# Seminars (NPC)

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Event When Speaker Title Presentation Material
NPCW01 9th January 2017
10:00 to 11:00
Pierre Py Cubulable Kähler groups
A Kähler group is the fundamental group of a compact Kähler manifold. We prove that if such a group is cubulable, it must have a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free Abelian factor. Similarly we prove that if an aspherical smooth projective manifold has a cubulable fundamental group, it must have a finite cover which is biholomorphic to a product of Riemann surfaces and complex tori. This is joint work with Thomas Delzant.
NPCW01 9th January 2017
11:30 to 12:30
Francois Dahmani The normal closure of a big Dehn twist in a mapping class group
NPCW01 9th January 2017
13:30 to 14:30
Richard Webb Polynomial-time Nielsen--Thurston type recognition
A cornerstone of the study of mapping class groups is the Nielsen--Thurston classification theorem. I will outline a polynomial-time algorithm that determines the Nielsen--Thurston type and the canonical curve system of a mapping class. Our approach uses the action on the curve complex. Time permitting, I shall discuss the conjugacy problem for the mapping class group. This is joint work with Mark Bell.
NPCW01 9th January 2017
14:30 to 15:30
Samuel Taylor Counting loxodromics for hyperbolic actions
Consider a nonelementary action by isometries of a hyperbolic group G on a hyperbolic metric space X. Besides the action of G on its Cayley graph, some examples to bear in mind are actions of G on trees and quasi-trees, actions on nonelementary hyperbolic quotients of G, or examples arising from naturally associated spaces, like subgroups of the mapping class group acting on the curve graph.
We show that the set of elements of G which act as loxodromic isometries of X (i.e those with sink-source dynamics) is generic. That is, for any finite generating set of G, the proportion of X-loxodromics in the ball of radius n about the identity in G approaches 1 as n goes to infinity. We also establish several results about the behavior in X of the images of typical geodesic rays in G. For example, we prove that they make linear progress in X and converge to the boundary of X. This is joint work with I. Gekhtman and G. Tiozzo.
NPCW01 9th January 2017
16:00 to 17:00
Camille Horbez Growth under automorphisms of hyperbolic groups
Co-authors: Rémi Coulon (Université de Rennes 1), Arnaud Hilion (Aix-Marseille Université), Gilbert Levitt (Université de Caen)

Given a torsion-free hyperbolic group G and an outer automorphism \Phi of G, we investigate the possible growth types of conjugacy classes of G under iteration of \Phi.
NPCW01 10th January 2017
09:00 to 10:00
Benjamin Beeker Cubical Accessibility and bounds on curves on surfaces.
NPCW01 10th January 2017
10:00 to 11:00
Michah Sageev Uniform exponential growth for groups acting on CAT(0) square complexes
We show that groups acting freely on CAT(0) square complexes satisfy uniform exponential growth. This is joint work with Aditi Kar.
NPCW01 10th January 2017
11:30 to 12:30
Daniel Woodhouse Understanding Tubular Groups
A group is tubular if it splits as a graph of groups with rank 2 free abelian vertex groups and infinite cyclic edge groups. Tubular groups have been an interesting source of counterexamples in geometric group theory. I will introduce the examples of Gersten and Wise before moving onto recent results relating to cubulating tubular groups. This will include a classification of virtually special tubular groups.
NPCW01 10th January 2017
13:30 to 14:30
Nir Lazarovich Stallings folds for CAT(0) cube complexes and quasiconvex subgroups
We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers. Joint work with Benjamin Beeker.
NPCW01 10th January 2017
14:30 to 15:30
Anthony Genevois Cubical-like geometry of graph products
In 1994, Bandelt, Mulder and Wilkeit introduced a class of graphs generalizing the so-called median graphs: the class of quasi-median graphs. Since the works of Roller and Chepoï, we know that median graphs and CAT(0) cube complexes essentially define the same objets, and because CAT(0) cube complexes play an important role in recent reseach in geometric group theory, a natural question is whether quasi-median graphs can be used to study some classes of groups. In our talk, we will show that quasi-median graphs and CAT(0) cube complexes share essentially the same geometry. Moreover, extending the observation that right-angled Artin and Coxeter groups have a Cayley graph which is median, arbitrary graph products turn out to have a Cayley graph (with respect to a natural, but possibly infinite, generating set) which is quasi-median. The main goal of this talk is to show how to use the quasi-median geometry of this Cayley graph to study graph products.
NPCW01 10th January 2017
16:00 to 17:00
Robert Kropholler Hyperbolic groups and their subgroups
NPCW01 11th January 2017
09:00 to 10:00
Jingyin Huang Commensurability of groups quasi-isometric to RAAG's
It is well-known that a finitely generated group quasi-isometric to a free group is commensurable to a free group. We seek higher-dimensional generalization of this fact in the class of right-angled Artin groups (RAAG). Let G be a RAAG with finite outer automorphism group. Suppose in addition that the defining graph of G is star-rigid and has no induced 4-cycle. Then we show every finitely generated group quasi-isometric to G is commensurable to G. However, if the defining graph of G contains an induced 4-cycle, then there always exists a group quasi-isometric to G, but not commensurable to G.
NPCW01 11th January 2017
10:00 to 11:00
Alessandro Sisto Bounded cohomology of acylindrically hyperbolic groups via hyperbolically embedded subgroups
I will discuss a few results about the bounded cohomology of an acylindrically hyperbolic group, focusing on quasimorphisms for concreteness. More specifically, the results relate the bounded cohomology of an acylindrically hyperbolic group to that of its hyperbolically embedded subgroups, which are special subgroups that generalise peripheral subgroups of relatively hyperbolic groups.
NPCW01 11th January 2017
11:30 to 12:30
Jason Behrstock Random graphs and applications to Coxeter groups
Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected.  We will explain some new threshold theorems for random graphs and focus in particular on applications to geometric group theory: these concern divergence functions, which provide quantifications of non-positive curvature. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.
NPCW01 11th January 2017
13:30 to 14:30
Urs Lang Group actions on spaces with a distinguished geodesic structure
Co-authors: Giuliano Basso (ETH Zurich), Dominic Descombes (ETH Zurich), Benjamin Miesch (ETH Zurich)

A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. The existence of a geodesic bicombing satisfying a suitable convexity condition may be viewed as a weak (but non-coarse) global notion of non-positive curvature that allows for non-unique geodesics. The talk will give a survey of recent results on group actions on spaces with such a structure.
NPCW01 11th January 2017
14:30 to 15:30
Bruno Duchesne Groups actions on dendrites
Co-author: Nicolas Monod (EPFL)

A dendrite is a compact metrizable space such that any two points are connected by a unique arc. Dendrites may appear as Julia sets, Berkovich projective lines and played in important role in the proof of the cut point conjecture for boundaries of hyperbolic groups by Bowditch.

In a common work with Nicolas Monod, we study groups acting on dendrites by homeomorphisms. In this purely topological context, we obtain rigidity results for lattices of algebraic groups, an analog of Tits alternative, simplicity and other topological results.

NPCW01 11th January 2017
16:00 to 17:00
Viktor Schroeder Boundaries and Moebius Geometry
We give a fresh view on Moebius geometry and show that the ideal boundary of a negatively curved space has a natural Moebius structure. We discuss various cases of the interaction between the geometry of the space and the Moebius geometry of its boundary. We discuss an approach how the concept of Moebius geometry can be generalized in order that it is usefull for the boundaries of nonpositively curved spaces like higher rank symmetric spaces, products of rank one spaces or cube complexes. In particular we describe a Moebius geometry on the Furstenberg boundary of a symmetric space.
NPCW01 12th January 2017
09:00 to 10:00
Petra Schwer Combinatorics of Coxeter groups and Affine Deligne Lusztig varieties
Co-authors: Liz Milicevic (Haverford College), Anne Thomas (University of Sydney)

