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Timetable (NPCW04)

Approximation, deformation, quasification

Monday 8th May 2017 to Friday 12th May 2017

Monday 8th May 2017
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director)
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).
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
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.
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.)
15:30 to 16:00 Afternoon Tea
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.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 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.
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.
11:00 to 11:30 Morning Coffee
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. 
12:30 to 13:30 Lunch @ Wolfson Court
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.
15:30 to 16:00 Afternoon Tea
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.
Wednesday 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.

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.
11:00 to 11:30 Morning Coffee
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.
12:30 to 13:30 Lunch @ Wolfson Court
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Trinity College
Thursday 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.

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. 
11:00 to 11:30 Morning Coffee
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- .
12:30 to 13:30 Lunch @ Wolfson Court
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. 
15:30 to 16:00 Afternoon Tea
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. 
Friday 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.
10:30 to 11:00 Morning Coffee
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. 
12:30 to 13:30 Lunch @ Wolfson Court
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.
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.
15:30 to 16:00 Afternoon Tea
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons