# Geometry and Topology Seminar[ Edit ]

## Moderator: Tobias Hartnick

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

Third and final lecture.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

Second in a series of three lectures.

*Abstract:*

In 1989, Pansu introduced the notion of the conformal dimension of the boundary at infinity of a negatively curved manifold. This notion, applied to the boundary at infinity of a Gromov hyperbolic group, gives a natural quasi-isometric invariant of the group. In these talks I'll survey some of what is known about conformal dimension and the challenge of calculating or even estimating its value.

First in a series of three lectures.

*Abstract:*

==== NOTE THE SPECIAL TIME ===

Let M be a compact complex manifold. Consider the action of the diffeomorphism group Diff(M) on the (infinite-dimensional) space Comp(M) of complex structures. A complex structure is called ergodic if its Diff(M)-orbit is dense in the connected component of Comp(M). I will show that on a hyperkaehler manifold or a compact torus, a generic complex structure is ergodic. If time permits, I would explain geometric applications of these results to hyperbolicity. I would try to make the talk accessible to non-specialists.

*Abstract:*

====== NOTE THE SPECIAL TIME ====

A subset S of a group G invariably generates G if for every choice of g(s) \in G,s \in S the set {s^g(s):s\in S} is a generating set of G. We say that a group G is invariably generated if such S exists, or equivalently if S=G invariably generates G. In this talk, we study invariable generation of Thompson groups. We show that Thompson group F is invariable generated by a finite set, whereas Thompson groups T and V are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

*Abstract:*

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.

*Abstract:*

Haglund showed that given an isometry of a CAT(0) cube complex that doesn't fix a 0-cube, there exists a biinfinite combinatorial geodesic axis.

I will explain how to generalize this theorem to show that given a proper action of Z^n on a CAT(0) cube complex, there is a nice subcomplex that embeds isometrically in the combinatorial metric and is stabilized by Z^n.

The motivation from group theory will also be given.

*Abstract:*

We prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a 2m-plat projection, where m is at least 3, each twist region of the plat contains at least three crossings, and n, the length of the plat, satisfies n > 4m(m − 2), then such a projection is unique up to obvious rotations. In particular, this projection gives a canonical form for such knots and links, and thus provides a classification of these links. This is joint work with Jessica S. Purcell.

*Abstract:*

There are two interesting norms on free groups and surface groups which are invariant under the group of all automorphisms:

A) For free groups we have the primitive norm, i.e., |g|_p = the minimal number of primitive elements one has to multiply to get g.

B) For fundamental group of genus g surface we have the simple curves norm, i.e., |g|_s = the minimal number of simple closed curves one need to concatenate to get g.

We prove the following dichotomy: either |g^n| is bounded or growths linearly with n. For free groups and surface groups we give an explicit characterisation of (un)bounded elements. It follows for example, that if g is a simple separating curve on a surface, then |g^n| growths linearly. However, if g is a simple non-separating curve, then |g^n| <= 2 for every n. This answers a question of D. Calegari.

The main idea of the proof is to construct appropriate quasimorphisms. M. Abert asked if there are Aut-invariant nontrivial homogeneous quasimorphisms on free groups. As a by-product of our technique we answer this question in the positive for rank 2. This is a joint work with M. Brandenbursky.

*Abstract:*

I will outline how one starts with a symplectic manifold and defines a category enriched in local systems (up to homotopy) on this manifold. The construction relies on deformation quantization and is related to other methods of constructing a category from a symplectic manifolds, such as the Fukaya category and the sheaf-theoretical microlocal category of Tamarkin. The talk will be accessible, with main examples being the plane, the cylinder, and the two-torus.

*Abstract:*

Geometric group theory arose from the study of periodic tilings of proper geodesic metric spaces, or equivalently the study of uniform lattices in isometry groups of such spaces. It provides a way to study finitely-generated infinite groups geometrically.

