Seminar in Geometric Group Theory

26 VI

Alexander Dranishnikov (University of Florida)

Title: On Lusternik-Schnirelmann category

Abstract: The Lusternik-Schnirelmann category cat(X) was introduced in the 1920s to provide a lower bound for the number of critical points of a smooth function on a manifold. It is a numerical homotopy invariant defined for all topological spaces. Usually, it is very difficult to compute cat(X). Even giving a reasonable estimate is a challenging problem. We will discuss some open problems in the LS-category theory and recent progress on them.

Talk place and time: room 403 at 14:00.

14 VI

Piotr Przytycki (McGill University)

Title: A CAT(0) space for the 3-dimensional tame automorphism group.

Abstract: The 3-dimensional tame automorphism group Tame(k^3) is the group of automorphisms of the affine space k^3 generated by affine maps and maps of the form (x,y,z)→(x+P(y,z),y,z) for P a polynomial in k[y,z]. We construct a simply-connected non-positively curved space with an action of Tame(k^3). This leads to the classification of finite subgroups of Tame(k^3). This is joint work with Stéphane Lamy.

7 VI

Herve Oyono-Oyono (Metz)

Title: K-theory, propagation and Novikov conjecture. (Joint work with G. Yu)

Abstract: We define for families of finite metric spaces quantitative assembly map estimates that take into account propagation phenomena for pseudo-differential calculus. We relate these estimates to the Novikov conjecture. As an application, we provide an elementary proof of the Novikov conjecture for groups with finite decomposition complexity.

19 IV 

5 IV

Michele Triestino (Université de Bourgogne) 

Ignacio Vergara (ENS Lyon)

Radial Schur multipliers

A Schur multiplier on a set X is a two-variable function on X, which acts on the algebra of bounded operators on l^2(X) by multiplication of the matrix coefficients. If X is (the set of vertices of) a connected graph, we can look at those multipliers which depend only on the distance between each pair of vertices. We say that such a function is radial. For homogeneous trees, Haagerup, Steenstrup, and Szwarc provided a characterization of these multipliers in terms of certain Hankel matrices. I will discuss some extensions of this result to products of trees, products of hyperbolic graphs, and CAT(0) cube complexes.

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 generalization of the criterion of Żuk and Ballmann-Świątkowski for property (T), and relies on eigenvalues of nonlinear p-Laplacians. Central to the proof is the authors' development of new bounds on spectral gaps of p-Laplacians 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 generalizations due to T. de Laat and M. de la Salle.

22 II

Eric Reckwerdt (IMPAN)

Title: Relative hyperbolicity and weak amenability

Abstract: This will be a slide-based talk discussing my (relatively) recent result with Guentner and Tessera that relatively hyperbolic groups (with polynomially growing peripherals) are weakly amenable. It's a run-through of a talk aimed at theoretical computer scientists, so it will be light on technical details and only run 45-50 minutes. Comments and critiques are requested.

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)

Title: Aut(F_5) has property (T)

Abstract: 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 that encode large-scale 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 epsilon-propagation, 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)

Title: Classifying space for proper actions for groups admitting a strict fundamental domain.

Abstract: For an infinite discrete group G, the classifying space for proper actions EG is a proper G-CW-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 well-suited 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 discuss 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 (McGill University)

Title: Quadric Complexes

Abstract: Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize 2-dimensional 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)

Title: Continuum of new counterexamples to the coarse Baum–Connes conjecture

Abstract: The coarse Baum–Connes conjecture predicts that the K-theory of the Roe algebra of a metric space can be computed as the image of a certain assembly map, and we will pinpoint a K-theory 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)

Title: 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 a finite index subgroup of G and verifying that 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. Using the work of T. de Laat and M. de la Salle, we also obtain fixed point properties for affine isometric actions on some non-Hilbert Banach spaces, including L^p spaces. Joint work with T. Odrzygóźdź and P. Przytycki.

