The concept of coarse embedding was introduced by Gromov in 1993. It plays a crucial role in the study of large-scale geometry of groups and the Novikov higher signature conjecture. Coarse amenability, also known as Guoliang Yu’s property A, is a weak amenability-type condition that is satisfied by many known metric spaces. It implies the existence of a coarse embedding into a Hilbert space. In this expository talk, we discuss the interplay between infinite expander graphs, coarse amenability, and coarse embeddings. We present several ’monster’ constructions, in the setting of metric spaces of bounded geometry, including a recent construction, jointly with Romain Tessera, of relative expander graphs which do not weakly contain any expander. This research was partially supported by my ERC grant ANALYTIC no. 259527.

We present new upper bounds for the height of elements in the cohomology of the unordered configuration space \(H^*(\mathrm{Conf}_n(\mathbb{R}^d)/\mathfrak{S}_n;\mathbb{F}_p)\) with coefficients in the field \(\mathbb{F}_p\). In the special case when \(d\) is a power of \(2\) and \(p=2\) we settle the original Vassiliev conjecture by proving that \(\mathrm{height}(H^*(\mathrm{Conf}_n(\mathbb{R}^d)/\mathfrak{S}_n;\mathbb{F}_2))=d\).

As applications of these results we obtain new lower bounds for the existence of complex \(k\)-regular maps as well as for complex \(\ell\)-skew maps \(\mathbb{C}^d\rightarrow\mathbb{C}^N\). (This is joint work with F. Cohen, W. Lück, G. M. Ziegler)

This talk will introduce a very natural and interesting differential complex associated with a CAT(0)-cube complex. The construction builds on ideas first introduced by Pytlik and Szwarc for the free group and extended by Julg and Valette in the case of groups acting on trees. We will extend ideas of Julg-Valette to show how this construction can be used to study K-amenability and K-homology of groups acting on CAT(0)-cube complexes. This talk is based on joint work with Erik Guentner and Nigel Higson.

Let \(G\) be a lattice in \(SO(n,1)\) and let \(h\colon G \to SO(n,1)\) be any representation. For cocompact lattices, the volume of a representation is an invariant whose maximal and rigidity properties have been studied extensively. We will show how to define the volume of a representation in the noncocompact case (a different definition by Francaviglia and Klaff also exists). In particular, we establish a rigidity result for maximal representations, recovering Mostow rigidity for hyperbolic manifolds.

In the cocompact case, the set of values for the volume of a representation is discrete. In even dimension, this follows from the fact that the volume form is an Euler class. In odd dimension, this was proven by Besson, Courtois and Gallot. The situation changes in the noncocompact case and for example the discreteness of the set of value is not valid anymore in dimension 2 and 3. We prove that in even dimension greater or equal to 4, the set of value of the volume of a representation is, up to a universal constant, an integer.

This is joint work with Marc Burger and Alessandra Iozzi.

The word problem for a given finitely generated group is the problem of telling whether a word in the generators represents the identity element. For any finitely presented group, this problem has an easy part: if the word is trivial, then it follows from the given finitely many relations; hence it is possible to algorithmically list all trivial words. Thus, in a group with unsolvable word problem, it is impossible to algorithmically list the non-trivial words.

For a finitely presented group, it is easy to list all actions on finite sets: for a proposed action of the generators, just check whether the relations hold. Hence, one can algorithmically list all finite quotients of a finitely presented group. This provides an obvious way of listing some non-trivial words: put down those, that represent a non-trivial element in some finite quotient. A group where this algorithm eventually finds each non-trivial word is called residually finite. Residually finite groups have an obvious solution to the word problem.

The conjugacy problem of telling which words represent conjugate elements allows for a similar treatment. In any finitely presented group, it is algorithmically easy to list all pairs of words representing conjugate elements. The hard part is to list the pairs of words representing non-conjugate elements. A group is called conjugacy separable, if any two non-conjugate elements stay non-conjugate in some finite quotient. Thus, finitely presented conjugacy separable groups admit an obvious solution to the conjugacy problem.

Other classical algorithmic problems can be treated analogously. Each leads to a corresponding notion of separability. The problem of telling whether two finitely generated subgroups are conjugate gives rise to the notion of subgroup conjugacy separability. A group is subgroup conjugacy separable if any two non-conjugate finitely generated subgroups have non-conjugate images in some finite quotient. We show that finitely generated free groups and fundamental groups of closed oriented surfaces are subgroup conjugacy separable.

We will use Fox derivatives to associate to a two-generator one-relator presentation a polytope with marked vertices. We will show that in most, possibly all, cases this polytope determines the Bieri-Neumann-Strebel invariant and the minimal complexity of HNN-decompositions of the group. This is joint work with Stephan Tillmann.

We explain how to construct a family of pairwise inequivalent irreducible unitary representations of an arbitrary hyperbolic group.

