Long-distance seminar on Geometric Group Theory in Mexico

Let Fn be a finitely generated free group, and let Out(Fn) be its outer automorphism group. In 2007, Farb and Handel proved that when n is at least 4, Out(Fn) is equal to its own abstract commensurator: in plain words, this means that every isomorphism between two finite-index subgroups of Out(Fn) is given by the conjugation by some element of Out(Fn). This can be viewed as a Mostow-type rigidity theorem for automorphisms of free groups. In a joint work with Ric Wade, we give a new proof of Farb and Handel's theorem, which allows us to extend it to the case where n=3, and to understand the abstract commensurator of some interesting normal subgroups of Out(Fn). In my talk, after reviewing the analogies with other more classical settings, I will explain the strategy of our proof.

Los espacios de Alexandrov son espacios métricos con curvatura seccional acotada inferiormente (en un sentido sintético) que generalizan a las variedades riemannianas completas de curvatura seccional acotada por debajo. 

En esta plática hablaré de la clasificación topológica de los espacios de Alexandrov de dimensión 3 que admiten una acción por isometrías del círculo y mencionaré algunas aplicaciones. Procederé a mencionar una generalización al caso de las acciones locales del círculo y su relación con el fenómeno del colapso (en el sentido de Gromov-Hausdorff) en la geometría de Alexandrov tridimensional. Los resultados aquí mencionados forman parte de un trabajo conjunto con Fernando Galaz-García y Luis Guijarro.

The Alexandrov spaces are metric spaces with sectional curvature bounded below (in a synthetic way) that generalises complete riemannian manifolds with sectional curvature bounded below.

In this talk I will talk about the topological classification of 3-dimensional Alexandrov space that admit an isometric action on the circle, and will mention some of its applications. Afterwards I will mention a generalisation to the case of local actions on the circle and its relation with the collapse phenomenom (in the Gromov-Hausdorff sense) in the 3-dimensional Alexandrov geometry. The result here mentioned are part of a joint work with Fernando Galaz-García and Luis Guijarro.

En esta plática nos enfocaremos en acciones de grupos en espacios (Gromov-)hiperbólicos y estructuras hiperbólicas (acilíndricamente hiperbólicas) de grupos. Esto enfocado a estudiar equivalencias de acciones en estos espacios métricos, y daremos una visión superficial de la prueba de que equivalencia gruesamente isoespectral es lo mismo que equivalencia por cuasi-isometrías gruesamente equivariantes. Obviamente empezaré dando las definiciones y resultados necesarios para entender lo arriba mencionado.

In this talk we will focus on group actions on (Gromov-)hyperbolic spaces and the (acylindrically) hyperbolic structures of groups. All this focused on studying the equivalence classes of actions on these metric spaces, and we will give a superficial view of the proof that coarsely isospectral equivalence is the same as coarsely-equivariant quasi-isometric equivalence. Obviously, we will start by giving the necessary definitions and results to understand the above-mentioned.

In order to prove the ending lamination conjecture for hyperbolic 3-manifolds, Minsky, Brock-Canary-Minsky, Brock-Bromberg and others related the geometry of a hyperbolic 3-manifold manifold to the combinatorics of the curve complex. Many of these theorems involve geometric limits and compactness arguments which make them very difficult to quantify effectively. We prove a theorem relating electric distance in a hyperbolic 3-manifold to curve complex distance, in a completely effective way that circumvents limiting arguments. As an application, we prove a quantitative version of a theorem of Rafi-Schleimer: a covering map between surfaces of negative Euler characteristic induces a quasi-isometric embedding of curve complexes, with constants bounded by explicit polynomials in the Euler characteristic of the base surface and the degree of the cover. This represents joint work with Priyam Patel and Sam Taylor.

Graph products of groups propose a common generalisation of free and direct products. In this talk, I will describe a geometric model to study groups which decompose as graph products. Various applications will be mentioned and explained.

We will talk about free groups of finite rank Fn and their group of symmetries, the group of outer automorphisms Out(Fn). The study of Out(Fn) is motivated by the study of the general linear group and the mapping class group of a surface. I will give an overview of the key geometric methods used to study Fn and Out(Fn). I will also mention some open problems.

A group is said to have a fixed point property if every isometric action of the group on a class of metric spaces has a fixed point. For the class of simplicial trees or CAT(0) cell complexes, fixed point properties are studied for many groups, since such properties are related to combinatorial structures or representations of groups. In this talk, we consider a fixed point property for Busemann spaces, which is a class of spaces containing trees and CAT(0) spaces. We construct a sufficient condition for groups to have a fixed point property for finite dimensional Busemann spaces. As an application, we show that Richard Thompson’s group T has fixed point properties for finite dimensional Busemann spaces, but not for infinite dimensional ones.

We study the mapping class group action on the curve complex and discuss the asymptotic translation length. Conjecturally, the asymptotic translation length can shed some light on the structure of normal subgroups of mapping class groups, and we report on some partial progress.

I will give a general introduction to right-angled Artin groups and their role in geometric group theory. I will illustrate more detailed interactions with mapping class groups and with other right-angled Artin groups, and their applications to group actions on the circle.

SL(2,Z) is a famous group:  it is, up to index 2,  the automorphism group of Z², the outer automorphism group of the free group F2, it acts on the modular tree in the hyperbolic plane, in such a way that it splits as a free product with amalgamation of Z/4Z and Z/6Z (over Z/2Z). Its infinite order elements exhibit some hyperbolic behavior. And it is rather easy to determine whether two of these elements are conjugate in SL(2,Z). For other automorphism groups, the situation can be more diverse. We will discuss automorphisms of free groups of higher rank, in particular a theorem of Brinkmann characterising hyperbolicity of the semi-direct product of Fn with Z, generalised by statement first stated by Gautero and Lustig, and to which Ghosh, and Li and myself provided a proof. In a number of cases this allows to propose an algorithm solving the conjugacy problem among certain outer-automorphisms of free groups.