We present combinatorial properties of Coxeter groups and buildings and explain how they can be used to study nonemptiness and dimensions of affine Deligne Lusztig varieties (ADLVs). These varieties are sub-varieties of the affine flag variety of an algebraic group. And their nonemptinedd can be stated in terms of galleries and their retracted images in the associated Bruhat-Tits building. In addition we will talk about the problem of exact computation of reflection length in affine Coxeter groups. Here reflection length means the minimal number of elements needed to write a given element as a product of reflections. For a particular class of elements the reflection length can be determined from the dimension of an ADLV.
NPCW01 12th January 2017
10:00 to 11:00
David Hume Coarse negative curvature in action
Admitting a "nice" action on a hyperbolic space has strong algebraic and geometric consequences for a group. For this talk we will focus on replacing "nice" by "non-trivial acylindrical". The goal of the talk is to open and discuss a number of interesting and important recognition problems for non-trivial acylindrical actions of groups on hyperbolic spaces and to discuss the relation between these problems and the existence of coarse negative curvature in the group. This is a summary of current progress on many projects with a variety of co-authors.
NPCW01 12th January 2017
11:30 to 12:30
Pierre Pansu Large scale conformal maps
Benjamini and Schramm's work on incidence graphs of sphere packings suggests a notion of conformal map between metric spaces which is natural under coarse embeddings. We show that such maps cannot exist between nilpotent or hyperbolic groups unless certain numerical inequalities hold.
NPCW01 12th January 2017
13:30 to 14:30
Emily Stark Commensurability classification of certain right-angled Coxeter groups and related surface amalgams
Co-authors: Pallavi Dani (Louisiana State University), Anne Thomas (University of Sydney)

I will describe the abstract commensurability classification within a class of hyperbolic right-angled Coxeter groups. I will explain the relationship between these groups and a related class of geometric amalgams of surface groups, and I will highlight the differences between the quasi-isometry classification and abstract commensurability classification in this setting. This is joint work with Pallavi Dani and Anne Thomas.
NPCW01 12th January 2017
14:30 to 15:30
Damian Osajda Group cubization
NPCW01 12th January 2017
16:00 to 17:00
Andrew Sale When the outer automorphism groups of RAAGs are vast
The outer automorphism groups of right-angled Artin groups (RAAGs) give a way to build a bridge between GL(n,Z) and Out(Fn). We will investigate certain properties of these groups which could be described as "vastness" properties, and ask if it possible to build a boundary between those which are "vast" and those which are not. One such property is as follows: given a group G, we say G has all finite groups involved if for each finite group H there is a finite index subgroup of G which admits a map onto H. From the subgroup congruence property, it is known that the groups GL(n,Z) do not have every finite group involved for n>2. Meanwhile, the representations of Out(Fn) given by Grunewald and Lubotzky imply that these groups do have all finite groups involved. We will describe conditions on the defining graph of a RAAG that are necessary and sufficient to determine when it's outer automorphism group has this property. The same criterion also holds for other properties, such as SQ-universality, or having a finite index subgroup with infinite dimensional second bounded cohomology. This is joint work with V. Guirardel.
NPCW01 13th January 2017
09:00 to 10:00
Yago Antolin Convex subgroups of orderable groups.
Let G be a group and H a subgroup of G. We will present a criterion for H to be convex in G for some left-order.  We will give several examples of groups on which maximal cyclic subgroups are always convex (for some order). We will also discuss the orderability of some groups acting on trees. This is a joint work with W. Dicks and Z. Sunic.
NPCW01 13th January 2017
10:00 to 11:00
Panos Papasoglu Cutpoints of CAT(0) groups
(Joint with Eric Swenson)
It is known that if the boundary of a 1-ended hyperbolic group G has a local cut point then G splits over a 2-ended group. We prove a similar theorem for CAT(0) groups, namely that if a finite set of points separates the boundary of a 1-ended CAT(0) group G
then G splits over a 2-ended group. Along the way we prove two results of independent interest: we show that continua separated
by finite sets of points admit a tree-like decomposition and we show a splitting theorem for nesting actions on R-trees.

NPCW01 13th January 2017
11:30 to 12:30
Dominik Gruber Small cancellation theory over Burnside groups
Co-author: Rémi Coulon (Université de Rennes 1)

I will discuss how combinatorics and geometry work together to provide a new and easy-to-apply tool for constructing infinite bounded torsion groups with prescribed properties. The main tools are acylindrical actions of (classical or graphical) small cancellation groups on hyperbolic spaces and the theory of periodic quotients of groups admitting such actions. As applications, we obtain Gromov monsters with bounded torsion, we show the unsolvability of numerous decision problems in categories of bounded torsion groups, and we obtain a Rips construction for bounded torsion groups. This is joint work with Rémi Coulon.

NPCW01 13th January 2017
13:30 to 14:30
Jack Button On the linearity of finitely presented groups and its connection to the existence of good geometric actions
The question of whether a finitely generated or presented group is linear has been looked at in various contexts, as has the existence of well behaved geometric actions of such groups by isometries on metric spaces although there is no single universally accepted definition of what such a well behaved action should be.

However at first sight there appears little connection between these two concepts, for instance one might consider some infinite simple groups or Baumslag-Solitar groups to make this point. In this talk we will first look at such groups to illustrate known results, before giving examples and evidence for the implication "linear implies good behaviour geometrically" if we use the appropriate notion of linearity.
NPCW01 13th January 2017
14:30 to 15:30
Alexandre Martin On the small cancellation geometry of certain graph products of groups
Graph products of groups generalise both right-angled Coxeter groups and right-angled Artin groups. While such groups are already known to act on right-angled buildings, I will explain how it is possible, when the underlying graph is a cycle, to construct a more "robust" action on a small cancellation polygonal complex. Such an action can be used to compute the automorphism group of such groups and understand their geometry. (joint work with A. Genevois)
NPCW01 13th January 2017
16:00 to 17:00
Christopher Cashen The topology of the contracting boundary of a group
Co-author: John Mackay (Bristol)

I will talk about a new way to topologize the space of hyperbolic directions in a finitely generated group and progress in generalizing boundary theory for hyperbolic groups to this space.
NPC 18th January 2017
14:00 to 16:00
Pierre Pansu 1. Lp-cohomology
Lp-cohomology is a quantitative version of cohomology of topological spaces, well suited for a geometric study of noncompact spaces and groups. It has been used with partial success in connection with problems in high-dimensional topology (Singer's conjecture on the Euler characteristic of aspherical manifolds), low-dimensional topology (Cannon's conjecture on groups whose boundary is a 2-sphere), Riemannian geometry (optimal curvature pinching) and coarse geometry (hyperbolic locally compact groups, coarse embeddings between nilpotent or hyperbolic groups). The course will provide background on these problems and be as self-contained as possible.

Duration: 9 lectures.

Contents:
1. What Lp-cohomology has been good for
2. L\infty cohomology and (Kähler) hyperbolic groups
3. Lp dimension and the ideal boundary of hyperbolic groups
4. Quasi-isometry invariance
5. Large scale conformal invariance
6. Classification of hyperbolic locally compact groups
7. Curvature pinching

NPC 19th January 2017
10:00 to 12:00
Andrzej Zuk Random walks on random symmetric groups.
NPC 25th January 2017
14:00 to 16:00
Pierre Pansu 2. Lp-cohomology
Lp-cohomology is a quantitative version of cohomology of topological spaces, well suited for a geometric study of noncompact spaces and groups. It has been used with partial success in connection with problems in high-dimensional topology (Singer's conjecture on the Euler characteristic of aspherical manifolds), low-dimensional topology (Cannon's conjecture on groups whose boundary is a 2-sphere), Riemannian geometry (optimal curvature pinching) and coarse geometry (hyperbolic locally compact groups, coarse embeddings between nilpotent or hyperbolic groups). The course will provide background on these problems and be as self-contained as possible.

Duration: 9 lectures.

Contents:
1. What Lp-cohomology has been good for
2. L\infty cohomology and (Kähler) hyperbolic groups
3. Lp dimension and the ideal boundary of hyperbolic groups
4. Quasi-isometry invariance
5. Large scale conformal invariance
6. Classification of hyperbolic locally compact groups
7. Curvature pinching

NPC 26th January 2017
10:00 to 12:00
Ian Leary Generalizing Bestvina-Brady groups using branched covers
In the 1990's Bestvina and Brady constructed groups that are FP but not finitely presented as the kernels of maps from right-angled Artin groups to Z.  I generalize this construction using branched coverings.  The main application is an uncountable family of groups of type FP.   A corollary is that every countable group embeds in a group of type FP_2.  I will explain the construction, and if time permits I will discuss the corollary and work with Ignat Soroko and Robert Kropholler on the quasi-isometry classification of the new groups.

NPC 1st February 2017
14:00 to 17:30
Pierre Pansu 3. Lp-cohomology
Lp-cohomology is a quantitative version of cohomology of topological spaces, well suited for a geometric study of noncompact spaces and groups. It has been used with partial success in connection with problems in high-dimensional topology (Singer's conjecture on the Euler characteristic of aspherical manifolds), low-dimensional topology (Cannon's conjecture on groups whose boundary is a 2-sphere), Riemannian geometry (optimal curvature pinching) and coarse geometry (hyperbolic locally compact groups, coarse embeddings between nilpotent or hyperbolic groups). The course will provide background on these problems and be as self-contained as possible.

