Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Dynamics, Geometry, & Groups Seminar

Dynamics, Geometry, & Groups - Hang Lu Su (ICMAT Madrid)

Thursday, November 26th, 2020

Time: 2:00 p.m Place:

Speaker: Hang Lu Su (ICMAT Madrid)

Title: Left-orderable groups and formal languages.

Abstract: I will introduce the notions of left-orderable groups, the topology on the space of left-orders, and formal language complexity with respect to the Chomsky hierarchy. I will give an idea of why it is intriguing to study left-orders using language complexity by introducing a toy example with the Klein bottle group. Finally, I will introduce some recent results concerning the closure properties of positive cone complexity. This work is joint with Yago Antolín and Cristóbal Rivas.

Dynamics, Geometry, & Groups - Elia Fioravanti

Thursday, November 19th, 2020

Time: 2:00 p.m Place:

Speaker: Elia Fioravanti (Max Planck Institute for Mathematics and at the University of Bonn)

Title: Coarse-median preserving automorphisms of special groups.

Abstract: We introduce the class of "coarse-median preserving" automorphisms of coarse median groups. For instance, we show that automorphisms of right-angled Artin groups are coarse-median preserving if and only if they are untwisted (in the sense of Charney-Stambaugh-Vogtmann), while all automorphisms of hyperbolic groups are coarse-median preserving. Our main result is that, for every special group G (in the sense of Haglund-Wise), every infinite-order, coarse-median preserving outer automorphism of G can be realised as a homothety of a finite-rank median space X equipped with a ``moderate'', isometric G-action. This generalises Paulin's result that every infinite-order outer automorphism of a hyperbolic group G can be realised as a homothety of a real tree equipped with a small, isometric G-action."]

Dynamics, Geometry, & Groups - Lei Chen (Caltech)

Thursday, November 12th, 2020

Time: 2:00 p.m Place:

Speaker: Lei Chen (California Institute of Technology)

Title: Nielsen realization problem for the mapping class group.

Abstract: Nielsen realization problem for the mapping class group Mod(Sg) asks whether the natural projection pg:Homeo+(Sg)→Mod(Sg) has a section. Moreover, we also care about the realization of subgroups of Mod(S_g). In this talk, I will explain some ideas towards this problem. It will use some rotation number results and fixed points theory.

Dynamics, Geometry, & Groups - Florian Richter (Northwestern U)

Thursday, November 5th, 2020

Time: 2:00 p.m Place:

Speaker: Florian Richter (Northwestern University)

Title: Additive and geometric transversality of fractal sets in the reals and integers.

Abstract: Using the language of fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the 1960s that explore the relationship between digit expansions of real numbers in distinct prime bases. While his famous x2 x3 conjecture remains open, recent solutions to some of his "transversality conjectures" have shed new light on old problems. In this talk we explore analogues of results surrounding Furstenberg's conjectures in the discrete setting of the integers, with the aim of understanding the independence of sets of integers that are structured with respect to different prime bases. This is based on joint work with Daniel Glasscock and Joel Moreira.

Dynamics, Geometry, & Groups - Ilya Gekhtman (Technion - Israel)

Thursday, October 22nd, 2020

Time: 2:00 p.m Place:

Speaker: Ilya Gekhtman (Technion-Israel Institute of Technology)

Title: Martin, Floyd and Bowditch boundaries of relatively hyperbolic groups.

Abstract: Consider a transient random walk on a countable group $G$. The Green distance between two points in the group is defined to be minus the boundary of the probability that a random path starting at the first point ever reaches the second. The Martin compactification of the random walk is a topological space defined to be the horofunction boundary of the Green distance. It is a topological model for the Poisson boundary. The Martin boundary typically heavily depends on the random walk; it is thus exciting when for some large class of random walks, the Martin boundary is equivariantly homeomorphic to some well known geometric boundary of the group. Ancona showed in 1988 that this is the case for finitely supported random walks on hyperbolic groups: the Martin boundary is identified with the Gromov boundary. We generalize Ancona's results to relatively hyperbolic groups: the Martin boundary equivariantly continuously surjects onto the Gromov boundary of any hyperbolic space on which the group acts geometrically finitely (called the Bowditch boundary), and the preimage of any conical limit point is a singleton. When the parabolic subgroups are virtually abelian (e.g. for Kleinian groups) we show that the preimage of a parabolic fixed point is a sphere of appropriate dimension, so the Martin boundary can be identified with a Sierpinski carpet. A major technical tool is a generalization of a deviation inequality due to Ancona saying the Green distance is nearly additive along word geodesics, which has various other applications, including to comparing harmonic and Patterson-Sullivan measures for negatively curved manifolds and to local limit theorems for random walks. We do all this using an intermediate construction called the Floyd metric obtaining by suitably rescaling the Cayley graph and considering the associated completion called the Floyd compactification. We show that for any finitely supported random walk on a finitely generated group, the Martin boundary surjects to the Floyd boundary, which in turn by work of Gerasimov covers the Bowditch boundary of relatively hyperbolic groups. This is based on several joint works with subsets of Dussaule, Gerasimov, Potyagailo, and Yang.

Dynamics, Geometry, & Groups - Mathew Cordes (ETH Zurich)

Thursday, October 15th, 2020

Time: 2:00 p.m Place:

Speaker: Mathew Cordes (ETH Zurich)

Title: Geometric approximate group theory.

Abstract: An approximate group is a group that is "almost closed" under multiplication. Finite approximate subgroups play a major role in additive combinatorics. Recently Breuillard, Green and Tao have established a structure theorem concerning finite approximate subgroups and used this theory to reprove Gromov’s growth theorem. Infinite approximate groups were studied implicitly long before the formal definition. Approximate subgroups of R^n that are Delone sets can be constructed using "cut-and-project" methods and are models for mathematical quasi-crystals. Recently, Björklund and Hartnick have begun a program investigating infinite approximate lattices in locally compact second countable groups using geometric and measurable structures. In the talk I will introduce infinite approximate groups and their geometric aspects. This is joint work with Hartnick and Tonic.

Dynamics, Geometry, & Groups - Chris Judge

Thursday, October 8th, 2020

Time: 2:00 p.m Place:

Speaker: Chris Judge (Indiana University Bloomington)

Title: Triangles have no interior hot spots in the long run.

Abstract: It well-known that the temperature distribution of a perfectly insulated, perfectly homogeneous body will tend to a constant temperature distribution as time tends to infinity. Jeff Rauch conjectured in the 1974 that, moreover, all local extrema of the temperature distribution will migrate towards the boundary as time tends to infinity. This is now believed to be true for convex domains. I will discuss joint work with Sugata Mondal that verifies the conjecture for triangles. This resolves Polymath 7.

Dynamics, Geometry, & Groups - Ivan Levcovitz

Thursday, October 1st, 2020

Time: 2:00 p.m Place:

Speaker: Ivan Levcovitz (Tecnhion-Israel Institute of Technology)

Title: Characterizing divergence in right-angled Coxeter groups.

Abstract: A main goal in geometric group theory is to understand finitely generated groups up to quasi-isometry (a coarse geometric equivalence relation on Cayley graphs). Right-angled Coxeter groups (RACGs) are a well-studied, wide class of groups whose coarse geometry is not well understood. One of the few available quasi-isometry invariants known to distinguish non-relatively hyperbolic RACGs is the divergence function, which roughly measures the maximum rate that a pair of geodesic rays in a Cayley graph can diverge from one another. In this talk I will discuss a recent result that completely classifies divergence functions in RACGs, gives a simple method of computing them and links divergence to other known quasi-isometry invariants.

Dynamics, Geometry, & Groups - Joseph Maher

Thursday, September 24th, 2020

Time: 2:00 p.m Place:

Speaker: Joseph Maher

Title: Random walks on WPD groups.

Abstract: The action of a group on a Gromov hyperbolic space X is WPD if the action is coarsely discrete along the quasi-axis of a loxodromic isometry. This is a generalization of acylindrical actions, and includes the actions of Out(F_n) on the free splitting and free factor complexes, and the action of the Cremona group on the Picard-Manin space. We show that for WPD actions the Gromov boundary for X is a geometric realization of the Poisson boundary for the random walk. We also show that generic elements are loxodromic with asymptotic probability one, with an explicit square root exponential bound on the rate of convergence, and show that the normal closure of random elements give infinitely many distinct normal subgroups of G. This is joint work with Giulio Tiozzo.

Dynamics, Geometry, & Groups - Ian Frankel (Queen's University)

Thursday, September 17th, 2020

Time: 2:00 p.m Place:

Speaker: Ian Frankel (Queen's University)

Title: Extremal length and Conformal Geometry.

Abstract: One of the key ideas in the study of moduli spaces of Riemann surfaces is the study of conformal invariants. We will describe one classical invariant, the extremal lengths of simple closed curves on a Riemann surface, and give applications connecting hyperbolic geometry to singular flat metrics on surfaces.