12 X

Yeong Chyuan Chung (IMPAN)

Title: Controlled K-theory for C*-algebras and beyond

Abstract: I will introduce a controlled version of K-theory for C*-algebras due to Oyono-Oyono and Yu, and explain how it can be extended to a larger class of Banach algebras that includes operator algebras on L^p spaces. Then I will formulate an L^p version of the Baum-Connes assembly map (with coefficients in C(X)), and outline how controlled K-theory can be used to show that the map is an isomorphism when we have a group action with finite dynamic asymptotic dimension.

28 IX

David Kerr (Texas A&M University)

Title: Dimension, comparison, and almost finiteness

21 IX

Kate Juschenko (Northwestern)

Title: Cycling amenable groups and soficity

Abstract: I will give an introduction to sofic groups and discuss a possible strategy towards finding a non-sofic 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 Stone-Cech compactification). I’ll try to focus mainly on examples and open questions.

6 VI

Erik Guentner (University of Hawaii)

Title: 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 quasi-isometric to an expander graph. This includes the strongest known expanders, that is, expanders with respect to Banach spaces of non-trivial type. More formally, we will prove that levels of the warped cone over a group action are quasi-isometric 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 non-trivially on the circle by homeomorphisms? If the homeomorphisms are assumed to be orientation-preserving, then it is equivalently the question about the existence of cyclically left-ordered (and non-trivial) quotients of random groups. We propose a combinatorial method for ruling out cyclic left-orderability and use it to show that random triangular groups don't admit cyclic left-orders at densities in the range (1/3, 1/2). We then discuss how the proof generalizes to the case of quotients, other density models with fixed-length relators (e.g., square model), and, finally, the Gromov density model.

11 V

Alice Libardi (Sao Paulo State University)

Title: The Borsuk-Ulam Property up to Cobordism

6 IV

Światosław Gal (Uniwersytet Wrocławski)

Title: Groups with proximal action are uniformly simple

Abstract: A group is called N-uniformly simple if for every nontrivial conjugacy class C, (C^±)^{≤N} covers the whole group. Every uniformly simple group is simple. It is known that many groups with geometric or dynamical origins are simple. In the talk, we prove that, in fact, many of them are uniformly simple. The results are due to the speaker, Kuba Gismatullin, and Nir Lazarovich.

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: In this lecture, I will introduce relatively hyperbolic groups and their geometric helper spaces. In particular, I'll cover the coned-off 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

Abstract: On group actions (by isometries) on metric spaces, the following question is fundamental: "knowing that the fixed point set for a small group is non-empty, under which conditions can we say more?" One extreme example is the Howe--Moore property of simple Lie groups G, which implies the following: "for a continuous unitary representation of G and the origin-excluded Hilbert space X, X admits a non-empty G-fixed point set if it has a non-empty fixed point set for a non-compact closed subgroup."

In this talk, we exhibit a new resolution to this question. It enables us to overcome "Bounded Generation" assumptions in Shalom's criteria (1999 Publ., IHES and 2006, ICM) to show Kazhdan's property (T). Furthermore, it bypasses "extrinsic" criteria by Dymara--Januszkiewicz and Ershov--Jaikin-Zapirain, which require tight estimates of certain spectral quantities. Thanks to it, we show fixed point properties on a certain class of Banach spaces "with unbounded wildness."

9 II

Piotr Nowak (IMPAN & UW)

Title: Geometric induction in equivariant KK-theory

Abstract: Given a quasi-isometry between groups G and H, I will construct a map from the G-equivariant KK-theory KK^G(A,B) to a certain H-equivariant KK-theory group. I will also discuss some motivations.

12 I

Piotr Nowak (IMPAN & UW)

Title: Fixed point properties and measure equivalence

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

22 XII

Damian Orlef (IMPAN)

Title: Non-orderability of random groups — work in progress

Abstract: A countable group admits a left-invariant linear order (resp. cyclic order) if and only if it acts faithfully by orientation-preserving homeomorphisms on the real line (resp. the circle). This provides a way to investigate questions connected to Gromov's conjecture that random groups should not have any non-trivial smooth actions on compact manifolds.