In joint work with Michael Björklund we propose a framework to study aperiodic tilings of proper geodesic metric spaces. This framework is based on three main ingredients:

1) Tao's notion of approximate subgroups (generalizing Meyer's notion of a model set in R^n)

2) Delone sets in locally compact groups

3) Classical geometric group theory

In this talk I will define the central notions of uniform and non-uniform approximate lattices arising in this framework, and explain some first steps towards a "geometric approximate group theory", i.e. a geometric theory of finitely generated (uniform) approximate lattice.

*Abstract:*

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 free 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.

*Abstract:*

I will overview how tubular groups have been studied over the past 30-40 years in geometric group theory before explaining recent results relating to the cubulation of tubular groups including my own work classifying which tubular groups are virtually special.

*Abstract:*

It is an old conjecture that closed (even dimensional) manifolds with nonzero Euler characteristic admit no flat structure. Although it turns out that there do exist manifolds with nonzero Euler characteristic that admit a flat structure, for closed aspherical manifolds this conjecture is still widely open. In 1958 Milnor proved the conjecture for surfaces through his celebrated inequality. Gromov naturally put Milnor’s inequality in the context of bounded cohomology, relating it to the simplicial volume.

I will show how to find upper and lower bounds for the simplicial volume of complex hyperbolic surfaces. The upper bound naturally leads to so-called Milnor-Wood inequalities strong enough to exclude the existence of flat structures on these manifolds.

*Abstract:*

Stable subgroups and the Morse boundary are two systematic approaches to collect and study the hyperbolic aspects of finitely generated groups. I will introduce a new quasi-isometry invariant of geodesic metric spaces which generalizes these strategies: the stable dimension. In the case of a proper Gromov hyperbolic space the stable dimension is the asymptotic dimension. Time permitting I will also discuss the stable dimension in the cases of right-angled Artin groups, mapping class groups, and Teichm¨uller space. This is joint work with David Hume.

*Abstract:*

It is well known that uniform spaces are inverse limits of pseudo-metric and, dually, that coarse spaces are direct limits of infinity-metric spaces. Usually, if, for example, a uniform space has some really nice covering property, then one can expect each of the metric spaces in the inverse approximation to also have that property. The dual statement for coarse spaces is also true. In a strong sense, both uniform spaces and coarse spaces are just special cases of groupoids. In a joint project with Joav Orovitz, we are employing an inverse approximation technique to topological groupoids that generalizes both of the above cases and we apply our metric approximations to extend the classical disintegration theorem of groupoid representations of John Renault. I plan on giving an overview of how the approximations work and how we use them in our proof of the aforementioned theorem.

*Abstract:*

In my talk I will give a review of the subject. I will present the steps of the classification of surfaces, using very nice methods and techniques, such as: degeneration of surfaces, braid monodromy, calculations of fundamental groups and Coxeter groups. We will see interesting examples of classification of known and significant surfaces, such as Hirzebruch surfaces.

*Abstract:*

We first give a short background on geometric structures. A geometry in the sense of Klein is given by a pair (Y, H) of a Lie group H acting transitively by diffeomorphisms on a manifold Y . Given a manifold of the same dimension as Y, a geometric structure modeled on (Y, H) is a system of local coordinates in Y with transition maps in H. For example, the geometrization conjecture (proved by Perelman) says that in dimension 3, every closed manifold can be cut into pieces, and each piece has one of 8 kinds of geometry.

A convex projective manifold C = Ω/Γ is the quotient of convex subset of projective space, Ω, by a discrete group of projective transformations Γ ⊂ P GL(n + 1, R). A generalized cusp in dimension 3 is a convex projective manifold that is the product of a ray and a torus. The holonomy centralizes a 1 parameter subgroup of PGL(n,R). I have shown : A generalized cusp on a properly convex projective 3 dimensional manifold is projectively equivalent to one of 4 possible cusps.