I will discuss various recent and not-so recent results that demonstrate the "particular beauty" of arithmetic congruence manifolds among the family of all hyperbolic manifolds of finite volume in low dimensions (2 and especially 3). In particular I will talk about when cusped arithmetic manifolds can be link complements.

Un grupo contable es ordenable si tiene un orden que es invariante por multiplicacion a izquierda. Otra caracterizacion es que no tienen una accion en \R por homeomorfismos. Cuantos ordenes posee un grupo contable?, tratare de explicar como uno puede hacer sentido de esta pregunta. Si el tiempo permite, explicare una estrategia para mostrar que reticulas de grado superior (por ejemplo ) no son ordenables. (Junto con Bertrand Deroin).

A countable group is orderable if it has an order which is invariant under left multiplication. Another characterisation is that they have an action on R by homeomorphisms. How many orders does an orderable group have? I will try to explain how one can make sense of this question. If the time allows it, I will explain a strategy to prove that higher degree lattices (for example) are not orderable. Joint work with Bertrand Deroin.

Todo grupo G tiene asociado su espacio de casi-morfismos QM(G). Cuando el grupo G actúa en un espacio métrico X por isometrías (con buenas propiedades) es posible calcular la dimensión de QM(G) estudiando la acción del grupo sobre la frontera de Gromov de X. En esta plática intentaremos motivar lo anterior en el caso de grupos modulares tanto de superficies de tipo finito como de tipo infinito. Al final revisaremos algunas de las preguntas abiertas.

Every group G has space of quasi-morphisms QM(G) associated to it. When the group G acts on a metric space X by isometries (with good properties) it is possible to compute the dimension of QM(G) studying the group action on the Gromov boundary of X. In this talk we will try to motivate the above-mentioned with the case of mapping class groups of finite- and infinite-type surfaces. At the end we will review some open questions.

Dado un grupo de Artin-Tits de tipo esférico A, un subgrupo parabólico P de A y dos elementos x, y en P conjugados en A, ¿es cierto que x e y son conjugados en P? Probamos que la respuesta es positiva en la mayoría de los casos (reencontrando en particular el resultado para grupos de trenzas, demostrado por González-Meneses en 2000) y describimos todos los casos en donde el resultado falla. Es un trabajo conjunto con Bruno Cisneros (Oaxaca) y María Cumplido (Dijon).

Given an Artin-Tits group A of spherical type, a parabolic subgroup P of A and two elements x and y in P which are conjugates in A, is it true that x and y are also conjugates in P? We prove that the answer is positive in most cases (rediscovering in particular the result for the braid groups, by González-Meneses in 2000) and we describe all the cases where the result fails. This is a joint work with Bruno Cisneros (Oaxaca) and María Cumplido (Dijon).

En esta plática daremos un vistazo introductorio a lo que es la teoría de Bass-Serre, es decir, al estudio de grupos a través de su acción en árboles. Empezaremos desde cero e iremos construyendo las herramientas necesarias, desde productos amalgamados y extensiones HNN, hasta grafos de grupos y espacios. Si el tiempo lo permite, veremos algunos resultados importantes que han sido obtenidos a través de la teoría de Bass-Serre.

In this talk we will give an introductory view to Bass-Serre theory, that is, to the study of groups through their actions on trees. We will start from zero and will build up the necessary tools, from amalgamated products and HNN extensions, to graph of groups and spaces. If the time allows it, we will see some important results that have been obtained through Bass-Serre theory.

En esta charla definiremos dimensión geométrica y dimensión cohomológica para familias. Después enunciaremos el Teorema de Eilenberg-Ganea para familias demostrado por Lück y Meintrup. Dicho teorema dice que ambas dimensiones coinciden siempre que alguna de ellas sea al menos 3. La conjetura de Eilenberg-Ganea dice que ambas dimensiones siempre coinciden. A continuación discutiremos algunos ejemplos de grupos construidos por Brady, Leary y Nucinkis de grupos que tienen dimensión cohomológica 2 y dimensión geométrica 3 respecto de la familia de subgrupos finitos. Finalmente discutiremos cómo, usando estos ejemplos y teoría de Bass-Serre, se pueden construir más contraejemplos a la conjetura de Eilenberg-Ganea para otras familias.

In this talk we will define the geometric and cohomological dimensions for families. Then we will state the Eilenberg-Ganea Theorem for families proved by Lück and Meintrup. Said theorem states that both dimensions coincide so long as one of them is at least three. The Eilenberg-Ganea conjecture says that both dimensions always coincide. We will discuss some examples built by Brady, Leary and Nucinkis of groups with cohomological dimension 2 and geometric dimension 3 with respect to the family of finite subgroups. Finally, we will discuss how, using these examples and Bass-Serre theory, more counterexamples can be built for the Eilenberg-Ganea conjecture for other families.

Explicaremos cómo una generalización de la construcción de Thurston nos permite construir elementos loxodrómicos de peso arbitrario que no dejan invariante una subsuperficie de tipo finito. En particular respondemos una pregunta abierta de Bavard & Walker. Trabajo en colaboración con Israel Jiménez. 

We will explain how a generalisation of Thurston's construction allows us to construct loxodromic elements with arbitrary weight that do not leave any finite-type subsurface invariant. In particular we will answer an open question of Bavard & Walker. Joint work with Israel Jiménez.

El primer ejemplo de un grupo finitamente generado pero sin cocientes finitos es el grupo de Higman H. Este grupo tampoco admite representaciones lineales, pues los grupos de matrices finitamente generados siempre admiten muchos cocientes finitos. En esta charla discutiremos sobre representaciones no lineales de H y también sobre sus casi-representaciones.

The first example of a finitely generated group without finite quotients is the Higman group H. This group does not admit linear representations, since finitely generated matrix groups always admit a lot of finite quotients. In this talk, we will discuss about non-linear representations of H and about its quasi-representations.