I will show that random groups in the Gromov density model do not have any left-orderable quotients at any density and how to use this fact, together with property (T), to show that there are no interesting smooth, orientation-preserving 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 left-invariant cyclic orders for some densities d>1/3, which is a joint result with Piotr Przytycki. An application of left-orders to the problem of finding the sharp critical threshold of freeness in the triangular binomial model will be discussed.

15 XII

Eric Reckwerdt (IMPAN)

Title: An Introduction to Graph Products of Groups

Abstract: 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 correspond to their inter-group 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)

Title: Coarse embeddability into a Hilbert space and containing expanders are two mutually exclusive properties; however, there exist metric spaces with neither.

Abstract: Looking at box spaces, we know quite a lot. Box spaces of amenable groups embed, while box spaces of groups with property (T) are expanders. For the free group, the answer is not that straightforward. 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)

Title: Towards a dynamical version of the Cuntz semigroup (joint with Bosa, Perera, Tikuisis, and Wu)

Abstract: The Cuntz semigroup is an invariant for C*-algebras combining refined K-theoretical 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 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)

Title: The amenability dimension for topological and C*-dynamics

Abstract: 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 Baum-Connes conjecture and the Farrell-Jones conjecture, as well as in 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 close relations with the Rokhlin dimension defined for C*-dynamics and often take finite values under reasonable assumptions. This talk includes work in collaboration with Finn-Sell, Hirshberg, Szabo, Winter, and Zacharias, as well as further recent developments.

10 XI

Mariusz Tobolski (UW)

Title: Borsuk-Ulam type theorems

Abstract: Assuming that both temperature and pressure are continuous functions, by the two-dimensional version of the celebrated Borsuk-Ulam theorem, we can conclude that there are always two antipodal points on Earth with exactly the same pressure and temperature. Recently, Baum, Dąbrowski, and Hajac formulated the following Borsuk-Ulam type conjecture: there is no equivariant map from X*G to X, where * is the join of topological spaces and both the free G-space X and the group G are compact Hausdorff. This conjecture was proved true by Chirvasitu and Passer. I will present the proof of the conjecture, and if time permits, I will also speak about generalizations to noncommutative geometry.

3 XI

Marek Kaluba (IMPAN)

Title: Computational aspects of property (T)

Abstract: Kazhdan's Property (T) is a well-known concept in the theory of group actions. Its numerous applications include finite generation of lattices, fixed-point properties of isometric actions, constructions of expanding graphs, and the product replacement algorithm. However, a complicated notion requires serious firepower to be established. Indeed, to prove that a group has property (T) requires a non-trivial effort even in the case of the most classical examples, such as SL(3,Z).

We hope to ease the effort by drawing from the field of semi-definite programming and cone optimization. Using the Positivstellensatz and following the work of Ozawa and Netzer & Thom, we will show how to cast property (T) into a semi-definite optimization problem. Given an explicit generating set $S$ of a finitely presented group $G$, this will (possibly) allow us to produce a "witness" for the property (T) and simultaneously estimate Kazhdan's constant for $(G,S)$.

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 that 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)

Title: 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 Bass-Serre 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 — right-angled hyperbolic buildings

Abstract: To a measure-class 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 negatively-curved 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.

We partially extend this result to groups acting on hyperbolic buildings and their boundaries. We do so by associating to the corresponding representation a representation of a certain Hecke algebra, which is a deformation of the classical representation of a hyperbolic reflection group. We show that the associated Hecke algebra representation is irreducible. This is joint work with Uri Bader.

4 X

15:00

room 403

Martin Finn-Sell (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 Benjamini-Schramm 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, that 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 the academic year 2015/2016. The former website is no longer maintained.