Duration: 9 lectures.

Contents:
1. What Lp-cohomology has been good for
2. L\infty cohomology and (Kähler) hyperbolic groups
3. Lp dimension and the ideal boundary of hyperbolic groups
4. Quasi-isometry invariance
5. Large scale conformal invariance
6. Classification of hyperbolic locally compact groups
7. Curvature pinching

NPC 2nd February 2017
10:00 to 12:00
Eric Swenson On infinite torsion subgroups of CAT(0) groups
It is know that every CAT(0) group contains a hyperbolic element, but it is still an open question if a CAT(0) group can contain a (needs be) finitely generated infinite torsion subgroup. (For cube complexes the answer is no by the strong Tits alternative of Caprace and Sageev.) I will demonstrate some of the restrictions on such a subgroup.

NPC 9th February 2017
10:00 to 12:00
Alessandro Sisto Quasi-flats in hierarchically hyperbolic spaces
The notion of hierarchically hyperbolic space provides a common framework to study mapping class groups, Teichmueller spaces with either the Teichmueller or the Weil-Petersson metric, CAT(0) cube complexes admitting a proper cocompact action, fundamental groups of non-geometric 3-manifolds, and other examples.   I will discuss the result that any top-dimensional quasi-flat in a hierarchically hyperbolic space lies within finite Hausdorff distance from a finite union of "standard orthants", a result new for both mapping class groups and cube complexes. Also, I will discuss how this can be used to reduce proving quasi-isometric rigidity results to much more manageable, (mostly) combinatorial problems that require no knowledge about the geometry of HHSs.   Joint work with Jason Behrstock and Mark Hagen.

NPC 16th February 2017
10:00 to 12:00
Rémi Coulon Monster groups acting on CAT(0) spaces
Since the beginning of the 20th century, infinite torsion groups have been the source of numerous developments in group theory: Burnside groups Tarski monsters, Grigorchuck groups, etc. From a geometric point of view, one would like to understand on which metric spaces such groups may act in a non degenerated way (e.g. without a global fixed point).   In this talk we will focus on CAT(0) spaces and present two examples with rather curious properties. The first one is a non-amenable finitely generated torsion group acting properly on a CAT(0) cube complex. The second one is a non-abelian finitely generated Tarski-like monster : every finitely generated subgroup is either finite or has finite index. In addition this group is residually finite and does not have Kazdhan property (T).   (Joint work with Vincent Guirardel)

NPC 22nd February 2017
14:00 to 16:00
Pierre Pansu 4. Lp-cohomology
Lp-cohomology is a quantitative version of cohomology of topological spaces, well suited for a geometric study of noncompact spaces and groups. It has been used with partial success in connection with problems in high-dimensional topology (Singer's conjecture on the Euler characteristic of aspherical manifolds), low-dimensional topology (Cannon's conjecture on groups whose boundary is a 2-sphere), Riemannian geometry (optimal curvature pinching) and coarse geometry (hyperbolic locally compact groups, coarse embeddings between nilpotent or hyperbolic groups). The course will provide background on these problems and be as self-contained as possible.

Duration: 9 lectures.

Contents:
1. What Lp-cohomology has been good for
2. L\infty cohomology and (Kähler) hyperbolic groups
3. Lp dimension and the ideal boundary of hyperbolic groups
4. Quasi-isometry invariance
5. Large scale conformal invariance
6. Classification of hyperbolic locally compact groups
7. Curvature pinching

NPC 23rd February 2017
10:00 to 12:00
Anastasia Khukhro Geometry of finite quotients of groups.
The study of graphs associated to groups has revolutionised group theory, allowing us to use geometric intuition to study algebraic objects. We will focus here on the case of groups admitting many finite quotients. Geometric properties of a collection of finite quotients of a group can provide information about the group if the set of finite quotients is sufficiently rich, and one can exploit the connections between the world of group theory and graph theory to give examples of metric spaces with interesting and often surprising properties. In this talk, we will describe some results in this direction, and then give recent results concerning the geometric rigidity of finite quotients of a group.   (Joint work with Thiebout Delabie). ​

NPC 1st March 2017
14:00 to 16:00
Emmanuel Breuillard Approximate groups: an introduction
NPC 2nd March 2017
10:00 to 12:00
Romain Tessera A Banachic generalization of Shalom's property H_FD.
A group has property H_FD if the first reduced cohomology of unitary representations is supported on finite sub-representations. Shalom has proved that this property is stable under quasi-isometry among amenable groups. We generalize this notion to the class of WAP representations, and we prove that this stronger property holds for a class of locally compact solvable groups including algebraic groups over local fields and their lattices. As a by-product we prove a conjecture of Shalom, namely that solvable finitely generated subgroups of GL(Q) have H_FD.   (Joint work with Yves Cornulier)

NPC 7th March 2017
10:00 to 12:00
Emmanuel Breuillard Approximate groups: basic definitions, structure theorem and geometric consequences
I will define approximate groups and discuss the definition and its variants. I will them state the general structure theorem for approximate groups and derive consequences such as improvements on Gromov's polynomial growth theorem and almost flat manifolds theorem and a generalized Margulis lemma.

NPC 8th March 2017
14:00 to 16:00
Pierre Pansu 5. Lp-cohomology
Lp-cohomology is a quantitative version of cohomology of topological spaces, well suited for a geometric study of noncompact spaces and groups. It has been used with partial success in connection with problems in high-dimensional topology (Singer's conjecture on the Euler characteristic of aspherical manifolds), low-dimensional topology (Cannon's conjecture on groups whose boundary is a 2-sphere), Riemannian geometry (optimal curvature pinching) and coarse geometry (hyperbolic locally compact groups, coarse embeddings between nilpotent or hyperbolic groups). The course will provide background on these problems and be as self-contained as possible.

Duration: 9 lectures.

Contents:
1. What Lp-cohomology has been good for
2. L\infty cohomology and (Kähler) hyperbolic groups
3. Lp dimension and the ideal boundary of hyperbolic groups
4. Quasi-isometry invariance
5. Large scale conformal invariance
6. Classification of hyperbolic locally compact groups
7. Curvature pinching

NPC 9th March 2017
10:00 to 11:00
Ursula Hamenstaedt Incompressible surfaces in closed locally symmetric manifolds
We construct Anosov representations of surface groups into cocompact lattices of simple Lie groups (with a few exceptions). The talk will begin with an account of closed maximal flats in compact higher rank locally symmetric spaces, recovering a result of Deitmar, and explain how to take advantage of such flats for the construction of incompressible surfaces.

NPC 9th March 2017
11:00 to 12:00
Koji Fujiwara Computing Kazhdan constants by computer.
Ozawa found a new characterization of property (T) of finitely generated groups. His theorem gives an algorithm to compute a lower bound of the Kazhdan constant. Netzer-Thom actually found a good positive lower bound for SL(3,Z). I will report on computer experiments on other examples. We found good lower bounds for lattice groups on A2-tilde buildings, which have property (T). This is an experimental math talk.

NPC 14th March 2017
10:00 to 12:00
Emmanuel Breuillard Approximate groups: structure theorem for special classes of groups
In the first part of the talk, I will recall the general structure theorem for approximate groups and derive a version of Gromov's almost flat manifolds theorem from it. In the second part I will explain how the structure theorem can be improved in two cases: linear groups and fully residually free groups.

NPC 15th March 2017
14:00 to 16:00
Pierre Pansu 6. Lp-cohomology
Lp-cohomology is a quantitative version of cohomology of topological spaces, well suited for a geometric study of noncompact spaces and groups. It has been used with partial success in connection with problems in high-dimensional topology (Singer's conjecture on the Euler characteristic of aspherical manifolds), low-dimensional topology (Cannon's conjecture on groups whose boundary is a 2-sphere), Riemannian geometry (optimal curvature pinching) and coarse geometry (hyperbolic locally compact groups, coarse embeddings between nilpotent or hyperbolic groups). The course will provide background on these problems and be as self-contained as possible.
Duration: 9 lectures.

Contents:
1. What Lp-cohomology has been good for
2. L\infty cohomology and (Kähler) hyperbolic groups
3. Lp dimension and the ideal boundary of hyperbolic groups
4. Quasi-isometry invariance
5. Large scale conformal invariance
6. Classification of hyperbolic locally compact groups
7. Curvature pinching

NPC 16th March 2017
10:00 to 11:00
Vladimir Markovic Caratheodory's metrics on Teichmuller spaces
Caratheodory's metrics on Teichmuller spaces

NPC 16th March 2017
11:00 to 12:00
Narutaka Ozawa Finite-dimensional representations constructed from random walks
Let an amenable group G and a probability measure \mu on it (that is finitely-supported, symmetric, and non-degenerate) be given. I will present a construction, via the \mu-random walk on G, of a harmonic cocycle and the associated orthogonal representation of G. Then I describe when the constructed orthogonal representation contains a non-trivial finite-dimensional subrepresentation (and hence an infinite virtually abelian quotient), and some sufficient  conditions for G to satisfy Shalom's property HFD. (joint work with A. Erschler, arXiv:1609.08585)

NPC 5th April 2017
14:00 to 15:30
Pierre-Emmanuel Caprace Exotic lattices and simple locally compact groups
The goal of this course is to give an overview of the construction, due to Marc Burger and Shahar Mozes, of a family of virtually simple groups acing properly and cocompactly on the product of two regular locally finite trees.
NPC 6th April 2017
10:00 to 11:00
John Mackay The Poincaré profile of a graph or group
Benjamini, Schramm and Timar introduced the notion of "separation profile" for a graph or finitely generated group.  I will discuss a family of invariants that generalise this idea, examples of the values obtained and applications to non-embedding results.   Joint work with David Hume and Romain Tessera.

NPC 6th April 2017
11:00 to 12:00
Shahar Mozes Topological finite generation of certain compact open subgroups of tree automorphisms.
Given a finite permutation group F on d letters, one can define a group of automorphism of a d-regular tree whose local action on the tree is given by the permutation group F. In a joint work with Marc Burger we determine when the maximal compact subgroup of this group is topologically finitely generated. This is motivated by studying uniform lattices in the group of automorphisms of a product of trees.

NPC 7th April 2017
16:00 to 17:00
Richard Schwartz Rothschild Lecture: Thomson's 5 point problem
Thomson's problem, which in a sense goes back to J.J. Thomson's 1904 paper, asks how N points will arrange themselves on the sphere (or the circle, or some other space) so as to minimize their total electrostatic potential.  Mathematicians and physicists have also considered this problem with respect to other potentials, such as power law potentials.  For special values of N, and the sphere of the appropriate dimension, there are spectacular answers which say that the potential minimizers are highly symmetric objects, such as the regular icosahedron or the E8 cell.  In spite of this work, very little has been proved about 5 points on the 2-sphere. In my talk I will explain my computer assisted but rigorous proof that there is a phase transition constant S=15.048... such that the triangular bi-pyramid is the minimizer with respect to a power-law potential if and only if the exponent is less or equal to S.  (This constant was conjectured to exist in 1977 by Melnyk-Knop-Smith.) The talk will have some colorful computer demos.
NPC 11th April 2017
10:00 to 12:00
Emmanuel Breuillard Approximate groups: Hrushovski's Lie model theorem
NPC 12th April 2017
14:00 to 15:30
Pierre-Emmanuel Caprace Exotic lattices and simple locally compact groups
NPC 13th April 2017
10:00 to 11:00
Indira Chatterji Median spaces and spaces with thin triangles.
A geodesic space has thin triangles if every triple of points belongs to the vertex set of a thin triangle. If such a space admits a structure of space with walls, then it is at finite Hausdorff distance of its associated median space. This shows for instance that the hyperbolic n-space is at finite Hausdorff distance from the associated median space.   Joint work with Cornelia Drutu and Frederic Haglund.

NPC 13th April 2017
11:00 to 12:00
Karen Vogtmann RAAG subgroups of RAAGs
Motivated by work of Grunewald and Lubotzky on virtual representations of Aut(F_n), we address the question of when a finite index normal subgroup of a RAAG is itself a RAAG.   Joint work with Beatrice Pozzetti.

NPC 18th April 2017
10:00 to 11:30
Pierre-Emmanuel Caprace Exotic lattices and simple locally compact groups
NPC 19th April 2017
10:00 to 11:00
Mikolaj Fraczyk Benjamini-Schramm convergence of arithmetic orbifolds.
Let X be the a symmetric space. We say that a sequence of locally symmetric spaces Benjamini-Schramm converges to X if for any real number R the fraction of the volume taken by the R-thin part tends to 0. In my thesis I showed that for a cocompact, congruence arithmetic hyperbolic 3-manifold the volume of the R-thin part is less than a power less than one of the total volume. As a consequence, any sequence of such manifolds Benjamini-Schramm converges to hyperbolic 3-space. I will give some topological applications of this result. Lastly, I will discuss Benjamini-Schramm convergence of congruence arithmetic orbifolds covered by the symmetric spaces of real rank 1.   (joint work with Jean Raimbault).

NPC 19th April 2017
14:00 to 16:00
Emmanuel Breuillard Hilbert's 5th problem and the Gleason-Yamabe theorem
NPC 20th April 2017
10:00 to 11:00
Andreas Aaserud Property (T) and approximate conjugacy of actions
I will define a notion of approximate conjugacy for probability measure preserving actions and compare it to the a priori stronger classical notion of conjugacy for such actions. In particular, I will spend most of the talk explaining the proof of a theorem stating that two ergodic actions of a fixed group with Kazhdan's property (T) are approximately conjugate if and only if they are actually conjugate. Towards this end, I will discuss some constructions from the theory of von Neumann algebras, including the basic construction of Vaughan Jones and a version of the Feldman-Moore construction. I will also provide some evidence that this theorem may yield a characterization of groups with Kazhdan's property (T).   (Joint work with Sorin Popa)

NPC 20th April 2017
11:00 to 12:00
Brian Bowditch Bounding genera of singular surfaces
If two simple closed curves on a closed surface lie in the same non-trivial homology class, then they bound a singular orientable surface. One can relate the minimal genus of such to distances in the curve graph. I will describe this, and some related results. The proofs are not hard, but call for some non-trivial input.  Maybe there is a simple combinatorial argument.

NPC 25th April 2017
10:00 to 11:00
Frédéric Paulin On diagonal group actions, trees and continued fractions in positive characteristic
If R, k and K are the polynomial ring, fraction field and Laurent series field in one variable over a finite field, we prove that the continued fraction expansions of Hecke sequences of quadratic irrationals in K over k behave in sharp contrast with the zero characteristic case. This uses the ergodic properties of the action of the diagonal subgroup of PGL(2,K) on the moduli space PGL(2,K)/PGL(2,R) and the action of the lattice PGL(2,R) on the Bruhat-Tits tree of PGL(2,K). (Joint work with Uri Shapira)
NPC 25th April 2017
11:00 to 12:00
John Parker Non-arithmetic lattices
In 1980 Mostow found the first examples of non-arithmetic lattices in PU(2,1). More examples were found and Deligne and Mostow gave a list of examples in 1986. Work of McMullen, based on work of Kappes and Moeller, showed there are 9 commensurability classes of non-arithmeticlattices in PU(2,1) on this list. No new examples were found until my recent work with Deraux and Paupert. We have constructed 13 new commensurability classes. I will give a history of this problem and then outline our recent results.

NPC 26th April 2017
14:00 to 16:00
Emmanuel Breuillard Approximate groups: nilprogressions and the structure theorem.
I will discuss nilprogressions and describe the strong form of the structure theorem for general approximate groups. I will also present an outline of the proof as well as some applications to the structure of vertex transitive graphs with large diameter.
NPC 27th April 2017
09:30 to 11:30
Pierre-Emmanuel Caprace Exotic lattices and simple locally compact groups
NPC 2nd May 2017
10:00 to 11:00
Brita Nucinkis Finiteness conditions for classifying spaces for the family of virtually cyclic subgroups.
A conjecture of Juan-Pineda and Leary states that any group admitting a cocompact model for the classifying space for the family of virtually cyclic subgroups has to be virtually cyclic already. This conjecture has been proved for large classes of groups. In this talk I will give an overview of some of these results and constructions,  will discuss a weakened condition for these spaces, and will give examples of groups satisfying this condition.

NPC 2nd May 2017
11:00 to 12:00
Adrien Le Boudec Uniformly recurrent subgroups and lattice embeddings
We study how certain countable groups containing amenable uniformly recurrent subgroups can be lattices in some locally compact groups.

