Seminar organizers: Nima Hoda, Kasia Jankiewicz, Piotr Przytycki, Daniel Wise

A Bi-Lagrangian structure in a manifold is the data of a symplectic form and a pair of transverse Lagrangian foliations. Equivalently, it can be defined as a para-Kähler structure, i.e. the para-complex equivalent of a Kähler structure. After discussing interesting features of bi-Lagrangian structures in the real and complex settings, I will show that the complexification of any Kähler manifold has a natural complex bi-Lagrangian structure. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which typically have a rich symplectic geometry. We will see that that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory while revealing new other features, and derive a few well-known results of Teichmüller theory. Time permits, I will present the construction of an almost hyper-Kähler structure in the complexification of any Kähler manifold. This is joint work with Andy Sanders.

Graph products of groups are a construction that interpolates between direct products and free products, and contain well-known examples such as right-angled Coxeter groups and right-angled Artin groups. In this talk, I will present a form of rigidity for the automorphism group of `cyclic products of groups', that is, the graph products over cycles on at least 5 vertices. I will recall a construction due to Davis that allows us to understand graph products through their action on CAT(0) cube complexes, and explain how this action extends to the whole automorphism group in the case of cyclic products of at least 5 groups. Such an action can be used to completely compute their automorphism group and to show their acylindrical hyperbolicity. This is joint work with Anthony Genevois.

Consider a residually finite group, then given a filtration we can construct the box space of this group with respect to the chosen filtration. It is a known result that the box space having property A is equivalent to the group being amenable. The same is true for hyperfiniteness. Firstly we will optimise this result by replacing filtrations by Farber sequences and secondly determine which arrows still hold when we replace box spaces by sofic approximations. Knowledge of the above concepts is not required since these will be introduced during the talk.

The Cremona group is the group of birational symmetries of the plane. This is a rather huge group, which is neither finitely generated nor finite dimensional. The abelianization of the Cremona group was recently studied: over the base field $\mathbb{C}$, the abelianization is trivial, but over $\mathbb{R}$, the abelianization is an uncountable direct sum of $\mathbb{Z}/2$. I will discuss similar results over more general fields, such as number fields or finite fields. The key is to use a natural action on a square complex, to exhibit various trees as quotients or subcomplexes, and then to apply Bass-Serre theory (joint work with Susanna Zimmermann).

I will explain Muehlherr's Twist Conjecture describing the hypothetical solution to the isomorphism problem in Coxeter Groups. I will mention recent progress in verifying it which is joint work with Jingyin Huang.

A graph is *geocyclic* if there is an upper bound on the
lengths of its isometrically embedded cycles. The 1-skeleta of
many combinatorially nonpositively curved spaces are geocyclic. I
will present these examples and explain why a finitely generated
group admitting a geocyclic Cayley graph is finitely presented and
has solvable word problem. Time permitting, I will describe a
strengthened form of geocyclicity and use it to prove that every
Cayley graph of $\mathbf{Z}^2$ is geocyclic. This is joint work
with Daniel Wise.

We prove that if $G$ is a finitely generated non-virtually-abelian group, then $(G \times \mathbb{Z}) \ast \mathbb{Z}$ does not embed into $\operatorname{Diff}^2(S^1)$. In particular, the class of subgroups of $\operatorname{Diff}^r(S^1)$ is not closed under taking free products for each $r \ge 2$. We complete the classification of RAAGs embeddable in $\operatorname{Diff}^r(S^1)$ for each $r$, answering a question in a paper of M. Kapovich. (Joint work with Thomas Koberda)

A tubular group is a group that splits as a graph of groups where all edge groups are $\mathbf{Z}$ and all vertex groups are $\mathbf{Z}\times\mathbf{Z}$. There are many interesting examples of tubular groups exhibiting subtle and pathological behaviour. In my talk I will characterize the f.g. tubular groups that are residually finite. This is (ongoing) joint work with Daniel Woodhouse.

There are several known constructions of actions as in the title (Yu, Nica, Bourdon, Alvarez - Lafforgue). I will discuss a new construction which, apart from a technical point, is particularly simple.

Sageev gave a very general construction of a CAT(0) cube complex dual to a `space with walls', and this construction has proved extraordinarily useful in recent celebrated work of Agol, Wise, and others. In one of the simplest nontrivial settings, this construction produces a non-positively curved cube complex dual to a finite collection of non-homotopic essential closed curves on a surface. I will show how this cube complex can be used to analyze the length function associated to a system of curves on the moduli space of hyperbolic structures on a surface of genus g, adding context to previous work of Basmajian.

A non-positively curved (NPC) cube complex is a combinatorial complex constructed by gluing Euclidean cubes along faces in a way that satisfies a combinatorial local non-positive curvature condition. A hierarchy is an inductive method of decomposing the fundamental group of a cube complex. Cube complexes and hierarchies of cube complexes have been studied extensively by Wise and feature prominently in Agol's proof of the Virtual Haken Conjecture for hyperbolic 3-manifolds. In this talk, I will give an overview of the geometry of cube complexes, explain how to construct a hierarchy for a NPC cube complex, and discuss applications of cube complex hierarchies to hyperbolic and relatively hyperbolic groups.

Small cancellation theory and simplicial nonpositive curvature are two combinatorial approaches to the study of nonpositively curved groups. The first theory enables one to construct numerous examples of groups, while the second one leads to many properties of ``metric flavour'', e.g., biautomaticity, existence of an EZ--structure, fixed point theorem. In this talk I shall briefly discuss these two theories and show how all small cancellation groups can be seen as groups of simplicial nonpositive curvature. Our construction, based on ideas of D. Wise, also carries through for graphical small cancellation groups. This is joint work with Damian Osajda.

I will discuss a variety of groups in the extended family of Thompson's groups, some natural CAT(0) cube complexes on which they act, and the Morse theoretic tools that lead to understanding their finiteness properties. This will span various projects of mine, often joint with combinations of Kai-Uwe Bux, Martin Fluch, Marco Marschler, Lucas Sabalka, Rachel Skipper, and Stefan Witzel. Prior knowledge of Thompson's groups will not be assumed.

This will consist of two lectures, the first by Wise and the second by Li on their joint work.

Let be G the fundamental group of a compact special cube complex. We show that there is a sequence of infinite index quasiconvex subgroups whose growth-rates converge to the growth-rate of G.

In the first talk, we will discuss growth-rates, prove the result in the case when G is a free group (work of Dahmani-Futer-Wise), sketch a proof of the general case. In the second talk, we will give details of the proof, including the automatic structure that it relies upon and the method for using automata to estimate growth-rates. We will also survey a collection of problems that are motivated by this work.

We will discuss the quasi-isometric rigidity of 3-manifold groups: A finitely generated groups that roughly (when viewed from far away) looks like the fundamental group of a compact 3-manifold contains a finite index subgroup isomorphic to the fundamental group of a compact 3-manifold. This a joint work with Peter Haissinsky.

This will consist of two lectures, the first by Wise and the second by Li on their joint work. Let be G the fundamental group of a compact special cube complex. We show that there is a sequence of infinite index quasiconvex subgroups whose growth-rates converge to the growth-rate of G. In the first talk, we will discuss growth-rates, prove the result in the case when G is a free group (work of Dahmani-Futer-Wise), sketch a proof of the general case. In the second talk, we will give details of the proof, including the automatic structure that it relies upon and the method for using automata to estimate growth-rates. We will also survey a collection of problems that are motivated by this work.

Classical translation surfaces can be obtained from gluing finitely many polygons along parallel edges of the same length. In recent years, people have asked what happens when you glue infinitely instead of finitely many polygons. From that question the field of infinite translation surfaces has evolved. It turns out that the behaviour of infinite translation surfaces is in many regards very different and more diverse than in the classical case. This includes that for classical translation surfaces, we have a Gauß–Bonnet formula which relates the cone angle of the singularities (coming from the corners of the polygons) to the genus of the surface. For infinite translation surfaces, we might observe so-called wild singularities for which the notion of cone angle is not applicable any more. In this talk, I will explain that there is still a relation between the geometry and the topology of infinite translation surfaces in the spirit of a Gauß-Bonnet formula. In fact, under some weak conditions, the existence of a wild singularity implies infinite genus.

Codimension 1 subjects such as surfaces play an essential role in the study of 3-manifolds. To some extent, their exsitence are "dual" to the actions of the 3-manifold groups on a dimension-1 space (e.g. a tree, R-tree ... ). In this talk, we will focus on the interconnection between the existence of co-dimension 1 foliations on a 3-manifold and the existence of actions of its fundamental group on certain simply connected 1-manifolds. Our discussion will include a seminal work of Roberts-Shareshian-Stein and my recent joint work with Steve Boyer on this topic.

In recent years one of the main tool in the study of the Cremona group is an action on an infinite dimensional hyperbolic space. In particular this was the key to get a Tits alternative for this group, and also for a proof of the non simplicity. My aim in this talk will be to give a gentle introduction to this construction, which relies on the notion of blowing-up a point in an algebraic surface (infinitely many times!).

The cubical dimension of a group $G$ is the infimum $n$ such that $G$ acts properly on an $n$-dimensional $\mathop{\rm CAT}(0)$ cube complex. In my talk I will discuss small cancellation groups and give a construction that for each $n$ provides an example of a $\mathop{\rm C}'(1/6)$ group with the cubical dimension greater than $n$.

In this talk we answer questions regarding the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop α, the length of α along a balanced folding path is not larger than the maximum of its lengths at the end points. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counterexamples.

There are various interesting statements that boil down to passing from an action of a group $G$ on a $\mathop{\rm CAT}(0)$ cube complex to an action on a tree. These include Stallings' ends theorem, the Nielsen realisation theorem for $\mathop{\rm Out}(F_n)$, and the Kropholler-Roller conjecture about almost-invariant subsets. I will discuss a method for converting a $G$-action on a $\mathop{\rm CAT}(0)$ cube complex $X$ to a $G$-action on a “lower complexity” $\mathop{\rm CAT}(0)$ cube complex $Y$ and describe conditions under which this can be used inductively to find a splitting of $G$. This leads to new proofs of the first two of the preceding statements, as well as a special case of the Kropholler-Roller conjecture. I will also briefly describe some possible generalisations of the construction. Most of this talk is on joint work with Nicholas Touikan; it will also touch on some joint work with Henry Wilton.

Let $F$ be a free group. If $H$ and $K$ are two finitely generated subgroups of $F$, then their intersection is also finitely generated. The Hanna Neumann Conjecture (HNC) is a statement relating the ranks of $H$ and $K$ to the rank of their intersection. It was posed by Hanna Neumann in 1957 and proved in 2011 by Joel Friedman and independently by Igor Mineyev. Both proofs were subsequently simplified by Warren Dicks. I will present the Mineyev-Dicks version of the proof.

A subgroup $H$ of a left ordered group $(G,\le)$ is $\le$-convex if for any $x,z \in H$ and $y \in G$ the inequalities $x \le y \le z$ imply $y \in H$. I will show that the family of $\le$-convex normal subgroup can be finite of arbitrary size bigger than 1, countably infinite, or of cardinality continuum. I will also point out there is no countable universal left-orderable group.

In the context of proving that the mapping class group has finite asymptotic dimension, Bestivina-Bromberg-Fujiwara exhibited a finite coloring of the curve graph, i.e. a map from the vertices to a finite set so that vertices of distance one have distinct images. In joint work with Josh Greene and Nicholas Vlamis we give more attention to the minimum number of colors needed. We show: The separating curve graph has chromatic number coarsely equal to $g \log(g)$, and the subgraph spanned by vertices in a fixed non-zero homology class is uniquely $(g-1)$-colorable. Time permitting, we discuss related questions, including an intriguing relationship with the Johnson homomorphism of the Torelli group.

The 3-dimensional tame automorphism group $\mathop{\rm Tame}(k^3)$ is the group of automorphisms of $k^3$ generated by affine maps and maps of form $(x,y,z) \to (x + P(y,z), y, z)$ for $P$ a polynomial in $k[y,z]$. We construct a simply-connected non-positively curved space with an action of $\mathop{\rm Tame}(k^3)$. That leads to the classification of finite subgroups of $\mathop{\rm Tame}(k^3)$. This is joint work with Stéphane Lamy.

Pick two (multi)-curves on a surface. What can be said about the subgroup generated by powers of Dehn twists about the two curves? For twists about single curves a complete classification up to isomorphism is known and is determined by intersection number and twist power. For multi-curves with more components, the fourth powers of the twists generate a free group. I will present analogous results for groups generated by two Dehn (multi-)twists in $\mathop{\rm Out}(F_r)$, where the analysis is complicated by the absence of a standard topological model.

In this talk I will be presenting some preliminary results discussing a new notion of curvature for groups. Motivated by Ollivier's interpretation of Ricci curvature in terms of optimal transport, we define a combinatorial analogue of Ricci curvature, and show some elementary properties. This talk will mostly be about exploring the new idea, and discussing some of its quirks and anomalities with respect to more well-known groups and group constructions. This is joint work with M. Duchin and R. Kropholler.

In this talk, I will survey the notion of Garside families in relation with the word problem for Artin-Tits groups. In particular, Dehornoy, Dyer and I proved that there exist a finite Garside family in any Artin-Tits group $G$. Then I will discuss how this family is (surprisingly) related to a well-known finite hyperplane sub-arrangement, the Shi arrangement, of the reflection arrangement of the Coxeter group associated to $G$.

The mapping class group of a compact surface $S$, denoted $\mathop{\rm MCG}(S)$, is the group of isotopy classes of elements of $\mathop{\rm Homeo}^{+}(S, \partial S)$, where isotopies are required to fix the boundary pointwise. The complex of curves, $C(S)$, is a simplicial complex where the vertices are free isotopy classes of essential simple closed curves in $S$ and there is an edge connecting two vertices if the isotopy classes have geometric intersection number zero. Choosing a generating set defines a word metric on the mapping class group. By using the combinatorial properties of $C(S)$ along with the action of $\mathop{\rm MCG}(S)$ on $C(S)$ we are able to study the geometry of the mapping class group.

Previously, Masur and Minsky constructed quasi-geodesics in $\mathop{\rm MCG}(S)$, but not much is known about geodesics. In joint work with Kasra Rafi, we have found some explicit examples of geodesics, denoted $\gamma$, in $\mathop{\rm MCG}(S_{0,5})$, where $S_{0,5}$ is the five-times punctured sphere. We were able to find these geodesics by having constructed an appropriate generating set and having found a homomorphism from $\mathop{\rm MCG}(S_{0,5})$ to $\mathbb{Z}$. In addition, we have constructed a pseudo-Anosov map whose axis is not strongly contracting. A geodesic is strongly contracting if its nearest point projection takes disjoint balls from the geodesic to sets of bounded diameter, where the bound should be independent of the ball. Strongly contracting geodesics show up in the study of growth tightness, for example, the strongly contracting property was used to prove growth tightness for actions on non relatively hyperbolic spaces in work by G. Arzhantseva, C. Cashen, J. Tao, which is why we pursued this result.