En esta charla explicaré cómo generalizar una construcción clásica de Thurston a superficies de tipo infinito. Hablaremos de aplicaciones de dicha construcción.

In this talk I will explain how to generalise the classical construction of Thurston to infinite-type surfaces. We will also talk about applications of such construction.

The curve graph of a finite type surface is a crucial tool for understanding the algebra and geometry of the corresponding mapping class group. Many of the applications that arise from this relationship rely on the fact that the curve graph is hyperbolic. We will describe actions of mapping class groups of infinite type surfaces on various graphs analogous to the curve graph. In particular, we will discuss the hyperbolicity of these graphs, some of their quasiconvex subgraphs, dynamical properties of the corresponding actions, and applications to bounded cohomology.

Se analizará  si son perfectos, simples, uniformemente perfectos y uniformemente simples los grupos de "intercambio de intervalos", "intercambio de intervalos afines" e "intercambio de intervalos con flip". Se definirán todos los conceptos mencionados.

We will analyse if the interval exchange transformation/affine interval exchange transformation/interval exchange and flip transformations groups are perfect, simple, uniformly perfect and uniformly simple. All the above-mentioned concepts will be defined in the talk.

En los últimos años el estudio de las superficies de tipo infinito ha ido en aumento, con la gran mayoría de los resultados siendo para superficies orientables y habiendo un fuerte enfoque en el grupo modular de estas superficies (llamados big mapping class groups). Entre estos resultados, Aramayona, Patel y Vlamis calcularon el primer grupo de cohomología del big mapping class group de una superficie orientable de género suficientemente grande. En esta plática veremos como obtener el resultado análogo para big mapping class groups de superficies no orientables; este resultado no es trivial pues si bien varias de sus técnicas pueden adaptarse para el caso no orientable, hay resultados clave que ellos usan en donde las técnicas para demostrarlos simplemente no pueden aplicarse a superficies no orientables.

In the last few years the study of infinite-type surfaces has grown, with the majority of results being for orientable surfaces and heavily-focused in the mapping class groups of these surfaces (called big mapping class groups). Among these results, Aramayona, Patel and Vlamis computed the first cohomology group of the big mapping class group of a surface with large-enough genus. In this talk we will see how to obtain the analogous result for big mapping class groups of non-orientable surfaces; this result is highly non-trivial given that while some of their techniques can be adapted to the non-orientable case, there are key results they use where the techniques to prove them simply cannot be applied to non-orientable surfaces.

Given a finite simplicial graph G, the right-angled Artin group (RAAG) associated to it is the group generated by the vertices of G, in which two generators commute when the corresponding vertices are joined by an edge. Bestvina and Brady considered the natural homomorphism that sends each generator to the integer number 1, and showed that its kernel can display some exotic finiteness properties depending on the topology of the graph G. In this talk we consider generalized Bestvina-Brady groups, i.e. kernels of arbitrary group homomorphisms to the integers. We show how a decomposition of the graph G always induces an explicit decomposition of these kernels as graphs of groups. As an application we will discuss a tame-vs-wild dichotomy for the generalized Bestvina-Brady groups defined over chordal graphs, as well as an explicit rank formula in the case of block graphs (e.g. trees), which we have used to construct examples of exotic behavior in the Bieri-Neumann-Strebel invariant of the corresponding RAAG. This is joint work with M. Barquinero and K. Ye.

Given a compact, connected, orientable surface, we can define many associated graphs whose vertices represent curves or multicurves in the surface. A first example is the curve graph, which has a vertex for every simple closed curve in the surface and an edge joining two vertices if the corresponding curves are disjoint. A key property of the curve graph is that it is Gromov hyperbolic, but this is not the case for all such graphs. I will introduce some graphs of multicurves, and, focusing on the example of the separating curve graph, I will present joint work with Jacob Russell classifying for which surfaces this graph is hyperbolic, for which it is relatively hyperbolic (a generalisation of hyperbolicity), and for which it is neither of these.

La propiedad T de Kazhdan es una propiedad de ciertos grupos topológicos, que se enuncia en términos de sus representaciones unitarias en espacios de Hilbert. Algunas de sus consecuencias es la generación compacta de grupos, generación finita de ciertos latices en grupos LCH, y cumplimiento de la propiedad FA para grupos discretos numerables. Esto último se verifica con la propiedad FH, la cual se da en términos de representaciones en espacios de Hilbert afines. En la presente platica hablaremos de estas dos propiedades, algunas de sus consecuencias (tanto en la Teoría de Grupos como en otras áreas de la matemática) y la relación entre estas dos mediante el teorema de Delorme-Guichardet.

Kazhdan's property (T) is a property of certain topological groups that can be stated in terms of unitary representations on Hilbert spaces. Some of its consequences is the compact generation of groups, finite generation of certain lattices in LCH groups, and the satisfaction of property FA for discrete countable groups. The latter is verified with the property FH, which is given in terms of representations on affine Hilbert spaces. In this talk we will speak about these two properties, some of its consequences (both in group theory and in other areas of mathematics) and the relation between these two due to the Delorme-Guichardet theorem.

En esta charla daremos una breve introducción a grupos de Artin, los cuales se definen mediante grupos de Coxeter; veremos algunas propiedades y sus representaciones geométricas vistas como  homomorfismos en "mapping class groups". 

In this talk we will give a brief introduction to Artin groups, which are defined via Coxeter groups. We will see some of their properties and geometric representations seen as homomorphisms into mapping class groups.

Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a universal version of these conjectures (and prove the existence part) by considering convex sets spanning quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. This work generalizes Alexandrov and Pogorelov’s results about the characterization metrics induced on the boundary of a compact convex subset of hyperbolic space. Time permitting, we will discuss why in hyperbolic space quasicircles can't be characterized by the width of their convex hulls, except when the convex hulls have small width. This is different than the anti-de Sitter setting, as Bonsante and Schlenker showed. (This is joint work with Bonsante, Danciger and Schlenker.) 

