# ENS Group Theory Seminar Videos

The current program of the ENS group seminar can be found here.

## December 8, 2020

### Robert Young (NYU Courant and IAS Princeton)

#### Hölder maps to the Heisenberg group

In this talk, we construct Hölder maps to the Heisenberg group H, answering a question of Gromov. Pansu and Gromov observed that any surface embedded in H has Hausdorff dimension at least 3, so there is no α-Hölder embedding of a surface into H when α > 2/3. Züst improved this result to show that when α > 2/3, any α-Hölder map from a simply-connected Riemannian manifold to H factors through a metric tree. We use new techniques for constructing self-similar extensions to show that any continuous map to H can be approximated by a (2/3 - ε)-Hölder map. This is joint work with Stefan Wenger.

### Matei Coiculescu (Brown University)

#### The Spheres of Sol

Sol, one of the eight Thurston geometries, is a solvable three-dimensional Lie group equipped with a canonical left invariant metric. Sol has sectional curvature of both signs and is not rotationally symmetric, which complicates the study of its Riemannian geometry.

Our main result is a characterization of the cut locus of Sol, which implies as a corollary that the metric spheres in Sol are topological spheres. This is joint work with Richard Schwartz.

### Richard Schwartz (Brown University and IAS Princeton)

#### The areas of metric spheres in Sol

This is a sequel talk, following Matei Coiculescu's talk about our joint work characterizing the cut locus of the identity in Sol. In this talk, I will explain my result that the area of a metric sphere of radius r in Sol is at most Cer for some uniform constant C. That is, up to constants, the sphere of radius r in Sol has the same area as the hyperbolic disk of radius r.

## November 24, 2020

### Alessandro Sisto (Heriot-Watt)

#### Cubulation of hulls and bicombings

It is well-known that the quasi-convex hull of finitely many points in a hyperbolic space is quasi-isometric to a tree. I will discuss an analogous fact in the context of hierarchically hyperbolic spaces, a large class of spaces and groups including mapping class groups, Teichmueller space, right-angled Artin and Coxeter groups, and many others. In this context, the approximating tree is replaced by a CAT(0) cube complex. I will also briefly discuss applications, including how this can be used to construct bicombings. Based on joint works with Behrstock-Hagen and Durham-Minsky.

### Thomas Haettel (Montpellier)

#### The coarse Helly property, hierarchical hyperbolicity and semihyperbolicity

For any hierarchical hyperbolic group, and in particular any mapping class group, we define a new metric that satisfies a coarse Helly property. This enables us to deduce that the group is semihyperbolic, i.e. that it admits a bounded quasigeodesic bicombing, and also that it has finitely many conjugacy classes of finite subgroups. This has several other consequences for the group. This is a joint work with Nima Hoda and Harry Petyt.

### Mark Hagen (Bristol)

#### Wallspaces, the Behrstock inequality, and ℓ1 metrics on asymptotic cones

From its hyperplanes, one can always characterise a CAT(0) cube complex as the subset of some (often infinite) cube consisting of the solutions to a system of "consistency" conditions. Analogously, a hierarchically hyperbolic space (HHS) can be coarsely characterised as a subset of a product of Gromov-hyperbolic spaces consisting of the "solutions" to a system of coarse consistency conditions.

HHSes are a common generalisation of hyperbolic spaces, mapping class groups, Teichmuller space, and right-angled Artin/Coxeter groups. The original motivation for defining HHSes was to provide a unified framework for studying the large-scale properties of examples like these.

So, it is natural to ask about the structure of asymptotic cones of hierarchically hyperbolic spaces.

Motivated by the above characterisation of a CAT(0) cube complex, we introduce the notion of an R-cubing. This is a space that can be obtained from a product of R-trees, with the ℓ1 metric, as a solution set of a similar set of consistency conditions. R-cubings are therefore a common generalisation of R-trees and (finite-dimensional) CAT(0) cube complexes. R-cubings are median spaces with extra structure, in much the same way that HHSes are coarse median spaces with extra structure.

The main result in this talk says that every asymptotic cone of a hierarchically hyperbolic space is bilipschitz equivalent to an R-cubing. This strengthens a theorem of Behrstock-Drutu-Sapir about asymptotic cones of mapping class groups. Time permitting, I will talk about an application of this result which is still in progress, namely uniqueness of asymptotic cones of various hierarchically hyperbolic groups, including mapping class groups and right-angled Artin groups. This is joint work with Montse Casals-Ruiz and Ilya Kazachkov.

## June 30, 2020

#### Free Q-groups are residually torsion-free nilpotent

A group G is called a Q-group if for any natural number n and any element g from G there exists a unique nth root of g in G. These groups were introduced by G. Baumslag in the sixties under the name of D-groups. The free Q-group on X can be constructed from the free group on X by applying an infinite number of amalgamations over cyclic subgroups. In this talk I will explain how to show that free Q-groups are residually torsion-free nilpotent. This solves a problem raised by G. Baumslag. A key ingredient of our argument is the proof of one instance of the Lueck approximation in characteristic p corresponding to an embedding of a finitely generated group into a free pro-p group. For more details see http://matematicas.uam.es/~andrei.jaikin/preprints/baumslag.pdf.

Slides for this talk are available here.

### David Conlon (Caltech)

#### Subset sums, completeness and colourings

(joint work with Jacob Fox and Huy Tuan Pham.)

We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the three problems of Burr and Erdos on Ramsey complete sequences, for which Erdos later offered a combined total of \$350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdos on the minimum number of colors needed to color the positive integers less than n so that n cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdos and Graham on sets of integers avoiding a given subset sum; and, answering a question of Tran, Vu and Wood, a strengthening of a seminal result of Szemeredi and Vu on long arithmetic progressions in subset sums.

### Harald Helfgott (Goettingen)

Video available soon.

#### Random walks on a divisibility graph

As one consequence, we establish a new bound for the logarithmic Chowla conjecture, viz., $\sum_{n\leq x} \frac{\lambda(n) \lambda(n+1)}{n} = O\left(\frac{1}{(\log \log x)^c}\right)$ for λ(n) the Liouville function and a fixed c>0. (Tao (2016) had proved a bound of o(1) by a different approach; his proof can be made to give a bound of O(1/log log log log x).)