Nansen Petrosyan we describe a procedure of constructing new models for EG out of the standard ones, provided the action of G on EG admits a strict fundamental domain. Our construction is of combinatorial nature, and it depends only on the structure of the fundamental domain. The resulting model is often much ‘smaller’ than the old one, and thus it is wellsuited for (co)homological computations. Before outlining the construction, I shall give some background on the space EG. I will also discuss some examples and applications in the context of Coxeter groups, graph products of finite groups and automorphism groups of buildings.


8 XII

Juhani Koivisto (University of Southern Denmark)

Title: Measure and coarse equivalence of unimodular lcsc groups Abstract: In this seminar I will tell about some recent developments in joint work with D. Kyed and S. Raum on measure equivalence of unimodular locally compact second countable (ulcsc) groups. Among other things, we will see that two compactly generated amenable ulcsc groups are uniformly measure equivalent if and only if they are coarsely equivalent. Welcome on a journey of measure theory and some coarse geometry!


30 XII

Nima Hoda (Mc Gill)

Title: Quadric Complexes Abstract: Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2dimensional CAT(0) cube complexes and may be viewed as the bisimplicial analog of systolic complexes. I will define and present some properties of quadric complexes and of groups which act on them. As time permits, I will discuss dismantlability, asphericity and other topics.


23 XI

Damian Sawicki (IMPAN)

Continuum of new counterexamples to the coarse Baum–Connes conjecture The coarse Baum–Connes conjecture predicts that the Ktheory of the Roe algebra of a metric space can be computed as the image of a certain assembly map and we will pinpoint a Ktheory class that cannot belong to this image. Counterexamples come from group actions with a spectral gap (typically: subgroup actions on compact simple Lie groups) via Kazhdan projections and the warped cone construction, which was conjectured by Druţu and Nowak. No familiarity with Baum–Connes conjectures will be assumed.


9 XI

Eric Reckwerdt (IMPAN)

Complements of Kazhdan projections in semisimple groups


26 X

Damian Orlef (IMPAN)

Title: Kazhdan's property (T) and more for random groups in the square model for d>5/12 Abstract: A random group in the square model is obtained by fixing a set of n generators and introducing at random about (2n)^(4d) relations of length 4 between them, where d is a fixed parameter called the density and n tends to infinity. By results of T. Odrzygóźdź, if d<1/2, then these groups are with overwhelming probability (w.o.p.) infinite and hyperbolic. We prove that for d>5/12 the random groups G in the square model have w.o.p. Kazhdan's property (T). The proof proceeds by constructing a triangular group H, which maps onto finite index subgroup of G and verifying that the Żuk's spectral criterion can be successfully applied to yield Kazhdan's property of H. The verification proceeds by analyzing random walks on the link of H, in the spirit of Broder and Shamir.


12 X

Yeong Chyuan Chung (IMPAN)

Title: Controlled Ktheory for C*algebras and beyond Abstract: I will introduce a controlled version of Ktheory for C*algebras due to OyonoOyono and Yu, and explain how it can be extended to a larger class of Banach algebras that includes operator algebras on L^p spaces.


28 IX

David Kerr (Texas A&M University)

Dimension, comparison, and almost finiteness


21 IX

Kate Juschenko (Northwestern)

Title: Cycling amenable groups and soficity Abstract: I will give introduction to sofic groups and discuss a possible strategy towards finding a nonsofic group. I will show that if the Higman group were sofic, there would be a map from Z/pZ to itself, locally like an exponential map, satisfying a rather strong recurrence property. The approach to (non)soficity is based on the study of sofic representations of amenable subgroups of a sofic group. This is joint work with Harald Helfgott.


8 VI

Rufus Willett (University of Hawaii)

Title: Dynamic Asymptotic Dimension: examples and questions. Abstract: I’ll introduce dynamic asymptotic dimension, a notion introduced in joint work with Guentner and Yu that generalises Gromov’s asymptotic dimension to general topological dynamical systems (Gromov’s notion is the special case of a group acting on its StoneCech compactification). I’ll try to focus mainly on examples and open questions.


6 VI

Erik Guentner (University of Hawaii)

Affine actions, cohomology and hyperbolicity


25 V

Damian Sawicki (IMPAN)

Title: How to fold a manifold and obtain a (super) expander Abstract: We will discuss how to warp the metric on a closed manifold so that the resulting metric space becomes quasiisometric to an expander graph. This includes strongest known expanders, that is, expanders with respect to Banach spaces of nontrivial type. More formally, we will prove that levels of the warped cone over a group action are quasiisometric to an expander with respect to a Banach space X if and only if the action has a spectral gap with respect to X. This extends earlier results of Piotr Nowak and myself and of Federico Vigolo.


