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Timetable (NPCW05)

Group actions and cohomology in non-positive curvature

Monday 19th June 2017 to Friday 23rd June 2017

Monday 19th June 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (INI Director)
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.
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
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.
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.
15:30 to 16:00 Afternoon Tea
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.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 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.
10:00 to 11:00 Dawid Kielak
Universal L2-torsion for free-by-cyclic groups
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.
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
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.
15:30 to 16:00 Afternoon Tea
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. 
Wednesday 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).

Related Links
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.
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Christ's College
Thursday 22nd June 2017
09:00 to 10:00 Vladimir Markovic
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.
11:00 to 11:30 Morning Coffee
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. 
12:30 to 13:30 Lunch @ Wolfson Court
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.
15:30 to 16:00 Afternoon Tea
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. 
Friday 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.)
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. 
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
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 (
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.
15:30 to 16:00 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons