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Seminars

7 de Junio de 2023, Aula 115. Facultad de Matemáticas, Universidad Complutense de Madrid

• 10:30 - Xabier Legaspi (UCM-ICMAT): (Prelectura Tesis) Growth in groups of non-positive curvature - Expand Abstract
  The aim of this thesis is to obtain a better understanding of the behaviour of exponential growth rates within the class of groups that act acylindrically in a hyperbolic space in the sense of Gromov. To do this, we will address two problems of a different nature.
  
  In the first problem we will study the exponential growth rates of quasi-convex subgroups. We will compare these rates with that of the ambient group and we will determine when it is possible to obtain strict equality/inequality. To do so, we will exploit proper actions on metric spaces that, a priori, are not hyperbolic, but that have isometries that behave like the loxodromic isometries of a hyperbolic space.
  
  The second problem revolves around uniform uniform exponential growth. We will prove that this property is preserved if we take small cancellation quotients of groups that act acylindrically on a hyperbolic space. As a corollary, we will obtain that there is a universal lower bound on the uniform exponential growth rate for the family of classical small cancellation quotients. This bound depends only on one of the two acylindricity parameters.
  
  
• 12:00 - Alan Reid (Rice University): Finite shadows of infinite groups: What do they hide? - Expand Abstract
  There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects — ie via their finite quotients. A natural question is therefore: to what extent do they determine the infinite group?
  In this colloquium style talk I will discuss the rich history of this problem and describe some of the major progress of recent years. In particular I will discuss results connected to the study of 3-manifold groups.
  
  

8 de Junio de 2023, Aula Naranja - ICMAT. Madrid

• 10:00 - Nicholas Vlamis (CUNY): Homeomorphism groups of self-similar 2-manifolds - Expand Abstract
  The study of homeomorphism groups and mapping class groups of infinite-type surfaces is in its infancy, and the deeper we dive into their structure we see that the class cannot be studied all at once. Recently, Mann and Rafi have introduced a very useful way to partition this class into several nice subclasses whose homeomorphism groups/mapping class groups often share many properties. In this talk, we will discuss recent results regarding the structure of homeomorphism groups of the class of self-similar 2-manifolds, and how these can be viewed as natural extensions of results regarding the homeomorphism group of the 2-sphere. The talk will largely be expository in nature, and we will discuss work of Anderson from the 50s and the recent work of Mann–Rafi, Malestein–Tao, Calegari–Chen, Calegari–Freedman, Hernández–Hrušák–Morales–Randecker–Sedano–Valdez, and Lanier and myself.
  
  
• 11:30 - Xiaolei Wu (Fudan University): On the poly-freeness of some classes of Artin groups - Expand Abstract
  Artin groups is an important class of groups under intensive study in recent years. It is a generalization of the braid groups. Bestvina asks whether all Artin groups are virtually poly-free. In this talk, we first give an introduction to poly-free groups and Artin groups. We explain some applications on the Farrell-Jones Conjecture. Then we explain some recent progress of Bestvina's question, in particular giving a short proof for the polyfreeness of even FC-type Artin groups. Part of this is joint work with Benjamin Brück and Dawid Kielak.
  
  

19 de Junio de 2023, ICMAT. Madrid

• 11:00 - Francesco Fournier Facio (ETH Zurich): Recent developments in Ulam stability
• 11:50 - Marcos Escartín (Universidad de Zaragoza): The $\Sigma^1$-conjecture for Artin groups

20 de Junio de 2023, Aula Gris 3 - ICMAT. Madrid

• 10:00 - Jun Ueki (Ochanomizu University): The $p$-adic torsions of knots - Expand Abstract
  First, I will briefly explain Ghys’s result on modular knots around the trefoil and the Rademacher symbol and its generalization for torus knots (joint work with Toshiki Matsusaka).
  Next, in view of the analogy between the Iwasawa theory and the Alexander--Fox theory, p being a prime number, we consider the system of cyclic $Z/p^nZ$-covers of a knot. We establish explicit formulas for the $p$-adic limit of the sizes of the 1st homology groups, exhibit several inspiring examples, and present further perspectives (joint work with Hyuga Yoshizaki + Sohei Tateno).
  
  
• 11:00 - Coffee break
• 11:30 - Misha Belolopetsky (IMPA): Subspace stabilisers in hyperbolic lattices - Expand Abstract
  In a joint work with Nikolay Bogachev, Alexander Kolpakov and Leone Slavich we show that immersed totally geodesic m-dimensional suborbifolds of an n-dimensional arithmetic hyperbolic orbifold correspond to finite subgroups of the commensurator whenever is m sufficiently large. In particular, for n = 3 this condition includes all totally geodesic suborbifolds. We call such totally geodesic subspaces by finite centraliser subspaces (or fc-subspaces for short) and use them to formulate an arithmeticity criterion for hyperbolic lattices. We show that a hyperbolic orbifold is arithmetic if and only if it has infinitely many fc-subspaces, while in the non-arithmetic case the number of fc-subspaces is finite and bounded in terms of the volume. The case of special interest is that of exceptional trialitarian 7-dimensional orbifolds -- we show that every such orbifold contains a totally geodesic arithmetic hyperbolic 3-orbifold of exceptional type. In the talk I will begin with an introduction to the topic and then discuss some results and their proofs.
  
  

21 de Junio de 2023, ICMAT. Madrid

• 11:00 - Christopher-Lloyd Simon (Penn State University): Linking numbers of modular knots [TBC]
• 11:50 - Julian Wykowski (University of Cambridge): Motivation for profinite Bass-Serre theory

22 de Junio de 2023, Aula Gris 2 - ICMAT. Madrid

• 10:00 - Jing Tao (University of Oklahoma): On surface homeomorphisms - Expand Abstract
  In the 1970s, Thurston generalized the classification of self-maps of the torus to surfaces of higher genus, thus completing the work initiated by Nielsen. This is known as the Nielsen-Thurston Classification Theorem. Since then, many alternative proofs using different aspects of surface theory have emerged. In this talk, I will overview the classical theory and sketch the ideas of a new proof, one that offers new insights into the hyperbolic geometry of surfaces. This is joint work with Camille Horbez.
  
  
• 11:00 - Coffee break
• 11:30 - Martin Kassabov (Cornell University): Tame automorphism groups of polynomial rings with property (T) and infinitely many alternating group quotients - Expand Abstract
  We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of Aut(Fp[x1,…,xn]) generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of Aut(Fp[x1,…,xn]) on the n-dimensional affine spaces over finite extensions of Fp. As a consequence, we obtain explicit presentations of Gromov hyperbolic groups with property (T) and infinitely many alternating group quotients. Our construction also yields an explicit infinite family of expander Cayley graphs of degree 4 for alternating groups of degree p7−1 for any odd prime p. This is a joint work with Pierre-Emmanuel Caprace.
  
  

23 de Junio de 2023, Department of Mathematics of Autonomous University of Madrid, 520

• 15:00 - Ph.D. defense of Jan Boschheidgen (UAM): Finite dimensional Approximations of Operators related to Groups and their Applications