18 V

Damian Orlef (IMPAN)

Title: Random groups, cyclic orders and actions on the circle Abstract: We consider the following question: can random groups in the Gromov density model, or other related models, act nontrivially on the circle by homeomorphisms? If the homeomorphisms are assumed to be orientationpreserving, then it is equivalently the question about the existence of cyclically leftordered (and nontrivial) quotients of random groups. We propose a combinatorial method for ruling out cyclic leftorderability and use it show that random triangular groups don't admit cyclic leftorders at densities in range (1/3, 1/2). We then discuss how the proof generalizes to the case of quotients, other density models with fixedlength relators (e.g. square model) and, finally, the Gromov density model.


11 V

Alice Libardi (Sao Paulo State University)

Title: The BorsukUlam Property up to Cobordism


6 IV

Światosław Gal (Uniwersytet Wrocławski)

Title: Groups with proximal action are uniformly simple Abstract: A group is called Nuniformly simple if for every nontrivial conjugacyclass C,


23 III

Eric Reckwerdt (IMPAN)

Title: Weak amenability of hyperbolic groups (and relatives) Abstract: In this talk I will discuss Ozawa's proof that hyperbolic groups are weakly amenable, with the aim of showing how the proof extends to the relatively hyperbolic case.


15 III

Eric Reckwerdt (IMPAN)

Title: Relatively Hyperbolic Groups Abstract: This lecture I will introduce relatively hyperbolic groups and their geometric helper spaces. In particular, I'll cover the conedoff Cayley graph, also known as the electric space, the cusped space of Groves and Manning, and the thinned cusped space, a new construction of Guentner, Tessera, and myself. In all cases, a group will be hyperbolic relative to a subgroup if and only if each of these three spaces is hyperbolic. Next week I will demonstrate applications of the new construction.


9 III

Masato Mimura (Tohoku/EPFL)

Title: Unbounded updates of fixed points


9 II

Piotr Nowak (IMPAN & UW)

Title: Geometric induction in equivariant KKtheory Abstract: Given a quasiisometry between groups G and H, I will construct a map from the Gequivariant KKtheory KK^G(A,B) to a certain Hequivariant KKtheory group.


12 I

Piotr Nowak (IMPAN & UW)

Title: Fixed point properties and measure equivalence Abstract: We show that property FLp is invariant under strong 2pmeasure equivalence. This implies certain invariance properties of the fixed point spectrum of groups. As a corollary we obtain that there exist many different strong pME classes among random hyperbolic groups. This is joint work with David Fisher.


22 XII 
Damian Orlef (IMPAN) 
Nonorderability of random groups — work in progress A countable group admits a leftinvariant linear order (resp. cyclic order) if and only if it acts faithfully by orientationpreserving homeomorphisms on the real line (resp. the circle). This provides a way to investigate questions connected to the Gromov's conjecture that random groups should not have any nontrivial smooth actions on compact manifolds. I will show that random groups in the Gromov density model do not have any leftorderable quotients at any density and how to use this fact, together with property (T), to show that there are no interesting smooth, orientationpreserving actions of random groups on the circle at densities d>1/3. I will also show that random groups in the triangular binomial model do not admit leftinvariant cyclic orders for some densities d>1/3, which is a joint result with Piotr Przytycki. An application of leftorders to the problem of finding the sharp critical threshold of freeness in the triangular binomial model will be discussed. 

15 XII

Eric Reckwerdt (IMPAN)

An Introduction to Graph Products of Groups In this talk I will define graph products — how to make them, how to play with them, and how to use them. In particular, I'll discuss the normal forms of their elements and the CAT(0)cube complexes that corresponds to their intergroup structure. I'll then show how to use this to prove that graph products of weakly amenable groups are again weakly amenable.


8 XII

Thiebout Delabie (Neuchatel)

Coarse embeddability into a Hilbert space and containing expanders are two mutually exclusive properties, however there exists metric space with neither. Looking at box spaces we know quite a lot. Box spaces of amenable group embed, while box spaces of groups with property (T) are expanders. For the free group the answer is not that straight forward. The free group has box spaces that are expanders, but there also exist box spaces that embed into a Hilbert space. An example of the first case is shown by Lubotzky. An example of the second case is shown by Arzhantseva, Guentner and Spakula. We will combine these results to provide an example that has neither property.