If S is a surface of finite-type and R is a subsurface, then R and S are homeomorphic only when the inclusion map is homotopic to the identity.  To prove this fact, one only needs to count the genus or number of punctures on the surface.  For surfaces of infinite-type, we will show that more "interesting" homeomorphic subsurfaces can occur.  In exploring this, we are led naturally to a subclass of arcs and a new graph on which many big mapping class groups act.

It is a classical theorem that every finite group can be realized as the isometry group of an orientable, closed hyperbolic surface.  This result was extended by Allcock who showed that every countable group can be realized as the isometry group of an orientable hyperbolic surface.  Allcock's construction does not concern itself with the underlying topology of the surface, which leads us to ask: given a topological surface S, provide a classification of the groups that are realized as the isometry group of some hyperbolic structure on S.  In general, this question is too difficult; however, there is a class of surfaces—the infinite-genus surfaces with no planar ends—in which we can provide a nearly complete answer.   I will discuss the solution in this setting and provide an application to mapping class groups.  This is joint work with Tarik Aougab and Priyam Patel.

The properties of a random walk on a group which acts on a hyperbolic metric space have been well-studied in recent years.  In this talk, I will focus on random walks on acylindrically hyperbolic groups, a class of groups which includes mapping class groups, Out(F_n), and right-angled Artin and Coxeter groups, among many others.  I will discuss how a random element of such a group interacts with fixed subgroups, especially so-called hyperbolically embedded subgroups.  In particular, I will discuss when the subgroup generated by a random element and a fixed subgroup is a free product, and I will also describe some of the geometric properties of that free product. This is joint work with Michael Hull.

Los grupos hiperbólicos se caracterizan por sus propiedades isoperimétricas: un grupo es hiperbólico si y solo si posee una función isoperimétrica lineal. Esto es, para un 2-complejo compacto X, la hiperbolicidad de su grupo fundamental tiene que ver con la relación que existe entre la longitud de curvas cerradas nullhomotópicas y el área mínima de los discos  – o "diagramas"-- que estas bordean en X.
Es posible definir otros tipos de funciones isoperimétricas – anulares, homológicas – que involucran otros tipos de "diagramas'' y proporcionan información interesante, tanto para grupos hiperbólicos como para otras clases de grupos.
En esta charla explicaré todas estas nociones y les contaré sobre una nueva familia de funciones isoperimétricas.
Este es trabajo en conjunto con Dani Wise.

Hyperbolic groups are characterised by their isoperimetric properties: A group is hyperbolic if and only if it admits a linear isoperimetric function. That is, given a compact 2-complex X, hyperbolicity of its fundamental group is related to the length of null-homotopic closed curves and the minimal area of discs -- or "diagrams" -- that they border in X.
It is possible to define other type of isoperimetric functions -- annular, homological -- that involve other type of "diagrams" and give us interesting information, both of the group and of other classes of groups.
In this talk I will explain all these notions and I will tell you about a new family of isoperimetric functions.
This is joint work with Dani Wise.

Dos grupos son cuasi-isométricos si tienen la misma geometría gruesa. En esta charla introduciremos lo que significa que dos grupo sean cuasi-isométricos relativo a dos colecciones de subgrupos. Luego definiremos lo que significa que una colección de subgrupos sea característica. Finalmente veremos un fenómeno de rigidez que se logra apreciar en este contexto, es decir, que dada una colección característica de G y una cuasi-isometría de G en H, existe una colección característica Q de H tal que las parejas de grupos con sus colecciones son cuasi-isométricas . Este es trabajo conjunto con Eduardo Martínez-Pedroza.

Two groups are quasi-isometric if they have the same coarse geometry. In this talk we introduce the meaning of two groups being quasi-isometric relative to two collections of subgroups. Then we define what a collection being characteristic means. Finally, we see a rigidity phenomenon that can be appreciated in this context, that is, given a characteristic collection of G and a quasi-isometry from G to H, there exists a characteristic collection Q of H such that these two groups with their respective collections are quasi-isometric. This is joint work with Eduardo Martínez-Pedroza.

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we prove certain graphs of manifold groups are action rigid. Consequently, we obtain examples of quasi-isometric groups that do not virtually have a common model geometry. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

Motivados por los grupos hiperbólicos y las propiedades que estos satisfacen, nos adentraremos a estudiar un concepto que relaja la definición de dichos grupos pero que conserva algunas de las propiedades que estos poseen; dicho concepto es el de grupos relativamente hiperbólicos. En esta sesión del seminario de distancia de Teoría Geométrica de Grupos hablaremos sobre los grupos relativamente hiperbólicos: su definición, cómo es que generalizan a los grupos hiperbólicos, la definición de su frontera y porqué es que ésta está bien definida.

Motivated by hyperbolic groups and the properties these satisfy, we study a concept that relaxes the definition of said groups but keeps several of their properties; said concept is that of relatively hyperbolic groups. In this session of the long-distance seminar in Geometric Group Theory we will talk about relatively hyperbolic groups: their definition, how these groups generalise hyperbolic groups, the definition of their boundary and why is this boundary well-defined.

The talk will center around a theorem of Gersten that states that finitely presented groups of hyperbolic groups of cohomological dimension at most two are hyperbolic. We will discuss different generalizations, some questions, and some of the tools involved.

Right-angled Artin groups (RAAGs) span a range of groups from free groups to free abelian groups.  Thus, their (outer) automorphism groups range from Out(F_n) to GL(n,Z).  Automorphism groups of RAAGs have been well-studied over the past decade from a purely algebraic viewpoint.  To allow for a more geometric approach, one needs to construct a contractible space with a proper action of the group.  I will talk about joint work with Bregman and Vogtmann in which we construct such a space, namely, an analogue of Culler-Vogtmann's Outer Space for arbitrary RAAGs.

Given a group Γ, we say a collection 𝓕 of subgroups of Γ is a family if it is non-empty, closed under conjugation and under taking subgroups. Fixing a group Γ and a family 𝓕 of subgroups of Γ, we say that a Γ-CW-complex X is a model for the classifying space  E_{𝓕}Γ if every isotropy group of X belongs to the family 𝓕 and the fixed point set X^H is contractible whenever H belongs to 𝓕. It can be shown that a model for the classifying space E_{𝓕}Γ always exists and it is unique up to Γ-homotopy equivalence.
We define the 𝓕-geometric dimension of Γ, denoted as gd_{𝓕}(Γ), as the minimal dimension of the models for the classifying space E_{𝓕}Γ.
Now, let Γ be the fundamental group of a 3-manifold. Define the family 𝓕_n as the family of virtualy Z^r subgroups for 0 ≤ r ≤ n.
In joint work with Luis Jorge Sánchez Saldaña we computed gd_{𝓕_n}(Γ) for all n≥2. In this talk I will give an explicit formula for this dimension. 

Abstract: A fundamental question in low dimensional topology asks what groups can arise as subgroups of the mapping class group of a surface. In this talk, we will address this question for groups that are indicable (recall a group is said to be indicable if it surjects onto ℤ) and for some surfaces of infinite type. In particular, we will concretely construct some classical examples of indicable groups as intrinsincally infinite type subgroups for certain surfaces and provide a combination theorem which inputs two groups and outputs their free product with commuting subgroups.

This is a joint work with Carolyn Abbott, Hannah Hoganson, Marissa Loving, and Priyam Patel. 

Abstract: We are motivated by looking for traces of hyperbolicity in a space or group which is not Gromov-hyperbolic. One previous approach in this direction is the notion of Morse quasigeodesics, which describes "negatively-curved" directions in the spaces; another previous approach is "higher rank hyperbolicity" with one example being that though triangles in products of two hyperbolic planes are not thin, tetrahedrons made of minimal surfaces are "thin". We introduce the notion of Morse quasiflats, which unifies these two seemingly different approaches and applies to a wider range of objects. In the talk, we will provide motivations and examples for Morse quasiflats, as well as a number of equivalent definitions and quasi-isometric invariance (under mild assumptions). We will also show that Morse quasiflats are asymptotically conical, and comment on potential applications. Based on joint work with B. Kleiner and S. Stadler. 

Abstract: A well-known open problem in group theory is the existence of a group with cohomological dimension 2 and geometric dimension 3, known as the Eilenberg-Ganea problem. For realitive versions of this problem there are examples of groups with relative cohomological dimension 2 and relative geometric dimension 3 due to the work of several authors. We call the aforementioned groups relative Eilenberg-Ganea groups. On the other hand, in geometric group theory, it is usual to try to classify groups up to quasi-isometry. In this talk we will explore roughly how many non quasi-isometric groups are among the class of relative Eilenberg-Ganea groups. Explicitely, using a quasi-isometry invariant introduced by Bowditch, called the Taut spectrum, we will sketch how to prove that there are continuously many non-quasi-isometric groups with proper (resp. virtually cyclic) cohomological dimension 2 and proper (resp. virtually cyclic) geometric dimension 3.  This is joint work with Eduardo Martínez-Pedroza.

Abstract:  A celebrated theorem of Margulis states that the first cohomology of an irreducible lattice in a product of rank one simple Lie groups vanishes.  In the more general $\mathrm{CAT}(0)$ setting one can find examples of irreducible lattices with non-trivial first cohomology.  A natural question is to ask is: do the kernels of these cohomology classes satisfy some finiteness property and whether this can characterise irreducibility?  We will show for the product of a Euclidean plane and a tree that virtually fibring is equivalent to being reducible.  We will also give examples of irreducible lattices in a product of a Salvetti complex and a Euclidean plane which exhibit arbitrary finiteness properties. 

Abstract:  Mackey functors for finite groups are well understood. In the early 2000s this was extended to infinite discrete groups, and their cohomological finiteness conditions have been expressed in terms of relative cohomology and finiteness conditions of classifying spaces for proper actions. In this talk I will indicate how one can extend the definition to totally disconnected groups, and will indicate some of the obstacles encountered here. This is ongoing work with Ilaria Castellano and Nadia Mazza.

Abstract:  Given a surface S, the Alexander method is a combinatorial tool used to determine whether two self-homeomorphisms of S are isotopic. This statement was formalized in the case of finite-type surfaces, which are surfaces with finitely generated fundamental groups. A version of the Alexander method was extended to infinite-type surfaces by Hernández-Morales-Valdez and Hernández-Hidber. We extend the remainder of the Alexander method to include infinite-type surfaces. 

In this talk, we will talk about several applications of the Alexander method. Then, we will discuss a technique useful in proofs dealing with infinite-type surfaces and provide a "proof by example" of an infinite-type analogue of the Alexander method.

Abstract: We will consider three families of groups - arithmetic groups, mapping class groups and groups of outer automorphisms of a free group. The study of arithmetic groups has had a profound influence on how we understand the latter two classes of groups. In this talk, we will specifically draw parallels between the associated simplicial complexes - Tits building, curve complex and free factor complex - by studying their homotopy type.

Abstract: Farb and Mosher defined convex cocompact subgroups of the mapping class group in analogy with convex cocompact Kleinian groups. These subgroups have since seen immense study, producing surprising applications to the geometry of surface group extension and surface bundles.  In particular, Hamenstadt plus Farb and Mosher proved that a subgroup of the mapping class groups is convex cocompact if and only if the corresponding surface group extension is Gromov hyperbolic. Among Kleinian groups, convex cocompact groups are a special case of the geometrically finite groups. Despite the progress on convex cocompactness, no robust notion of geometric finiteness in the mapping class group has emerged.  Durham, Dowdall, Leininger, and Sisto recently proposed that geometric finiteness in MCG(S) might be characterized by the corresponding surface group extension being hierarchically hyperbolic instead of Gromov hyperbolic. We provide evidence in favor of this hypothesis by proving that the surface group extension of the  stabilizer of a multicurve is hierarchically hyperbolic.