NPC 3rd May 2017
14:00 to 16:00
Richard Schwartz Pappus's Theorem and the Modular Group
NPC 4th May 2017
09:30 to 11:30
Pierre-Emmanuel Caprace Exotic lattices and simple locally compact groups
NPC 5th May 2017
15:45 to 17:15
Pierre-Emmanuel Caprace Exotic lattices and simple locally compact groups
NPCW04 8th May 2017
10:00 to 11:00
Goulnara Arzhantseva Approximations of infinite groups
We discuss various (still open) questions on approximations of finitely generated groups, focusing on finite-dimensional approximations such as residual finiteness and soficity. We begin with a survey of our prior results and then introduce a new type of approximations: constraint metric approximations. We study their existence and stability. In particular, we investigate the constraint soficity. We characterize the stability of the commutator in permutations, with constraints. This answers a question of Gorenstein-Sandler-Mills (1962).

Based on joint works with Liviu Paunescu (Bucharest).
NPCW04 8th May 2017
11:30 to 12:30
Alessandra Iozzi Degenerations of maximal representations, non-Archimedean upper half space and laminations
We study degenerations of maximal representations into Sp(2n,R) and identify phenomena already present in the Thurston boundary of Teichmüller apace as well as new geometric features.  We give equivalent conditions for the existence of measured laminations in term of an appropriate notino of length.  This is joint work with Marc Burger, Anne Parreau and Beatrice Pozzetti.
NPCW04 8th May 2017
13:30 to 14:30
Wouter Van Limbeek Towers of regular self-covers and linear endomorphisms of tori
Let M be a closed manifold that admits a nontrivial cover diffeomorphic to itself. What can we then say about M? Examples are provided by tori, in which case the covering is homotopic to a linear endomorphism. Under the assumption that all iterates of the covering of M are regular, we show that any self-cover is is induced by a linear endomorphism of a torus on a quotient of the fundamental group. Under further hypotheses we show that a finite cover of M is a principal torus bundle. We use this to give an application to holomorphic self-covers of Kaehler manifolds.
NPCW04 8th May 2017
14:30 to 15:30
Andreas Thom On finitarily approximable groups
Starting with the work of Gromov on Gottschalk’s Surjunctivity Conjecture, the class of sofic groups has attracted much interest in various areas of mathematics. Major applications of this notion arose in the work Elek and Szabo on Kaplansky’s Direct Finiteness Conjecture, Lück’s Determinant Conjecture, and more recently in joint work with Klyachko on generalizations of the Kervaire-Laudenbach Conjecture and Howie’s Conjecture. Despite considerable effort, no non-sofic group has been found so far. In view of this situation, attempts have been made to provide variations of the problem that might be more approachable. Using the seminal work of Nikolov-Segal, we prove that the topological group SO(3) is not weakly sofic and describe the class of discrete groups that is approximable by finite solvable groups. (This is joint work with Jakob Schneider and Nikolay Nikolov.)
NPCW04 8th May 2017
16:00 to 17:00
Mikael de la Salle Characterizing a vertex-transitive graph by a large ball
The subject of the talk will be vertex-transitive infinite connected graphs with bounded degree and with a property of large scale simple connectedness. The most classical examples of such objects are Cayley graphs of finitely presented groups, but I will explain that there are (uncountably) many other, and I will study some topological questions on the space of all such graphs. In particular I will give some answers to a question of Benjamini and Georgakopoulos asking which Cayley graphs are isolated. Based on joint works with Romain Tessera.
NPCW04 9th May 2017
09:00 to 10:00
Alex Lubotzky Local testability in group theory I
A finitely generated group  G is be called TESTABLE ( or stable w.r.t. to the symmetric groups) if every almost homomorphism from G into a symmetric group Sym(n) is "close" to a real homomorphism. In the talk (which is a first in a series of two; the second will be given by Oren Becker), we will present this notion, its relation to local testability in computer science and its connections with other group theoretic concepts such as sofic groups, amenability, residual finiteness, the profinite topology, LERF and Kazhdan's property (T).
The goal is to develop methods to distinguish between testable and non testable groups. Some results and some conjectures will be presented.
Joint work with Oren Becker.
NPCW04 9th May 2017
10:00 to 11:00
Oren Becker Local testability in group theory II
This talk is a continuation of Alex Lubotzky's talk with a similar title (but an effort will be made to keep it independent).
We will describe a combinatorial/geometric method to prove testability (or non-testability) in various cases.
For certain amenable groups, we present a method of "tiling" every Schreier graph by finite Schreier graphs. This is an extension of the work of Weiss on monotileable groups. We then use the tilings to prove testability for those groups by a method which has its origins in the work of Ornstein-Weiss on amenable groups. This enables us to answer some questions posed in a paper by Arzhantseva and Paunescu and extend some of their results. It also suggests many more questions for further research.
NPCW04 9th May 2017
11:30 to 12:30
Aditi Kar Ping Pong on CAT(0) cube complexes
Joint work with Michah Sageev, in which we consider structural aspects of CAT(0) cubed groups that allow us establish properties like P_naive and uniform exponential growth.
NPCW04 9th May 2017
14:30 to 15:30
Alan Reid Arithmetic of Dehn surgery points
Associated to a finite volume hyperbolic 3-manifold is a number field and quaternion algebra over that number field.
Closed hyperbolic 3-manifolds arising from Dehn surgeries on a hyperbolic knot complement provide a family of number fields and
quaternion algebras that can be viewed as varying over the canonical component of the character variety of the knot.  This talk will investigate this, and  give examples of different behavior. The main results will show how this can be explained using the language of Azumaya algebras over a curve. This is joint work with Ted Chinburg and Matthew Stover.
NPCW04 9th May 2017
16:00 to 17:00
Stefaan Vaes Negative curvature and rigidity for von Neumann algebras
Popa's deformation/rigidity theory led to numerous classification and structure theorems for von Neumann algebras coming from groups and their actions on measure spaces. Negative curvature phenomena like hyperbolicity have played a key role in several of these results. I will first give an introduction to von Neumann algebras and then present a number of rigidity theorems, highlighting the usage of negative curvature type concepts.
NPCW04 10th May 2017
09:00 to 10:00
Anne Parreau Vectorial metric compactification of symmetric spaces and affine buildings
In higher rank symmetric spaces and affine buildings, the natural
projection of segments in a closed Weyl chamber may be regarded as a
universal metric with vectorial values. It refines all Finsler
metrics.  Remarkably, many of the traditional basic properties of
CAT(0) spaces still hold for the vectorial metric, providing similar
properties for all Finsler metrics in a unified way.  We will show
that the classical Busemann compactification construction can be
directly conducted in this context, giving a natural compactification
by vector-valued horofunctions.  These functions correspond to
strongly asymptotic classes of facets.  This compactification is
naturally homeomorphic to the maximal Satake compactification and
dominates all linear Finsler compactifications.

NPCW04 10th May 2017
10:00 to 11:00
Nikolay Nikolov Homology torsion growth of higher rank lattices
The asymptotic behaviour of Betti numbers and more generally, representation multiplicities associated to lattices in Lie groups have been extensively studied. In this talk I will discuss the asymptotic behaviour of two related invariants: rank and homology torsion in higher rank lattices. In the nonuniform case this is well understood due to the validity of the Congruence subgroup property but the uniform (cocompact) case is wide open. With M. Abert and T. Gelander we resolved this for right angled lattices. A group is right angled if it can be generated by a sequence of elements of infinite order each of which commutes with the previous one. However not all lattices are right angled and I will survey the many open questions in this area.
NPCW04 10th May 2017
11:30 to 12:30
Fanny Kassel Convex cocompactness in real projective geometry
We will discuss a notion of convex cocompactness for discrete groups preserving a properly convex open domain in real projective space. For hyperbolic groups, this notion is equivalent to being the image of a projective Anosov representation. For nonhyperbolic groups, the notion covers Benoist's examples of divisible convex sets which are not strictly convex, as well as their deformations inside larger projective spaces. Even when these groups are nonhyperbolic, they still share some of the good properties of classical convex cocompact subgroups of rank-one Lie groups; in particular, they are quasi-isometrically embedded and structurally stable. This is joint work with J. Danciger and F. Guéritaud.
NPCW04 11th May 2017
09:00 to 10:00
Uri Bader Unitary representations of reflection groups and their deformations.
The first two parts of my talk will consist of independent surveys of exciting theories.
1. I will discuss the theory of "boundary representations", which consists of the study of a class of unitary representations arising naturally in geometric group theory.
2. I will describe the deformation theory of a right-angled Coxeter group W, known as the Iwahori-Hecke algebra H.
Later I will combine the two and focus on the boundary representation of W and explain how it deforms into a representation of H.
If time permits, I will relate the above to the boundary representations of groups which act on hyperbolic buildings.