29 XI

Joachim Zacharias (Glasgow)

Towards a dynamical version of the Cuntz semigroup Abstract: The Cuntz semigroup is an invariant for C*algebras combining refined Ktheoretical and tracial properties of the algebra in question. It carries important information and therefore plays an important role in the classification programme. It is however notoriously difficult to determine. Based on ideas of Winter and others we consider a dynamical version of the Cuntz semigroup for discrete groups acting on compact spaces which appears much easier to determine. It models the Cuntz semigroup of the crossed product C*algebra of the action. For classification one is mostly interested in actions of amenable groups. In this this case the dynamical Cuntz semigroup of the action is closely connected to Rokhlin tower approximations. The existence of such approximations in the topological setting has only been established for a restricted class of amenable groups and appears as an important open problem in general. We will give an introduction to Cuntz semigroup methods and present some basic results concerning the dynamical Cuntz semigroup and connections to the classification programme.


24 XI

Jianchao Wu (Penn State)

The amenability dimension for topological and C*dynamics We showcase a number of recently emerged concepts of dimensions defined for topological dynamical systems, such as the dynamical asymptotic dimension and the amenability dimension. Roughly speaking, these dimensions measure the complexity of the topological dynamical system in terms of partial orbits. They turn out to be of great use to the study of the BaumConnes conjecture and the FarrellJones conjecture, as well as the bounding the nuclear dimensions for crossed product C*algebras, which is a crucial regularity property in the recent breakthrough in the classification program of simple separable amenable C*algebras. They have


10 XI 
Mariusz Tobolski (UW) 
Title: BorsukUlam type theorems


3 XI

Marek Kaluba (IMPAN)

Title: Computational aspects of property (T).


20 X

Chris Cave (University of Copenhagen)

Title: Exactness of locally compact second countable groups Abstract: In this talk we will show that a locally compact second countable group is exact (in the sense that the reduced cross product preserves short exact sequences) if and only if the group admits a topological amenable action on a compact Hausdorff space (usually called amenable at infinity). A metric lattice is a discrete metric subspace that is "large enough" in the group. We take badly behaved operators and extend them to the whole group to show a particular sequence of reduced cross products is not exact whenever the group is not amenable at infinity. This is joint work with Jacek Brodzki and Kang Li. I will also present a work in progress with Jacek Brodzki and Jianchao Wu by showing the close relationship between the uniform Roe algebra of the metric lattice and its counterpart in the locally compact second countable group.


13 X

Eric Reckwerdt (IMPAN)

Weak amenability of free products. Abstract: In this talk I shall show that free products of weakly amenable groups are again weakly amenable. Highlights will include how to extend positive definite and completely bounded functions from factor groups to their free products and a little BassSerre theory for free groups. If time permits, I will introduce graph products and the generalization of the construction for free products to graph products. 

6 X

Jan Dymara (Uniwersytet Wrocławski)

Title: Boundary unitary representations  rightangled hyperbolic buildings. Abstract: To a measureclass preserving action of a group $G$ on a space $X$ with measure $\mu$ one can associate in a natural way a unitary representation of $G$ on $L^2(X,\mu)$. For the fundamental group $\Gamma$ of a compact negativelycurved manifold $M$ one can consider the action of $\Gamma$ on the universal cover of $M$ and on its ideal boundary. The corresponding representation was shown to be irreducible by Bader and Muchnik. 

4 X 15:00 room 403 
Martin FinnSell (Universitaet Wien) 
Title: A coarse geometric approach to understanding general finite approximations of discrete groups. Abstract: In this talk I will describe what it means for a finitely generated discrete group to be sofic in terms of BenjaminiSchramm convergence of finite graphs, and discuss a few questions connected with this concept. From this definition I will talk about a host of new results, which fit an old scheme, which link coarse geometric properties of the approximation directly to properties of the group  this is the beginning of a program to understand how coarse properties match up directly with what are typically measurable properties of graphs. The main technique used here is a construction originally due to Skandalis, Tu and Yu and involves a topological groupoid  consequently, I will spend some time describing how to obtain this groupoid and discuss groupoids in general (for instance, to try and motivate why they are interesting in their own right and not only as a vehicle for proving results about groups). 

Here is a webpage that we used during academic year 2015/2016. The former website in no longer maintained. 