I will present some results on the arithmetic structure within the set of lengths of primitive geodesics in spaces of negative (or non-positive) curvature. For example, one can always find "almost" arithmetic progressions. In contrast, genuine arithmetic progressions seem quite rare. I will explain how to find such arithmetic progressions in the special case of arithmetic locally symmetric spaces. This is joint work with Ben McReynolds.

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. Using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical manifolds and give refined upper bounds of integral foliated simplicial volume in terms of stable integral simplicial volume. This allows us to compute the integral foliated simplicial volume of hyperbolic 3-manifolds. This is joint work with Cristina Pagliantini.

In this talk I will discuss homology theories used in large scale geometry such as Roe’s coarse homology, uniformly finite homology of Block and Weinberger or Dranishnikov’s coarsely equivariant homology. These homology theories are usually difficult to compute. In particular, there are no natural, general Kunneth-type formulas for such theories. I will present a geometric method for killing large scale homology theories of certain products. As an application we completely determine the uniformly finite homology of lattices in products of 2 and 3 trees. This is joint work with Francesca Diana (Regensburg).

The Singer conjecture states that \(L^2\)-homology of the fundamental group of an aspherical closed manifold vanishes except possibly in the middle dimension.

The cocompact action dimension of a group \(G\), \(\mathrm{cadim}(G)\), is the least dimension of a contractible manifold (possibly with boundary) which admits a proper cocompact \(G\)-action. The cadim conjecture states that \(L^2\)-homology of any group \(G\) vanishes above \(\mathrm{cadim}(G)/2\).

We prove that the Singer conjecture and the cadim conjecture are equivalent. We also prove the Singer conjecture for 4-dimensional Coxeter group manifolds.

Both proofs are using the notion of a hierarchy for a group action, which generalizes the idea of a Haken manifold. This is a joint work with Kevin Schreve.

I will describe my recent construction of small cancellation labellings for some infinite sequences of finite graphs of bounded degree. They are used to define infinite graphical small cancellation presentations of groups.This technique allows one to provide examples of groups with exotic properties:

I will describe the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu's Property A) that are coarsely embeddable into a Hilbert space. Moreover, these groups act properly on CAT(0) cubical complexes. (This is based in a large part on a joint work with Goulnara Arzhantseva.)

I will show the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs - in contrast, for the Gromov monster only a weak embedding is established. Such examples have important applications to the Baum-Connes conjecture. I will indicate further applications.

Under abstract group theoretic conditions on a countable group \(\Gamma\) we classify all lattice embeddings of \(\Gamma\) into a locally compact group \(G\). Under these, relatively mild, conditions the only possible non-uniform lattice embeddings turn out to be lattices in semisimple Lie groups, generalized S-arithmetic or S-adelic lattices.

The proofs rely on a number of results about rigidity of classical lattices (Margulis' arithmeticity and normal subgroup theorem) as well as about quasi-isometry rigidity (Kleiner-Leeb).

This is based on joint work with Uri Bader and Alex Furman.

In general not many methods are know to test simplicial complexes for being CAT(\(\kappa\)), not even if they come as a nice sub-complex of some CAT(\(\kappa\)) complex. In part one of my talk we will get to know this problem as well as some attempts how to solve it.

What got me interested in this question is the following: There is a close connection between braid groups, orthoscheme complexes of non-crossing partitions and the embeddability of the diagonal links of these complexes into spherical buildings of type A. This connection allows one to reduce the question whether braid groups are CAT(0) to showing that a certain subcomplex of a spherical building ist CAT(1). We use this reduction to show that braid groups with at most 6 strands are CAT(0). This will be the second part of the talk.

We present new examples of closed smooth locally CAT(0) manifolds which do not support Riemannian metrics of nonpositive sectional curvature.

I will introduce the operation, called dense amalgam, which to any tuple \(X_1,\dots,X_k\) of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted \(\widetilde\sqcup(X_1,\dots,X_k)\), in which there are many appropriately distributed copies of the spaces \(X_1,\dots,X_k\). I will also present a convenient characterization of dense amalgams, in terms of a list of properties, similar in spirit to the well known characterization of the Cantor set. I will explain that, in various settings, the ideal boundary of the free product of groups (amalgamated along finite subgroups) is homeomorphic to the dense amalgam of boundaries of the factors. For example, the boundary of a Coxeter group which has infinitely many ends, and which is not virtually free, is the dense amalgam of the boundaries of the maximal 1-ended special subgroups.

The Baumslag-Solitar groups are a particular class of two-generator one-relation groups which have played a surprisingly useful role in combinatorial and geometric group theory. They have provided examples which mark boundaries between different classes of groups and they often provide test-cases for theories and techniques. In this talk, I will illustrate the ideas behind the proof of the Farrell-Jones Conjecture for them. This is a joint work with F. Thomas Farrell.