Based on a joint work with Jan Dymara.

NPCW04 11th May 2017
10:00 to 11:00
Kathryn Mann Large scale geometry in large transformation groups
In this talk I will survey some recent work on coarse geometry of transformation groups, specifically, groups of homeomorphisms and diffeomorphisms of manifolds. Following a framework developed by C. Rosendal, many of these groups have a well defined quasi-isometry type (despite not being locally compact or compactly generated). This provides the right context to discuss geometric questions such as boundedness and subgroup distortion -- questions which have already been studied in the context of actions of finitely generated groups on manifolds.
NPCW04 11th May 2017
11:30 to 12:30
Phillip Wesolek Approximating simple locally compact groups by their dense subgroups
Co-authors: Pierre-Emmanuel Caprace (Université catholique de Louvain), Colin Reid (University of Newcastle, Australia )  The collection of topologically simple totally disconnected locally compact (t.d.l.c.) groups which are compactly generated and non-discrete, denoted by , forms a rich and compelling class of locally compact groups. Members of this class include the simple algebraic groups over non-archimedean local fields, the tree almost automorphism groups, and groups acting on cube complexes.

In this talk, we study the non-discrete t.d.l.c. groups which admit a continuous embedding with dense image into some group  ; that is, we study the non-discrete t.d.l.c. groups which approximate
groups  . We consider a class which contains all such t.d.l.c. groups and show enjoys many of the same properties previously established for . Using these more general results, new restrictions on the members of are obtained. For any , we prove that any infinite Sylow pro- subgroup of a compact open subgroup of is not solvable. We prove further that there is a finite set of primes such that every compact subgroup of is virtually pro- .
NPCW04 11th May 2017
14:30 to 15:30
Harald Helfgott The diameter of the symmetric group: ideas and tools
Given a finite group and a set of generators, the diameter of the Cayley graph is the smallest such that every element of can be expressed as a word of length at most in ^(-) . We are concerned with bounding .

It has long been conjectured that the diameter of the symmetric group of degree is polynomially bounded in . In 2011, Helfgott and Seress gave a quasipolynomial bound (exp((log n)^(4+epsilon))). We will discuss a recent, much simplified version of the proof.
NPCW04 11th May 2017
16:00 to 17:00
David Fisher Subexponential growth, measure rigidity, strong property (T) and Zimmer's conjecture
Co-authors: Aaron Brown (University of Chicago), Sebastian Hurtado (University of Chicago)

Lattices in higher rank simple Lie groups, like SL(n,R) for n>2, are known to be extremely rigid. Examples of this are Margulis' superrigidity theorem, which shows they have very few linear represenations, and Margulis' arithmeticity theorem, which shows they are all constructed via number theory. Motivated by these and other results, in 1983 Zimmer made a number of conjectures about actions of these groups on compact manifolds. After providing some history and motivation, I will discuss a very recent result, proving many cases of the main conjecture. The proof has many surprising features, including that it uses hyperbolic dynamics to prove an essentially elliptic result, that it uses results on homogeneous dynamics, including Ratner's measure classification theorem, to prove results about inhomogeneous system and that it uses analytic notions originally defined for the purposes of studying the K theory of C^* algebras.
NPCW04 12th May 2017
09:30 to 10:30
Indira Chatterji Old results and new questions on the rapid decay property
A discrete group has the rapid decay property if its group law behaves well enough with respect to its large scale geometry. I will define the rapid decay property and discuss a few questions that I couldn’t answer while writing a survey on that subject.
NPCW04 12th May 2017
11:00 to 12:00
Danny Calegari Laminations and external angles for similarity pairs
The Barnsley-Harrington Mandelbrot set for similarity pairs has many interesting affinities with the “usual” Mandelbrot set. In particular, there is a “coding” of boundary points by data analogous to the “external angle” for points on the boundary of the usual Mandelbrot set. Instead of a single real number - an external angle - there is another parameter, a “scale factor”, which can be between 1 and 2, and is 2 when the similarity pair is quasiconformally conjugate (as a conformal dynamical system on its limit set) to (the inverse of) a degree 2 rational map on its Julia set. As with the ordinary external angle, there is associated to the pair (angle, scale factor) a lamination of the circle which parameterizes cut points for the limit set. This is joint work with Alden Walker.
NPCW04 12th May 2017
13:30 to 14:30
Emmanuel Breuillard How to quickly generate a nice hyperbolic element
In the 60's Rota and Strang defined the notion of joint spectral radius of a finite set of matrices. This adequately generalizes the spectral radius of a single matrix to several matrices, and the relation between the limit norm of powers and the maximal eigenvalue (spectral radius formula) can be extended to this setting. In this talk I will present a general geometric formulation in which one considers a finite set of isometries S and the joint minimal displacement L(S), which is closely related to the joint spectral radius of Rota and Strang. The main result is a spectral radius formula for isometric actions on spaces with non-positive curvature (in particular symmetric spaces of non-compact type and \delta-hyperbolic spaces) which extends the previously known results about matrices. Applications to uniform exponential growth will be given. Joint work with Koji Fujiwara.
NPCW04 12th May 2017
14:30 to 15:30
Peter Kropholler A random walk around soluble group theory
Co-authors: Karl Lorensen (Penn State Altoona), Armando Martino (Southampton), Conchita Martinez Perez (Zaragoza), Lison Jacoboni (Orsay)

This talk is about new developments in the theory of soluble (aka solvable) groups. In the nineteen sixties, seventies, and eighties, the theory of infinite solvable groups developed quietly and unnoticed except by experts in group theory. Philip Hall's work was a major impact and inspiration but before that there had been pioneering work of Maltsev and Hirsch. In the eighties, new vigour was brought to the subject through the work of Bieri and Strebel: the BNS invariant was born and for the first time there appeared a connection between the abstract algebra of Maltsev, Hirsch and Hall, and the topological and geometric insights of Thurston, Stallings and Dunwoody.

Nowadays, solvable groups are vital for a number of reasons. They are a primary source of examples of amenable groups, exhibiting a rich display of properties as shown in work of, for example, Erschler. There is an intimate connection with 3 manifold theory: we imagine that 3 manifolds revolve around hyperbolic geometry. But if hyperbolic geometry is the sun at the centre of the 3 manifold universe then Sol Nil S^3 S^2xR and R^3 (5 of the remaining 7 geometries identified by Thurstons geometrization programme must be the outlying planets: all virtually solvable and very much full of life. We might think of these solvable geometries as in some way the trivial cases. But they have also been an inspiration both in algebra and in geometry.

In this talk I will take a survey that leads in a meandering way through solvable infinite groups and culminates in a study of random walks on Cayley graphs including recent work joint with Lorensen as well as independent results of Jacoboni.
NPC 16th May 2017
10:00 to 11:00
Michael Davis The action dimensions of some discrete groups
The geometric dimension of a torsion-free group G is the minimum dimension of a model for BG by a CW complex.  Its action dimension is the minimum dimension of a model for BG by a manifold. I will discuss some recent work with Kevin Schreve and Giang Le in which we compute the action dimension of Artin groups, graph products of groups and other examples of groups which are given as simple complexes of groups.

NPC 16th May 2017
11:00 to 12:00
Yves de Cornulier Commensurating actions of groups of birational transformations
A commensurating action is an action of a group on a set along with a subset that is commensurated, in the sense that it has finite symmetric difference with each of its translates. There are natural ways to go from commensurating actions to actions on CAT(0) cube complexes and vice versa. For each variety X, we construct a natural commensurating action of the group of birational transformations of X. The commensurated subset turns out to be the set of irreducible hypersurfaces in X. Under the assumption that the acting group has Property FW (which means that every action on a CAT(0) cube complex has a fixed point), we deduce restrictions on its birational actions.   (Joint work with Serge Cantat)

NPC 16th May 2017
14:00 to 16:00
Viktor Schroeder Moebius Geometry of Boundaries''
NPC 17th May 2017
14:00 to 16:00
Richard Schwartz Iterated barycentric subdivision and steerable semigroups of SL_n(R)
NPC 19th May 2017
15:45 to 17:15
Richard Schwartz Thurston's Shapes of Polyhedra
NPC 23rd May 2017
10:00 to 11:00
Erik Guentner Affine actions, cohomology and hyperbolicity
Many groups admit an affine action on a Hilbert, or suitable Banach space which is proper, or at least has an unbounded orbit. For example, CAT(0) cubical groups act properly on Hilbert space; and a hyperbolic group acts properly on an Lp-space, although some cannot act on a Hilbert space with an unbounded orbit (or even without a global fixed point). In the talk I will describe these results and will discuss some recent work, joint with Eric Reckwerdt and Romain Tessera, on the existence of affine actions of relatively hyperbolic groups.