Abstract: Asymptotic mapping class group of genus zero was first introduced by Funar and Kapoudjian. It is a finitely presented group which contains the mapping class groups of all genus zero surfaces. Later the definitions were generalized to allow the genus to be any finite number (Funar--Aramayona) and infinite  (Funar--Kapoudjian). We will discuss how  these groups are constructed and show that they are in fact of type F_infinity. The proof boils down to prove certain subsurface complexes are highly connected. This is based on joint work with  Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Nansen Petrosyan.

Abstract: Divergence of a metric space is an interesting quasi-isometry invariant of the space which measures how geodesic rays diverge outside of a ball of radius r, as a function of r. Divergence of a finitely generated group is defined as the divergence of its Cayley graph. For symmetric spaces of non-compact type the divergence is either linear or exponential, and Gromov suggested that the same dichotomy should hold in a much larger class of non-positively curved CAT(0) spaces. However this turned out not to be the case and we now know that the spectrum of possible divergence functions on groups is very rich. In a joint project with Pallavi Dani, Yusra Naqvi, and Anne Thomas, we initiate the study of the divergence in the general Coxeter groups. We introduce a combinatorial invariant called the "hypergraph index", which is computable from the Coxeter graph of the group, and use it to characterize when a Coxeter group has linear, quadratic or exponential divergence, and also when its divergence is bounded by a polynomial.

Abstract: The Generalized Andrews-Curtis Conjecture states that if K and L are simple homotopy equivalent 2-dimensional complexes, then they are equivalent via a 3-deformation (expansions and collapses involving only complexes of dimension at most 3). There is an equivalent formulation in terms of group presentations connected via a sequence of elementary transformations. One of them is the stabilization transformation which consists of adding a new generator x and a new relator r=x. We will construct an example of two presentations (both with two generators and two relators) which are connected by elementary transformations, and we will show that the stabilization move cannot be avoided. Examples of this kind are known, but this is the first which occurs in non-minimal Euler characteristic, where most invariants fail to distinguish such situations. This example is an application of the Winding invariant, a map from F_2’ to the ring of Laurent polynomials in two variables.

Abstract: Recently, Algom-Kfir and Bestvina introduced mapping class groups of locally finite graphs as a proposed analog of Out(F_n) in the infinite-type setting. In this talk we will introduce the classification of infinite-type graphs, their mapping class groups, and some important types of elements in these groups. Using a framework established by Rosendal, we will then discuss the coarse geometry of the pure mapping class groups. Our techniques lead to a variety of corollaries, including results on asymptotic dimension. This is joint work with George Domat and Sanghoon Kwak.

Abstract: Vamos a comenzar definiendo la clase de variedades de gráficas de dimensiones superiores, las cuales son una extensión de las ideas de Frigerio, Lafont y Sisto en 2015. Estas variedades se forman con una cantidad finita de variedades (piezas) identificando sus fronteras difeomorfas hasta obtener una variedad suave de dimensión n. Presentaremos algunas propiedades de estas variedades y estudiaremos el árbol de Bass-Serre asociado a su cubriente universal para probar que un isomorfismo entre dos variedades de gráficas nos preserva la estructura de estas.

Una subfamilia de las variedades de gráficas de dimensiones superiores que vamos a usar, incluye a la familia de variedades con descomposiciones cuspidales que define Tam Nguyen Phan en 2012. Usando las propiedades del espacio eléctrico, estudiaremos también algunos invariantes bajo cuasi-isometrías de esta subfamilia de variedades.

We are going to start by defining a higher graph manifold, which are a generalisation of the ideas of Frigerio, Lafont and Sisto (2015). These manifolds are formed by finitely many manifolds (pieces) identified by their diffeomorphic boundaries to obtain a smooth n-dimensional manifold. We will introduce several of its properties and we'll study the Bass-Serre tree associated to its universal cover to prove that an isomorphism between two graph manifolds preserves the graph structure.

A subfamily of higher graph manifolds that we will use, includes the family of manifolds with cusp decomposition introduced by Tam Nguyen Phan (2012). Using the properties of the electric space, we'll also study some of the quasi-isometric invariants of this subfamily of manifolds.

Abstract: Configuration spaces are fascinating objects, and appear for example in the study of (surface) braid groups, hyperplane arrangements, dynamical systems and topological robotics, In this talk, we study some of their topological aspects. The configuration spaces F_n(M), where M is either the 2-sphere S^2 or the real projective plane RP^2, are particularly interesting, as their higher homotopy groups coincide with those of S^2 (and S^3). We study the natural inclusion i of F_n(M) into the n-fold Cartesian product M^n, and we prove that its homotopy fibre is the Cartesian product of an (orbit) configuration space with a product of loop spaces of S^2. This enables us to determine the homomorphisms that occur in the long exact sequence of the homotopy fibration of i. This is joint work with Daciberg Gonçalves.

Abstract: Los grupos de Artin forman una importante familia de grupos en teoría geométrica de grupos. En esta charla hablaremos de algunas propiedades de una subfamilia que contiene a los RAAGs (grupos de Artin de ángulo recto). En particular se discutira la intersección de subgrupos parabólicos. La charla esta basada en un trabajo con Islam Foniqi.

The Artin groups form an important family of group in Geometric Group Theory. In this talk we'll talk about some properties of a subfamily that includes RAAGs (Right-Angled Artin Groups). In particular, the intersection of parabolic subgroups will be discussed. This talk is based on joint work with Islam Foniqi.

Trabajo con Octave Lacourte. Estudiamos un análogo del grupo de intercambios de intervalos en dimensión superior: el grupo de intercambios de rectángulos, cortando el rectángulo en subrectángulos, y ensamblando las piezas con traslaciones. Obtenemos un subconjunto generador (por restricted shuffles), y probamos que el subgrupo derivado es simple, y describimos el homomorfismo de abelianización.

Joint work with Octave Lacourte. We study the analogue of the group of interval exchange transformations for higher dimensions: The group of rectangle exchange transformations, by cuttin the rectangle into subrectangles and assembling the pieces back together via translations. We obtain a generating set (by restricted shuffles), we prove that the derived subgroup is simple, and we describe the abelianization homomorphism.

 In the late 1990's Bestvina and Brady constructed the first examples of groups of type FP that are not finitely presented.  (Being of type FP is a sort of 'algebraic shadow' of having a finite classifying space.)  There have been many generalizations, but all previous constructions have used the same circle of ideas: Morse theory on cubical complexes and Brown's criterion.  I will present a new construction which instead uses 'graphical small cancellation'. No previous knowledge of small cancellation theory or Bestvina-Brady groups required!  (Joint work with Tom Brown.)

Abstract: Bounded cohomology of groups is the cohomology of the cochain complex of bounded bar cochains. Bounded cohomology inherits a cup-product structure from the cup-product in ordinary cohomology. In general, bounded cohomology is difficult to compute and knowledge on the cup-product structure is rather limited. In this talk, I will survey recent developments, computations, open problems, and potential applications. The talk is based on joint work with Benjamin Br\"uck,  Francesco Fournier-Facio, and Marco Moraschini.

Abstract: Artin groups, also called Artin-Tits groups, are important examples in geometric group theory. For various non-positively curved or negatively curved properties on discrete groups, Artin groups are interesting targets. In this talk, we treat acylindrical hyperbolicity of Artin groups. Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and the actions on them. By developing their study and formulating some additional discussion, we demonstrate that acylindrical hyperbolicity holds for more general Artin groups. Indeed, we are able to treat Artin groups of infinite type associated with graphs that are not cones. This talk is based on a joint-work with Shin-ichi Oguni (Ehime University).

Abstract: Birman--Chillingworth firstly proved that each mapping class group of a nonorientable surface is a subgroup of the mapping class group of the doule covering orientable surface. We prove that this natural injective homomorphism is a quasi-isometric embedding by using semihyperbolicity of the (extended) mapping class groups of the orientable surfaces. This is a joint work with Takuya Katayama.

Abstract: Sigma invariants are geometric invariants associated to a finitely generated group that can be used to determine the homological and homotopical finiteness properties of coabelian subgroups. The Sigma invariants for right angled Artin groups have been computed by Meier, Meinert and VanWyck. In this talk we will see how to generalize their computations to some Artin groups.

This is a joint work with Ruben Blasco, José Ignacio Cogolludo and Marcos Escartín. 

Abstract: A classical theorem of Y. Ihara states that a torsion-free discrete subgroup of SL_2(ℚ_p) must be free. The aim of this talk is to show how Ihara's result can be extended to a more general type of buildings whose Coxeter systems decompose as free products with amalgamation.

Joint work with Bianca Marchionna and Thomas Weigel. 

Abstract: The fine curve graph is a hyperbolic graph on which the homeomorphism group of a surface acts (in an interesting way). It is motivated by, and shares many properties with, the wildly successful curve graph machinery for mapping class groups — but it also shows new behaviour not encountered in the classical setting.

In this talk, we will explore some of this new behaviour by describing (certain) Gromov boundary points and their stabilisers.

This is joint work with Jonathan Bowden and Richard Webb

Abstract: El grupo modular y el grafo de curvas son dos objetos que se asocian a una superficie S.  En esta charla se dará una breve introducción de estos conceptos, se hablará de una acción  del grupo modular en el complejo de curvas  y se probará, usando una cuasiisometría al grafo de curvas, que el  grupo modular de una superficie orientable de género al menos dos es débilmente hiperbólico relativo a una colección finita de estabilizadores de curvas.

Abstract: We study some circularly orderable groups called laminar groups. Under natural assumptions on the invariant laminations, one can construct a simply connected 3-manifold on which the given group acts on, and consequently show that the given group is a 3-manifold group. This talk is based on the joint work with KyeongRo Kim and Hongtaek Jung.

Abstract: Los grupos de Artin (o Artin-Tits) son generalizaciones naturales de los grupos de trenzas desde un punto de vista algebraico: de la misma forma que las trenzas se obtienen a partir de la presentación del grupo simétrico, otros grupos de Coxeter dan lugar a diferentes grupos de Artin. Hay pocos resultados probados en general para grupos de Artin, lo que hace de ellos una familia de grupos especialmente misteriosa. Para estudiarlos, los especialistas se han centrado en el estudio de uno subgrupos especiales, denominados "subgrupos parabólicos". Estos subgrupos se han usado para construir importantes complejos simpliciales, como el complejo de Deligne o el complejo de subgrupos parabólicos irreducibles. La pregunta básica "¿Es la intersección de subgrupos parabólicos un subgrupo parabólico?" ha resultado crucial para su estudio y, hasta el año 2018, solo estaba probada para la familia de grupos de Artin de ángulo recto. En esta charla, hablaremos del progreso en esta cuestión, empezando por el importante progreso que supuso responder en positivo a la pregunta en grupos de Artin de tipo esférico (con técnicas combinatorias) y como este descubrimiento ha precedido otros avances que usan diferentes técnicas de teoría geométrica de grupos. Esta charla está basada en trabajos con Volker Gebhardt, Juan González-Meneses, Bert Wiest, Alexandre Martin y Nicolas Vaskou.

Abstract: It is a classical result of Grossman that mapping class groups of finite type surfaces are residually finite. In recent years, residual finiteness growth functions of groups have attracted much interest; these are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity. Residual finiteness growth functions detect many subtle properties of groups, including linearity. In this talk , I will discuss some recent joint work with Thomas Koberda on residual finiteness growth for mapping class groups, adapted to nilpotent and solvable quotients of the underlying surface group.

