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Timetable (NPCW01)

Non-positive curvature in action

Monday 9th January 2017 to Friday 13th January 2017

Monday 9th January 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (INI Director)
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.
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Francois Dahmani
The normal closure of a big Dehn twist in a mapping class group
12:30 to 13:30 Lunch @ Wolfson Court
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.
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.
15:30 to 16:00 Afternoon Tea
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.
17:00 to 18:00 Welcome Wine Reception
Tuesday 10th January 2017
09:00 to 10:00 Benjamin Beeker
Cubical Accessibility and bounds on curves on surfaces.
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.
11:00 to 11:30 Morning Coffee
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. 
12:30 to 13:30 Lunch @ Wolfson Court
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.
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.
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Robert Kropholler
Hyperbolic groups and their subgroups
Wednesday 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. 
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.
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
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.
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.

Related Links
15:30 to 16:00 Afternoon Tea
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.
Thursday 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.  
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.
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
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. 
14:30 to 15:30 Damian Osajda
Group cubization
15:30 to 16:00 Afternoon Tea
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.
19:30 to 22:00 Formal Dinner at Emmanuel College
Friday 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.
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.

11:00 to 11:30 Morning Coffee
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.

Related Links
12:30 to 13:30 Lunch @ Wolfson Court
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. 
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)
15:30 to 16:00 Afternoon Tea
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.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons