Nonpositive curvature in action
Monday 9th January 2017 to Friday 13th 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. 
INI 1  
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 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Richard Webb Polynomialtime NielsenThurston type recognition
A cornerstone of the study of mapping class groups is the NielsenThurston classification theorem. I will outline a polynomialtime algorithm that determines the NielsenThurston 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. 
INI 1  
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 quasitrees, 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 sinksource dynamics) is generic. That is, for any finite generating set of G, the proportion of Xloxodromics 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. 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Camille Horbez Growth under automorphisms of hyperbolic groups
Coauthors: Rémi Coulon
(Université de Rennes 1), Arnaud Hilion
(AixMarseille Université), Gilbert Levitt
(Université de Caen)
Given a torsionfree 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. 
INI 1  
17:00 to 18:00  Welcome Wine Reception 
09:00 to 10:00 
Benjamin Beeker Cubical Accessibility and bounds on curves on surfaces. 
INI 1  
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.

INI 1  
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. 
INI 1  
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 cocompactly on CAT(0) cube complexes via finiteness properties of
their hyperplane stabilizers.
Joint work with Benjamin Beeker. 
INI 1  
14:30 to 15:30 
Anthony Genevois Cubicallike geometry of graph products
In 1994, Bandelt, Mulder and Wilkeit introduced a class of graphs
generalizing the socalled median graphs: the class of quasimedian
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 quasimedian
graphs can be used to study some classes of groups. In our talk, we will
show that quasimedian graphs and CAT(0) cube complexes share
essentially the same geometry. Moreover, extending the observation that
rightangled 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
quasimedian. The main goal of this talk is to show how to use the
quasimedian geometry of this Cayley graph to study graph products.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Robert Kropholler Hyperbolic groups and their subgroups 
INI 1 
09:00 to 10:00 
Jingyin Huang Commensurability of groups quasiisometric to RAAG's
It is wellknown that a finitely generated group quasiisometric to a
free group is commensurable to a free group. We seek higherdimensional
generalization of this fact in the class of rightangled Artin groups
(RAAG). Let G be a RAAG with finite outer automorphism group. Suppose in
addition that the defining graph of G is starrigid and has no induced
4cycle. Then we show every finitely generated group quasiisometric to G
is commensurable to G. However, if the defining graph of G contains an
induced 4cycle, then there always exists a group quasiisometric to G,
but not commensurable to G.

INI 1  
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.

INI 1  
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 nonpositive curvature. Some of this talk will
be on joint work with Hagen and Sisto; other parts are joint work with Hagen,
Susse, and FalgasRavry.

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Urs Lang Group actions on spaces with a distinguished geodesic structure
Coauthors: 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 noncoarse) global notion of nonpositive curvature that allows for nonunique geodesics. The talk will give a survey of recent results on group actions on spaces with such a structure. 
INI 1  
14:30 to 15:30 
Bruno Duchesne Groups actions on dendrites
Coauthor: 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

INI 1  
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.

INI 1 
09:00 to 10:00 
Petra Schwer Combinatorics of Coxeter groups and Affine Deligne Lusztig varieties
Coauthors: 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 subvarieties 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 BruhatTits 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. 
INI 1  
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 "nontrivial acylindrical". The goal of the talk is to open and discuss a number of interesting and important recognition problems for nontrivial 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 coauthors.

INI 1  
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. 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Emily Stark Commensurability classification of certain rightangled Coxeter groups and related surface amalgams
Coauthors: Pallavi Dani
(Louisiana State University), Anne Thomas
(University of Sydney)
I will describe the abstract commensurability classification within a class of hyperbolic rightangled 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 quasiisometry classification and abstract commensurability classification in this setting. This is joint work with Pallavi Dani and Anne Thomas. 
INI 1  
14:30 to 15:30 
Damian Osajda Group cubization 
INI 1  
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 rightangled 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 SQuniversality, or
having a finite index subgroup with infinite dimensional second bounded
cohomology.
This is joint work with V. Guirardel.

INI 1  
19:30 to 22:00  Formal Dinner at Emmanuel College 
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 leftorder. 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.

INI 1  
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 1ended hyperbolic group G has a local cut point then G splits over a 2ended group. We prove a similar theorem for CAT(0) groups, namely that if a finite set of points separates the boundary of a 1ended CAT(0) group G then G splits over a 2ended group. Along the way we prove two results of independent interest: we show that continua separated by finite sets of points admit a treelike decomposition and we show a splitting theorem for nesting actions on Rtrees. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Dominik Gruber Small cancellation theory over Burnside groups
Coauthor: Rémi Coulon
(Université de Rennes 1)
I will discuss how combinatorics and geometry work together to provide a new and easytoapply 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

INI 1  
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 BaumslagSolitar 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. 
INI 1  
14:30 to 15:30 
Alexandre Martin On the small cancellation geometry of certain graph products of groups
Graph products of groups generalise both rightangled Coxeter groups and
rightangled Artin groups. While such groups are already known to act
on rightangled 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) 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Christopher Cashen The topology of the contracting boundary of a group
Coauthor: 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. 
INI 1 