Abstract: The class of right-angled Artin groups (RAAGs for short) and especially their subgroups have proved to be fundamental in group theory in view of recent results showing that many important families of groups, for instance, Coxeter, one-relator groups with torsion and fundamental groups of hyperbolic 3-manifolds, can be seen virtually as subgroups of RAAGs. On the other hand, some subgroups of RAAGs are known to exhibit a wild structure and algorithmic properties, for instance, there are finitely presented subgroups of RAAGs with undecidable conjugacy and membership problems.

In this talk, we will discuss subgroups of right-angled Artin groups and more precisely how the finiteness properties of the subgroup can determine a tame structure and a good algorithmic behaviour of these subgroups. More concretely, we will present a generalisation of Schreier's theorem to finitely generated normal subgroups of RAAGs and use it to study subdirect subgroups of the direct product of coherent RAAGs.

This is joint work with Jone Lopez de Gamiz Zearra.

Abstract:  Let G be a hyperbolic-by-cyclic group. In this talk, we consider the homology groups of sequences of finite index subgroups of G. As I will explain, the study of such sequences is linked with the L^2 homology of the group. We prove that, when the monodromy is polynomially growing, the homology of such sequences grows sublinearly with the index of the subgroups. The proof requires the construction of trees on which G acts nicely. 

This is a joint work with Andrew, Hughes and Kudlinska.

Abstract:  In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.

Abstract: Parabolic subgroups play a central role in the theory of Artin groups. By a classical result of van der Lek we know that parabolic  subgroups of Artin groups are themselves Artin groups. Let A be an Artin group and P and Q parabolic subgroups of A. In this talk we will show that if P is contained in Q, then P is a parabolic subgroup of Q, answering a question of Godelle. The contents of this talk are joint work with Luis Paris.Yvon Verbene

Abstract: Among the techniques provided by geometric group theory in the study of groups, constructing and studying isometric) actions on hyperbolic spaces and their boundaries is one of the most fruitful and has received a lot of attention in the last decades. In this talk, I focus on groups that are not reachable by such a strategy.

I will introduce Property (NL), which indicates that a group does not admit any (isometric) action on a hyperbolic space with loxodromic elements. It turns out that many groups satisfy this property; including many Thompson-like groups. In particular, every finitely generated group quasi-isometrically embeds into a finitely generated simple group with Property (NL). I will also talk about the stability of the property under group operations and explore connections to other fixed point properties and the poset of hyperbolic structures. This talk is based on a paper co-authored with F.Fournier and A.Genevois, with an appendix by A.Sisto.

Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, most cubulated groups, and others. In this talk I'll provide an introduction to studying the quasi-isometric geometry of groups and spaces from this point of view, both describing new tools to use to study these groups and applications of those results.  This talk will include joint work with Mark Hagen and Alessandro Sisto.

Abstract:  A rank n generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to ℤ^n.  This class of groups extends the class of  (generalized) Baumslag-Solitar groups.

In this talk we first present what has been known about residual finiteness and subgroup separability for certain subclasses of this class, and then we completely classify these groups in terms of their separability properties. Specifically, we determine when they are residually finite, subgroup separable and cyclic subgroup separable.

Abstract:  Translation surfaces arise naturally in many different contexts such as the theory of Teichmüller spaces, of mathematical billiards, or of stability conditions of categories. Most visually, they can be described by polygons that are glued along edges which are parallel and have the same length. 

In this talk, we will be interested in the Veech groups of translation surfaces, that is, the stabilizer of the GL(2,R) action on the moduli space for a given translation surface. In general, it is not known whether a given abstract group can be realized as the Veech group of a translation surface.

After introducing the realization problem for Veech groups, I will speak about some recent progress in this direction for translation surfaces of infinite type. This is joint work with Mauro Artigiani, Chandrika Sadanand, Ferrán Valdez, and Gabriela Weitze-Schmithuesen.

Abstract:  A raíz del celebrado teorema de Gromov sobre el crecimiento polinomial, los grupos nilpotentes han sido un objeto de estudio central en la teoría geométrica de grupos. Un aspecto interesante es la conjeturada clasificación quasiisométrica de los grupos nilpotentes. Un importante invariante quasiisométrico de un grupo finitamente presentable es su función de Dehn. Esta función nos brinda una medida cuantitativa para detectar si una palabra en el conjunto generador del grupo representa al elemento neutro del grupo.

Gracias al trabajo de Gersten, Holt y Riley sabemos que la función de Dehn de un grupo nilpotente de clase c está acotada superiormente por un polinomio de grado c+1. En esta plática veremos resultados recientes que nos permiten determinar las funciones de Dehn de grandes familias de grupos nilpotentes. Una interesante consecuencia es la obtención de una colección no numerable de parejas de grupos nilpotentes con conos asintóticos bilipschitz equivalentes pero con funciones de Dehn diferentes.

Esta plática esta basada en trabajo conjunto con Claudio Llosa Isenrich y Gabriel Pallier.Anja Randecker

Abstract: In this talk we present an analysisi of the Baum-Connes isomorphism for some extensions of the special and general linear group. After describing with some detail the structure of these groups, introducing some crystallographic groups which appear in our computations and briefly reviewing Baum-Connes conjecture, we will compute the source of the assembly map in this case. We will also comment some remarkable differences between the general and the special case. This is joint work with Sanaz Pooya (Potsdam) and Alain Valette (Neuchâtel).

Abstract: The interest in the geometry of group extensions started with the geometrization theorem of Thurston for compact irreducible atoroidal 3-manifolds. We will talk about the geometry of group extensions and the motivations behind such studies in the cases of closed surface groups and free groups. More specifically, we will talk about the most general case so far and, we will give necessary and sufficient conditions for a free extension of a (non-Abelian) free group given by a subgroup of the outer automorphism group of the free group (Out(F_n)) to be hyperbolic and relatively hyperbolic. Joint work with Pritam Ghosh.