For a generalized cusp C = Ω/Γ in dimension n, we require that ∂C is compact and strictly convex (contains no line segment) and that there is a diffeomorphism h : [0, ∞) × ∂C → C. Together with Sam Ballas and Daryl Cooper we have classified generalized cusps in dimension n, and explored new geometries arising from such cusps. We show the holonomy of a generalized cusp is a lattice in one of a family of Lie groups G(λ) parameterized by a point λ = (λ1, ..., λn) ∈ R n . More generally a maximal-rank cusp in a hyperbolic n-orbifold is determined by the similarity class of lattice in Isom(E^{ n−1} )

*Abstract:*

===== Time and Room changed because of the Special Lecture Series !!! ===

A countable group G is homogeneous if any two finite tuples of elements which satisfy the same first-order properties are in the same orbit under Aut(G). We give some conditions for a torsion free hyperbolic group to be homogeneous in terms of its JSJ decomposition. This is joint work with Ayala Byron.

*Abstract:*

The mapping class group is an example of a perfect group; its abelianization is trivial. In particular, every element can be written as a product of commutators. Endo and Kotschik showed that the mapping class group is not uniformly perfect; there is no bound on the number of commutators required to represent a given element. To prove this they showed that there are elements with positive "stable commutator length." Their proof uses rather sophisticated results on the symplectic geometry of 4-manifolds. In this talk we will use more elementary methods to give a complete characterization of when the stable commutator length is positive in the mapping class group. The is joint work with M. Bestvina and K. Fujiwara.

*Abstract:*

Starting from a word in the standard generators in the mapping class group of a surface, we construct a weighted planar graph. Braid relations in the mapping class group correspond to the well-known Y-Delta transform of electric networks. Heegaard decompositions of closed 3-manifolds lead to similar planar graphs.

Counting critical points and closed orbits of discrete vector fields on such a graph, we obtain simple formulas for some celebrated 3-manifold invariants. A combinatorial counterpart of a certain complicated duality (between Chern-Simons theory and closed strings on a resolved conifold) turn out to be a generalization of the Matrix-Tree Theorem.

(This is an extended version of my IMU talk.)

*Abstract:*

This is a joint work with Gili Golan. I will talk about maximal subgroups of F, Stallings 2-cores of subgroups, and the generation problem for F.

*Abstract:*

NOTE THE SPECIAL DATE AND TIME!

There are several constructions of quasimorphisms on the Hamiltonian groups of surfaces that were proposed by Gambaudo-Ghys, Polterovich, Py, etc. These constructions are based on topological invariants either of individual orbits or of orbits of finite configurations of points and the quasimorphism computes the average value of these invariants along the surface. We show that many quasimorphisms that arise this way are not Hofer continuous. This allows to show non-equivalence of Hofer's metric and some other metrics on the Hamiltonian group.

NOTE THE SPECIAL DATE AND TIME!

*Abstract:*

I will describe a class of groups that act freely on the product of two trees. Consequently such groups are the fundamental groups of nonpositively curved square complexes.

The class of groups contains free groups and is closed under amalgamated free products along cyclic subgroups.

In a related result we show that every word-hyperbolic limit groups acts freely on the product of two trees. This is joint work with Frederic Haglund.

*Abstract:*

Joint work with Marc Soret. In a (N,q)-torus knot, a particle goes q times around a vertical planar circle which is being rotated N times around a central axis. On a Lissajous toric knot K(N,q,p), the particle goes through a Lissajous curve parametrized by (sin(qt), cos(pt+u)) while we rotate this curve N times around a central axis; we assume (N,q)=(N,p)=1. Christopher Lamm first defined these knots as billiard knots in the solid torus and we encountered them as singularity knots of minimal surfaces in R^4. They are naturally presented as closed braids which we write precisely: we derive that they are all ribbon or periodic, as stated by Lamm. Finally we give an upper bound for the 4-genus of K(N,q,p) in the spirit of the 4-genus of the torus knot.

*Abstract:*

I will discuss a recent construction by Pedroza and Przytycki of a dismantlable classifying space for the parabolic subgroups of a relatively hyperbolic group. I will include some basic exposition on hyperbolic groups before describing the construction and presenting some of the ideas involved in the proof that it yields a classifying space.

*Abstract:*

Garside groups have been first introduced by P.Dehornoy and L.Paris in 1990. In many aspects, Garside groups extend braid groups and more generally finite-type Artin groups. These are torsion-free groups with a word and conjugacy problems solvable, and they are groups of fractions of monoids with a structure of lattice with respect to left and right divisibilities. It is natural to ask if there are additional properties Garside groups share in common with the intensively investigated braid groups and finite-type Artin groups. In this talk, I will introduce the Garside groups in general, and a particular class of Garside groups, that arise from certain solutions of the Quantum Yang-Baxter equation. I will describe the connection between these theories arising from different domains of research, present some of the questions raised for the Garside groups and give some partial answers to these questions.

*Abstract:*

In the late 60s, Ottmar Loos gave a surprising and beautiful characterization of affine symmetric spaces as smooth reflection spaces with a weak isolation property for fixed points. The first half of this talk is intended as a survey on the structure of Riemannian and affine symmetric spaces from this reflection space point of view. In particular, we explain how geometric representations of finite reflection group arise from the local geometry of flats in such spaces. The second half of this talk is then devoted to exotic examples of topological reflection spaces, which satisfy all of Loos' axioms except for smoothness. This part is based on ongoing joined work with W. Freyn, M. Horn and R. Köhl. We show that for any 2-spherical Coxeter group W there exists an infinite-dimensional such reflection space of finite rank whose local geometry is governed by the geometric representation of W. Our examples are based on split-real Kac-Moody groups and have a number of geometric properties not observed in this context before. For example, any two points in the reflection space can be joined by a piecewise geodesic curve, but the reflection space is not midpoint convex. Time permitting we will discuss further properties of the construction, such as the classification of automorphisms and its relation to the natural boundary action of elliptic subgroups of the automorphism group.

*Abstract:*

Let G be a finitely generated group, and let dG be the word metric with respect to some finite generating set. let H be a subgroup of G. We say that H has \emph{ bounded packing } in G if for all R>0, there is an upper bound M(D) on the number of left cosets that are D-close. That is to say that if g1H,…,gM(D)H are distinct left cosets, then there exists 1≤i<j≤M(D) such that dG(giH,gjH)>D. We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. The main ingredient of the proof is a cubical flat torus theorem. This is joint work with Dani Wise.

*Abstract:*

We show that the boundary of a one-ended hyperbolic group that has enough codimension-1 surface subgroups and is simply connected at infinity is homeomorphic to a 2-sphere. Together with a result of Markovic, it follows that these groups are Kleinian groups. In my talk, I will describe this result and give a sketch of the proof.This is joint work with N. Lazarovich.

*Abstract:*

We exhibit a class of Artin groups that are the fundamental groups of nonpositively curved compact cube complexes. We show that 2-dimensional or 3-generator Artin groups outside this class are not the fundamental groups of nonpositively curved compact cube complexes, even if we pass to a finite index subgroup. In particular, this includes the braid group on 4 strands. This is joint work with Jingyin Huang and Piotr Przytycki.

*Abstract:*

In a recent publication, R. Willett, E. Guentner, and G. Yu develop a new concept which predicates connections between asymptotic dimension, amenable actions, transformation groupoids, nuclear dimension of C*-algebras and more. Our aim in giving these two talks is to go over what we think are the main points of this paper and to provide the proper context for these results. For the most part, the talks are independent of each other.

*Abstract:*

We show how quantization of families with values in K-theory can detect non-trivial Hamiltonian fibrations, yielding examples that are not detected by previous methods (the characteristic classes of Reznikov for example). We also upgrade a theorem of Spacil on the cohomology-surjectivity of a natural map of classifying spaces by providing it with an "almost" weak retraction. Joint work with Yasha Savelyev.

