Braids: our past informing our future

Braids have been objects of interest for humans for many millennia, appearing in our hair, our art, and our clothing, for example. In 1925, Emil Artin introduced the notion of a `braid group', launching the study of braids as algebraic objects. In this talk we will describe some important types of braid groups that appear widely in geometry and topology, survey recent developments in our understanding of their fundamental structures, and highlight some contemporary problems that inform these investigations.

Harmonic functions on the Heisenberg group

A harmonic function on a group G is a function which is equal to the average of its translates, average for a measure m whose support is finite and spans G. We want to describe the extremal positive harmonic functions h on G.

By a theorem of Choquet and Deny, when G is abelian, h is a character. By a theorem of Margulis, when G is nilpotent and m is symmetric, h is also a character. By a theorem of Ancona, when G is hyperbolic and m is symmetric, these functions h are parametrized by the boundary of G.

We will see that, when the group G is the Heisenberg group, the classical partition function occurs as a function h and that this function is the only one beyond characters.

Apollonian circles and their properties

Starting from four mutually tangent circles in the plane one can successively inscribing more circles between any three tangent circles. This gives infinitely circles forming an Apollonian circle packing. We will describe properties of the radii of these circles. This will draw upon ideas from Fractal Geometry, Number Theory and Probability Theory.

An invitation to dilation surfaces

A dilation surface is a geometric object obtained from gluing together Euclidean polygons along parallel sides, allowing some form of stretching when performing such gluings. These are interesting geometric objects in their own right : they carry natural complex structures which makes for a connection with Teichmüller theory. From that point one can define moduli spaces of such objects and start asking geometry-oriented questions.

Dilation surfaces also carry natural families of foliations; they are therefore objects of interest from a dynamical viewpoint as well.

In this talk I will try to give a gentle introduction to these objects and motivate a series of questions of importance in the topic. If time permits, I will allow myself to advertise my own work on the subject.

Effective equidistribution of expanding horospheres

I will overview results of effective equidistribution of horospheres in local symmetric spaces of rank one and higher rank, and will end with a presumably sharp result of effective distribution for a special family of expanding horospheres (modulo SL(d, ℤ)) in the locally symmetric space of positive definite quadratic forms of determinant one on ℝ^d modulo SL(d, ℤ) (joint work with N. Peyerimhoff).

Critical exponents of subgroups of groups acting on Gromov hyperbolic spaces, and amenability

This is joint work with R. Coulon, B. Schapira and S. Tapie.

One of the first things we learn about a (proper) Gromov hyperbolic geodesic space X is the construction of the visual boundary ∂X. An ergodic theorist then learns that for a non-elementary discrete group of isometries G acting properly on X, there is an interesting family of s-quasi-conformal measures on ∂X, called Patterson-Sullivan measures. And the smallest such s coincides with the critical exponent δ_G of G: that is, δ_G is the exponential growth in R of Gx ∩ B(x,R) ⊂ X. A general question might be, given a subgroup H of G, how do we compare δ_H and δ_G? For G with "good dynamics", we expect that δ_G = δ_H if and only if H is co-amenable in G. Indeed, we may see Grigorchuk's co-growth criterion as an instance of this question where X is a Cayley graph of a free group. Differently, in the case that X is a rank 1 symmetric space, some results were known using Brooks' characterisation of amenability in terms of the bottom of the spectrum of the Laplacian. I will motivate the question, and say something about our new approach, for which we have an optimal conclusion, that if the action of G is SPR, then we have δ_G = δ_H if and only if H is co-amenable in G.

Quasi-isometric diversity of marked groups

Quasi-isometries are maps between metric spaces that perturb distances only by multiplicative and additive constants. Such maps are especially important in Geometric Group Theory as they give rise to a natural equivalence relation between finitely generated groups, equipped with their word metrics. The main goal of Asymptotic Group Theory, initiated by Gromov, is to study finitely generated groups up to quasi-isometries and look for group-theoretic properties that are invariant under this equivalence relation.

In the talk I will discuss a method, based on Descriptive Set Theory, which allows to show that various classes of finitely generated groups contain continuously many quasi-isometry classes. We will apply this technique to show that there is a continuum of 2-generated pairwise non-quasi-isometric groups which satisfy various exotic properties (such are being simple, torsion and divisible, or being a Tarski monster, or having asymptotic dimension 1). The talk will be based on a joint work with Denis Osin and Stefan Witzel.

Geometry of graphs of multicurves

Given a compact, orientable surface, we can define many associated graphs which have multicurves as vertices. A first example is the curve graph, which has a vertex for every isotopy class of curves and an edge between two vertices if the curves are disjoint. I will present joint work with Jacob Russell on certain such graphs, giving a classification of when they are hyperbolic and relatively hyperbolic.

Hyperbolic groups with locally contractible boundaries

It is an fundamental principle of geometric group theory that large scale geometric properties of metric spaces on which a group acts should be considered properties of the group itself. If a given group acts on a hyperbolic metric space then a particularly natural example of such a large scale property is the Gromov boundary of that space: this is a topological space that compactifies the hyperbolic space at infinity. These spaces can be quite complicated, and I will begin giving some examples to illustrate this. I will then describe a new result that restricts the topological pathologies of a Gromov boundary by characterising the circumstances under which the boundary is locally contractible. This work generalises a theorem of Bestvina and Mess.

Weil-Petersson geometry / dynamics for exceptional moduli

This talk will survey some recent results (such as the analogue of Sullivan, Masur, Kleinbock-Margulis log laws) for Weil-Petersson geometry/dynamics with a particular focus on exceptional moduli spaces. This is joint work with Carlos Matheus (CNRS Ecole Polytechnic).

Three dichotomies for connected unimodular Lie groups

Using the Levi decomposition theorem, Lie groups are usually studied in two separate classes: semisimple and solvable. Both these classes further divide into two subclasses with very different behaviour: semisimple groups split into the rank 1 and higher rank cases; while solvable groups divide into those of polynomial growth and those of exponential growth.

Amongst connected unimodular Lie groups, let us say that G is "small" if it shares a cocompact subgroup with some direct product of a rank one simple Lie group and a solvable Lie group with polynomial growth. Otherwise, we say G is "large". We present three strong dichotomies which distinguish "small" and "large" Lie groups; which are respectively algebraic, coarse geometric, and local analytic in nature. As an application we will show that Baumslag-Solitar groups admit a similar "small"/"large" dichotomy. This is part of a joint project with John Mackay and Romain Tessera.

Cohomology fractals

This is joint work with David Bachman and Henry Segerman. I'll tell the story of how we discovered ``cohomology fractals''. We wrote an online app available at:

https://henryseg.github.io/cohomology_fractals/

that allows us (and you!) to explore these fractals, for a wide range of cohomology classes on a wide range of hyperbolic three-manifolds. You use the arrow keys and WASD keys to move around.