In a joint work with Matt Tointon, we study the fine structure of approximate groups. We deduce various applications on growth, isoperimetry and quantitative estimates for the the simple random walk on finite vertex transitive graphs.

# Past Topology Seminar

A conjecture from the '80s claims that the isometry type of a closed, negatively curved Riemannian manifold should be uniquely determined by the lengths of its closed geodesics. By work of Otal, this is essentially equivalent to the problem of extending cross-ratio preserving maps between Gromov boundaries of simply connected, negatively curved manifolds. Progress on the conjecture has been remarkably slow, with only the 2-dimensional and locally symmetric cases having been solved so far (Otal '90 and Hamenstädt '99).

Still, it is natural to try leaving the world of manifolds and address the conjecture in the general context of non-positively curved metric spaces. We restrict to the class of CAT(0) cube complexes, as their geometry is both rich and well-understood. We introduce a new notion of cross ratio on their horoboundary and use it to provide a full answer to the conjecture in this setting. More precisely, we show that essential, hyperplane-essential cubulations of Gromov-hyperbolic groups are completely determined by their combinatorial length functions. One can also consider non-proper non-cocompact actions of non-hyperbolic groups, as long as the cube complexes are irreducible and have no free faces.

Joint work with J. Beyrer and M. Incerti-Medici.

Stable commutator length (scl) is a well established invariant of elements g in the commutator subgroup (write scl(g)) and has both geometric and algebraic meaning. A group has a \emph{gap} in stable commutator length if for every non-trivial element g, scl(g) > C for some C > 0.

SCL may be interpreted as an 'algebraic translation length' and such a gap may be thus interpreted an 'algebraic injectivity radius'.

Many classes of groups have such a gap, like hyperbolic groups, mapping class groups, Baumslag-Solitar groups and graph of groups.

In this talk I will show that Right-Angled Artin Groups have the optimal scl-gap of 1/2. This yields a new invariant for the vast class of subgroups of Right-Angled Artin Groups.

Symmetric spaces and lattices are important objects to model spaces in geometry and topology. They have been studied from many different viewpoints. We will concentrate on their coarse geometry view point in this talk. I will first quickly go over some well-known results about quasi-isometry of those spaces. Then I will move to the study about quasi-isometric embeddings. While results in this direction are far less complete and well-studied, there are some rigidity phenomenons still happening here.

In his famous book on partial differential relations Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental assertion and sketch a proof.

In the sequel we will apply this to extend local deformations of closed $G_2$ structures, and to construct

$C^{1,1}$-Riemannian metrics which are positively curved "almost everywhere" on arbitrary manifolds.

This is joint work with Christian Bär (Potsdam).

We recall the impact of stabilizing a 4-manifold with $S^2 \times S^2$. The corresponding local situation concerns knots in the 3-sphere which bound (nullhomotopic) discs in a stabilized 4-ball. We explain how these discs arise, and discuss bounds on the minimal number of stabilizations needed. Then we compare this minimal number to the 4-genus.

This is joint work with A. Conway.

## Further Information:

The algebraic K-theory of a space encodes important invariants of the space which are of interest in both homotopy theory and geometric topology.

In this talk, I will discuss properties of transfer maps in the algebraic K-theory of spaces ('wrong-way' maps) in connection with index theorems for (smooth or topological) manifold bundles and also compare these maps with other related constructions such as the Becker-Gottlieb transfer and the Waldhausen trace.

I will describe an extremely easy construction with formal group laws, and a

slightly more subtle argument to show that it can be done in a coordinate-free

way with formal groups. I will then describe connections with a range of other

phenomena in stable homotopy theory, although I still have many more

questions than answers about these. In particular, this should illuminate the

relationship between the Lambda algebra and the Dyer-Lashof algebra at the

prime 2, and possibly suggest better ways to think about related things at

odd primes. The Morava K-theory of symmetric groups is well-understood

if we quotient out by transfers, but somewhat mysterious if we do not pass

to that quotient; there are some suggestions that dilation will again be a key

ingredient in resolving this. The ring $MU_*(\Omega^2S^3)$ is another

object for which we have quite a lot of information but it seems likely that

important ideas are missing; dilation may also be relevant here.

The lower central series of a group $G$ is defined by $\gamma_1=G$ and $\gamma_n = [G,\gamma_{n-1}]$. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: $\delta_n = \{g: g-1\text{ belongs to the $n$-th power of the augmentation ideal}\}$.

It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has $\delta_n\ge\gamma_n$, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with $\delta_4/\gamma_4$ cyclic of order 2. On the positive side, Sjogren showed that $\delta_n/\gamma_n$ is always a torsion group, of exponent bounded by a function of $n$. Furthermore, it was believed (and falsely proven by Gupta) that only $2$-torsion may occur.

In joint work with Roman Mikhailov, we prove however that for every prime $p$ there is a group with $p$-torsion in some quotient $\delta_n/\gamma_n$.

Even more interestingly, I will show that the dimension quotient $\delta_n/gamma_n$ is related to the difference between homotopy and homology: our construction is fundamentally based on the order-$p$ element in the homotopy group $\pi_{2p}(S^2)$ due to Serre.

I will introduce two obstructions for a rational homology 3-sphere to smoothly bound a rational homology 4-ball- one coming from Donaldson's theorem on intersection forms of definite 4-manifolds, and the other coming from correction terms in Heegaard Floer homology. If L is a nonunimodular definite lattice, then using a theorem of Elkies we will show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives a rephrasing of the obstruction coming from Donaldson's theorem. Furthermore, from this perspective it is easy to see that if the obstruction to bounding a rational homology ball coming from Heegaard Floer correction terms vanishes, then (under some mild hypotheses) the obstruction from Donaldson's theorem vanishes too.