*Abstract:*

*NOTE THE CHANGE IN TIME AND PLACE*

I will describe a “cubical flat torus theorem” for a group G acting properly and cocompactly on a CAT(0) cube complex.This states that every “highest” free abelian subgroup of G acts properly and cocompactly on a convex subcomplex that is quasi-isometric to a Euclidean space.I will describe some simple consequences, as well as the original motivation which was to prove the “bounded packing property” for cyclic subgroups of G.This is joint work with Daniel Woodhouse.

*Abstract:*

The dual graph of a collection of disjoint simple closed curves is a useful invariant for distinguishing mapping class group orbits of curves. When the collections of curves are allowed intersections, however, the dual graph is not a well-defined invariant. Sageev's dual cube complex construction -- coming from a much more general context -- can be thought of as a fix for this problem. We will explore this invariant in general, in the context of a counting problem for simple curves (joint work with Tarik Aougab), and we will also describe a first step towards gleaning geometric data from its structure.

*Abstract:*

The contact mapping class group of a contact manifold V is the set of contact isotopy classes of diffeomorphisms of V preserving the contact structure. In this talk I'll show that for certain V the mapping class group contains an isomorphic copy of the full braid group on n strands. As a byproduct of the construction a result related to the contact isotopy problem is obtained, namely that there are contactomorphisms which are smoothly isotopic to the identity, but not so through contactomorphisms. In fact, the pure braid group embeds into the part of the contact mapping class group consisting of classes which are smoothly trivial. Joint work with Frol Zapolsky.

*Abstract:*

Two problems in geometric group theory are to characterize the abstract commensurability and quasi-isometry classes among finitely generated groups and to understand for which classes of groups the classifications coincide. I will discuss these questions and present a solution within the setting of certain amalgamated free products of surface groups.

*Abstract:*

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the positive diagonal Cartan subgroup in SL(n,R). For n = 3, it turns out the diagonal Cartan subgroup has precisely 5 limits, and for n = 4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n ≥ 7, there is a continuum of non conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers.

This second talk will focus on the general case.The first talk is not a prerequisite to attend this second talk, and both should be very accessible.

*Abstract:*

I will discuss a number of results on the interrelation between the L^p -metric on the group of Hamiltonian diffeomorphisms of surfaces and the subset A of autonomous Hamiltonian diffeomorphisms. In particular, I will show that there are Hamiltonian diffeomorphisms of all surfaces of genus g ≥ 2 or g = 0 lying arbitrarily L^p -far from the subset A, answering a variant of a question of Polterovich for the L^p -metric. This is a joint work with Egor Shelukhin.

*Abstract:*

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the positive diagonal Cartan subgroup in SL(n,R). For n = 3, it turns out the diagonal Cartan subgroup has precisely 5 limits, and for n = 4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n ≥ 7, there is a continuum of non conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers.

This first talk will focus on n=3 and hyperreal techniques. It should be very accessible.

*Abstract:*

The goal of these two talks is to explain a result concerning the quasiconformal properties of the boundary of right-angled hyperbolic buildings.

In this second talk I will describe the geometry of right-angled buildings and explain how, under some good geometric assumptions, we can prove that the boundary of such a building satisfy the combinatorial Loewner Property (CLP).

Throughout both talks I will insist on some geometric ideas and examples in order to avoid technicality.

*Abstract:*

The goal of these two talks is to explain a result concerning the quasiconformal properties of the boundary of right-angled hyperbolic buildings.

In this first talk I will recall classical questions, conjectures and results that link the quasiconformal structure of the boundary of a hyperbolic space to rigidity phenomenon inside the space. Some basic tools of this theory, such as the conformal dimension, the Loewner property and the Combinatorial Loewner Property (CLP), will be introduced and explained.

Throughout both talks I will insist on some geometric ideas and examples in order to avoid technicality.