NPC 23rd May 2017
11:00 to 12:00
Nicolas Matte Bon Uniformly recurrent subgroups and rigidity of non-free minimal actions
A uniformly recurrent subgroup is a closed minimal invariant subset in the Chabauty space of a group. After explaining the relationship between uniformly recurrent subgroups and stabilisers of minimal actions on compact spaces, I will illustrate some examples in which a lack of uniformly recurrent subgroups leads to rigidity phenomena for non-free minimal actions.   (Joint works with Adrien Le Boudec and Todor Tsankov)

NPC 23rd May 2017
14:00 to 16:00
Viktor Schroeder Moebius Geometry of Boundaries''
NPC 24th May 2017
14:00 to 16:00
Richard Schwartz The pentagram map and discrete integrable systems
NPC 26th May 2017
10:00 to 12:00
Richard Schwartz PETs, pseudogroup actions, and renormalisation
NPC 30th May 2017
10:00 to 11:00
Alina Vdovina Buildings, surfaces and quaternions
Buildings are exciting objects which have geometric, algebraic and number theoretical aspects. We will give elementary constructions of several classes of buildings as universal covers of finite complexes. Then (as a geometric application) we will show a connection of our results with Gromov's surface subgroup question and (as an arithmetic application) we will give explicit examples of quaternionic lattices.

NPC 30th May 2017
11:00 to 12:00
Ruth Charney Are geodesic metric spaces determined by their Morse boundaries?
Boundaries of hyperbolic spaces have played a key role in low dimensional topology and geometric group theory. In 1993, Paulin showed that the topology of the boundary of a hyperbolic space, together with its quasi-mobius structure, determines the space up to quasi-isometry. One can define an analogous boundary, called the Morse boundary, for any proper geodesic metric space. I will discuss an analogue of Paulin’s theorem for Morse boundaries of CAT(0) spaces.   (Joint work with Devin Murray)

NPC 30th May 2017
14:00 to 16:00
Viktor Schroeder Moebius Geometry of Boundaries''
NPC 13th June 2017
11:00 to 12:00
David Fisher Strong property (T), subexponential growth of derivatives and invariant metrics
I will discuss how one uses the strong property (T) of Lafforgue to find invariant smooth metrics for actions with subexponential growth of derivatives. This is the "easier half" of the recent proof of many cases of Zimmer's conjecture by myself, Brown and Hurtado.  I will begin by motivating and explaining strong property (T) and move on to the application.

NPC 14th June 2017
13:00 to 14:00
Vaughan Jones Knots and links from the Thompson groups
We will begin with a general procedure for constructing actions ofgroups of fractions of certain categories and give a few examples of this procedure.We then realise the Thompson groups F, T and V as groups of fractions of categories of forestsand obtain many actions of these groups on many spaces. By representing the category of forests on Conway tangles one obtains constructions of knots and links from F and T and we can show that any link can be obtained in this way. Applying a TQFT gives unitary representations on Hilbert spacewhose coefficients are the TQFT link invariants.
NPCW05 19th June 2017
10:00 to 11:00
Marc Burger Compactifications of spaces of maximal representations and non archimedean geometry
Maximal representations form certain components of the variety of Sp(2n,R)-representations of
a compact surface group. These components coincide with Teichmueller space for SL(2,R). As in
the case of SL(2,R), one can use length functions to compactifiy these components thereby generalizing the
Thurston boundary of Teichmueller space. We will present recent results concerning the structure of
these boundaries and the properties of the length functions forming them.

The general picture that emerges is that this boundary decomposes into a closed subset formed of length
functions vanishing on subsurfaces or associated to R-tree actions with small stabilizers, and an open complement
on which the mapping class group acts properly discontinuously. The latter part of the boundary is non empty if and
only if n is at least 2.

The approach is based on the study of an  analogue of maximal representations over ordered, non archimedean fields.

This is joint work with A. Iozzi, A. Parreau and B. Pozzetti.
NPCW05 19th June 2017
11:30 to 12:30
Roberto Frigerio Bounded cohomology and combinatorial volume forms
Co-authors: Federico Franceschini (KIT Karlsruhe), MAria Beatrice Pozzetti (University of Warwick), Alessandro Sisto (ETH Zurich)

In this talk we describe a family of 3-dimensional combinatorial volume forms on non-abelian free groups. These forms define non-trivial classes in bounded cohomology, and they may be exploited to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional.

If time is left, as another application of combinatorial volume forms, we provide a purely cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmuller translation distance.
NPCW05 19th June 2017
13:30 to 14:30
Clara Löh The uniform boundary condition and simplicial volumes
Co-author: Daniel Fauser (Universität Regensburg)

The uniform boundary condition on a normed chain complex requires the existence of controlled fillings for all boundaries. The uniform boundary condition naturally comes up in the context of glueing results for simplicial volume. Matsumoto and Morita showed that the singular chain complex of spaces with amenable fundamental group satisfies the uniform boundary condition, using bounded cohomology. We give a direct geometric proof of this fact in the aspherical case. This proof admits generalisations to integral foliated simplicial volume, which provides upper bounds for -Betti numbers and the rank gradient.
NPCW05 19th June 2017
14:30 to 15:30
Juliette Bavard Around a big mapping class group
Co-author: Alden Walker (Center for Communications Research, La Jolla)

The mapping class group of the plane minus a Cantor set naturally appears in many dynamical contexts, including group actions on surfaces, the study of groups of homeomorphisms on a Cantor set, and complex dynamics. In this talk, I will present the 'ray graph', which is a Gromov-hyperbolic graph on which this big mapping class group acts by isometries (it is an equivalent of the curve graph for this surface of infinite topological type). If time allows, I will explain how we can use this action to construct non trivial quasimorphisms on this group, although it is not acylindrically hyperbolic. This might involve joint work with Alden Walker.
NPCW05 19th June 2017
16:00 to 17:00
Kasra Rafi Geodesic currents and counting problems
We show that, for every filling geodesic current, a certain scaled average of the mapping class group orbit of this current converges to multiple of the Thurston measure on the space of measured laminations. This has applications to several counting problems, in particular, we count the number of lattice points in the ball of radius R in Teichmüller space equipped with Thurston’s asymmetric metric. This is a joint work with Juan Souto.
NPCW05 20th June 2017
09:00 to 10:00
Thomas Delzant Kaehler groups and CAT(0) cubic complexes
Join work with Pierre PyWe apply the work of Caprace Sageev on group acting on CAT(O) cube complexes, and the work of Bridson Howie Miller Short on subproducts of surface groups to prove that if a Kahler groups is cubulable, it contains a subgroup of finite index which is a product of surface groups and abelian groups.
NPCW05 20th June 2017
10:00 to 11:00
The universal L2-torsion, introduced by Friedl and Lück, allows for an extension of the Thurston norm from the setting of 3-manifolds to that of free-by-cyclic groups. We will discuss this extension, and show that this norm and the Alexander norm for F2-by-Z satisfy an inequality analogous to the one satisfied by the Thurston and Alexander norms on 3-manifolds. We will also discuss the relationship between the universal L2-torsion and the Bieri-Neumann-Strebel invariants.

This is joint work with Florian Funke.
NPCW05 20th June 2017
11:30 to 12:30
Eriko Hironaka Polynomial invariants of graph maps and applications to Out(Fn) and Mod(Sgn)
Mapping classes on surfaces and outer automorphisms of the free group define dynamical systems that can often be described in terms of graph maps.   In this  talk we present algebraic and geometric invariants of graph maps and relate these to  the corresponding invariants in the geometric setting,  In particular, we present a unified way to view  Alexander polynomials and the lesser known Teichmueller polynomials.  This talk is based on joint work with Kasra Rafi and Yael Algom-Kfir.
NPCW05 20th June 2017
14:30 to 15:30
Jing Tao Effective quasimorphisms on right-angled Artin groups
In this talk, I will describe joint work with T. Fernos and M. Forester in which we construct quasimorphisms on right-angled Artin groups that can "see" every nontrivial element. As consequence, there is a uniform lower bound of 1/24 for stable commutator lengths in right-angled Artin groups.
NPCW05 20th June 2017
16:00 to 17:00
Roman Sauer The evolution of L2-Betti numbers
L2-Betti numbers of Riemannian manifolds were introduced by Atiyah in the 1970s. Cheeger and Gromov extended their scope of definition to all countable discrete groups in the 1980s. Nowadays, there are L2-Betti numbers of arbitrary spaces with arbitrary discrete group actions, of locally compact groups, of quantum groups, of von Neumann algebras, of measured equivalence relations and of invariant random subgroups. Their relation to classical homology comes via a remarkable theorem of Lück, the approximation theorem. We sketch the remarkable extension of the  definition of L2-Betti numbers and present some results about totally disconnected groups. The latter is based on joint work with Henrik Petersen and Andreas Thom.
NPCW05 21st June 2017
09:00 to 10:00
Ian Leary Generalized Bestvina-Brady groups and their applications
Co-authors: Robert Kropholler (Tufts University), Ignat Soroko (University of Oklahoma)

In the 1990's Bestvina and Brady used Morse theory to exhibit (as subgroups of right-angled Artin groups) the first examples of groups that are but not finitely presented.

The speaker has generalized this construction, via branched coverings, to construct continuously many groups of type , including groups of type FP that do not embed in any finitely presented group.

I shall discuss the construction and some applications, including the theorem that every countable group embeds in a group of type and the construction of continuously many quasi-isometry classes of acyclic 4-manifolds admitting free, cocompact, properly discontinuous discrete group action (the latter joint with Robert Kropholler and Ignat Soroko).

NPCW05 21st June 2017
10:00 to 11:00
Yael Algom Kfir The boundary of hyperbolic free-by-cyclic groups
Given an automorphism $\phi$ of the free group $F_n$ consider the HNN extension $G = F_n \rtimes_\phi \Z$. We compare two cases:
1. $\phi$ is induced by a pseudo-Anosov map on a  surface with boundary and of non-positive Euler characteristic. In this case $G$ is a CAT(0) group with isolated flats and its (unique by Hruska) CAT(0)-boundary is a Sierpinski Carpet (Ruane).
2. $\phi$ is atoroidal and fully irreducible. Then by a theorem of Brinkmann $G$ is hyperbolic. If $\phi$ is irreducible then Its boundary is homeomorphic to the Menger curve (M. Kapovich and Kleiner).
We prove that if $\phi$ is atoroidal then its boundary contains a non-planar set. Our proof highlights the differences between the two cases above.
This is joint work with A. Hilion and E. Stark.
NPCW05 21st June 2017
11:30 to 12:30
Henry Wilton Surface subgroups of graphs of free groups
A well known question, usually attributed to Gromov, asks whether every hyperbolic group is either virtually free or contains a surface subgroup. I’ll discuss the answer to this problem for the class of groups in the title when the edge groups are cyclic.  The main theorem is a result about free groups F which is of interest in its own right: whether of not an element w of F is primitive can be detected in the abelianizations of finite-index subgroup of F.  I’ll also mention an application to the profinite rigidity of the free group.
NPCW05 22nd June 2017
09:00 to 10:00
NPCW05 22nd June 2017
10:00 to 11:00
Matt Clay L2-torsion of free-by-cyclic groups
I will provide an upper bound on the L2-torsion of a free-by-cyclic group, in terms of a relative train-track representative for the monodromy. This result shares features with a theorem of Luck-Schick computing the L2-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the L2-torsion is determined by the exponential dynamics of the monodromy. In light of the result of Luck-Schick, a special case of this bound is analogous to the bound on the volume of a 3-manifold that fibers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima-McShane.
NPCW05 22nd June 2017
11:30 to 12:30
Michelle Bucher Vanishing simplicial volume for certain affine manifolds
Affine manifolds, i.e. manifolds which admit charts given by affine transformations, remain mysterious by the very few  explicit examples and their famous open conjectures: the Auslander Conjecture, the Chern Conjecture and the Markus Conjecture. After reviewing the current state of knowledge on these conjectures, I will present an intermediate conjecture, somehow between the Auslander Conjecture and the Chern Conjecture, involving the simplicial volume, a topological invariant of manifolds introduced by Gromov in the beginning of the 80’s. In a joint work with Chris Connell and Jean-François Lafont, we prove the latter intermediate conjecture under some hypothesis, thus providing further evidence for the veracity of the Auslander and Chern Conjectures.
NPCW05 22nd June 2017
14:30 to 15:30
Jean-Francois Lafont Hyperbolic groups with boundary an n-dimensional Sierpinski space
Let G be a torsion-free Gromov hyperbolic group, whose boundary at infinity is an n-dimensional Sierpinski space. I'll explain why, if n>4, the group G is in fact the fundamental group of a (unique) aspherical (n+2)-manifold with non-empty boundary. Time permitting, various related results will also be discussed. This is joint work with Bena Tshishiku.
NPCW05 22nd June 2017
16:00 to 17:00
Denis Osin Extending group actions on metric spaces
I will discuss the following natural extension problem for group actions: Given a group G, a subgroup H of G, and an action of H on a metric space, when is it possible to extend it to an action of the whole group on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of H? We begin by formalizing this problem and present a construction of an induced action which behaves well when H is hyperbolically embedded in G. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups.
NPCW05 23rd June 2017
09:00 to 10:00
Bill Goldman The dynamics of classifying geometric structures
The general theory of locally homogeneous geometric structures (flat Cartan connections) originated with Ehresmann. Their classification is analogous to the classification of Riemann surfaces by the Riemann moduli space. In general, however, the analog of the moduli space is intractable, but leads to a rich class of dynamical systems.

For example, classifying Euclidean geometries on the torus leads to the usual action of the SL(2,Z)  on the upper half-plane. This action is dynamically trivial, with a quotient space the familiar modular curve.  In contrast, the classification of other simple geometries on  on the torus leads to the standard linear action of SL(2,Z) on R^2,  with chaotic dynamics and a pathological quotient space.

This talk describes such dynamical systems, and we combine Teichmueller theory to understand the geometry of the moduli space when the topology is enhanced with a  conformal structure. In joint work with Forni, we prove the corresponding extended Teichmueller flow is strongly mixing.

Basic examples arise when  the moduli space  is described by the nonlinear symmetries of cubic equations like Markoff’s equation x^2 + y^2 + z^2 = x y z.  Here both trivial and chaotic dynamics arise simultaneously, relating to possibly singular hyperbolic-geometry structures on surfaces. (This represents joint work with McShane-Stantchev- Tan.)
NPCW05 23rd June 2017
10:00 to 11:00
Kevin Schreve Action dimension and L^2 Cohomology
Co-authors: Michael Davis (Ohio State University), Giang Le ()

The action dimension of a group G is the minimal dimension of contractible manifold that G acts on properly discontinuously. Conjecturally, if a group has nontrivial cohomology in dimension n, the action dimension of G is bounded below by 2n. I will describe examples where this conjecture holds, including lattices in Euclidean buildings, graph products, and fundamental groups of some complex hyperplane complements. This will involve joint work with Mike Davis and Giang Le, as well as Grigori Avramidi, Mike Davis, and Boris Okun.
NPCW05 23rd June 2017
11:30 to 12:30
Christopher Leininger Free-by-cyclic groups and trees
Given a hyperbolic free-by-cyclic group G, I will explain how to assign an action of G on a topological R-tree T_U for certain components U of the BNS invariant.   For every element x in U, there is a metric on T_U so that G acts by homotheties and the kernel of x acts by isometries.  This is part of ongoing joint work with Spencer Dowdall and  Ilya Kapovich.
NPCW05 23rd June 2017
13:30 to 14:30
Grigori Avramidi Topology of ends of nonpositively curved manifolds
Co-author: Tam Nguyen Phan (Binghamton University)

The structure of ends of a finite volume, nonpositively curved, locally symmetric manifold M is very well understood. By Borel-Serre, the thin part of the universal cover of such a manifold is homotopy equivalent to a rational Tits building. This is a simplicial complex built out of the algebra of the locally symmetric space which turns out to have dimension = dim M/2. Another application is that the group cohomology with group ring coefficients of the fundamental group of M vanishes in low dimensions (
NPCW05 23rd June 2017
14:30 to 15:30
Karen Vogtmann The borders of Outer space
Outer space is an analog for the group Out(F_n) of the symmetric space associated to an algebraic group.  Motivated by work of Borel and Serre, Bestvina and Feighn defined a bordification of Outer space; this is an enlargement of outer space which is highly-connected at infinity and on which the action of Out(F_n) extends, with compact quotient. We realize this bordification as a deformation retract of Outer space instead of an extension.  We use this to give a simpler connectivity proof, and to give a description of the boundary nicely analogous to that of the Borel-Serre boundary of a symmetric space. This is joint work with Kai-Uwe Bux and Peter Smillie.