Universal $$L^2$$-torsion, $$L^2$$-Euler characteristic, Thurston norm and polytopes (Wolfgang Lück)

The aim of the talk is to define an invariant (« universal $$L^2$$-torsion ») from which many others (usual $$L^2$$-torsion, $$L^2$$-Alexander invariant and Euler characteristic,…) can be derived, as well as the relations between them.

Classifying spaces for families and their finiteness properties (Brita Nucinkis)

Let $$G$$ be a group and $$\mathcal F$$ a family of subgroups such that:

• For all $$H \in \mathcal F, g \in G$$ we have also $$H^g := gHg^{-1} \in \mathcal F _); • For all \( H, H’ \in \mathcal F$$ we have $$H \cap H’ \in \mathcal F$$.

For example $$\mathcal F$$ can be :

• The trivial subgroup;
• Finite subgroups;
• Virtually cyclic subgroups;
• Free abelian, nilpotent, … subgroups

A model for the classifying space \e E_{\mathcal F}G \) is then a CW-complex $$X$$ with a $$G$$-action such that:

• If $$H \in \mathcal F$$ then the subset $$X^H$$ of points fixed by $$H$$ is a contractible subcomplex of $$X$$;
• Otherwise $$X^H$$ is empty.

For example, if $$\mathcal F = \{\{e\}\}$$ then $$E_{\mathcal F}G$$ is the classifying space $$EG$$ for $$G$$. The classifying space can also be defined by the following universal property: it is the only G-complex $$Y$$ such that for every $$G$$-action whose point stabilisers are in $$F$$, there exists a $$G$$-map $$E_{\mathcal F}G \to Y$$ (which is unique up to $$G$$-homotopy).

Analytic aspects of locally compact groups (Corina Ciobotaru)

The objects of interest in this talk are the unitary representations of a locally compact group $$G$$. These are homomorphisms $$\pi: G \to \mathrm U(\mathcal H)$$ where $$\mathcal U(\mathcal H$$ is the group of unitary operators on a Hilbert space $$\mathcal H$$. It will be required that they be continuous in the following sense: for every $$v \in \mathcal H$$ the map $$G \to \mathcal H, g \mapsto \pi(g)v$$ is continuous. Basic examples are the following:

1. The trivial representation;
2. The left-regular representation $$\lambda: G \to \mathcal U(L^2(G, \mu_{Haar}))$$ acting by $$\lambda(g)f(x) = f(g^{-1}x)$$.

$$L^2$$-Alexander torsions of 3–manifolds (Yi Liu)

Let $$N$$ be an irreducible, compact, orientable 3–manifold whose boundary is either empty or contains only tori as connected components. Call a triple $$(G, \gamma, \phi)$$ admissible if $$G$$ is a discrete group, $$\gamma : \pi_1(N) \to G$$ and $$phi : \pi = \pi_1(N) \to {\mathbb Z}$$ are homomorphisms such that there exists a commutative diagram:
$\begin{array}{ccc} \pi_1(N) & \overset{\gamma}{\rightarrow} & G \\ & \underset{\phi}{\searrow} & \downarrow \\ & & \mathbb Z \end{array}$
Fix a cellulation of $$N$$ and let $$C_*(\widetilde N)$$ be the chain complex of the universal cover. Let $$t > 0$$ and define a representation:
$\kappa(\gamma, \phi, t) :\left\{ \begin{array}{ll} {\mathbb Z} \pi \to {\mathbb R} G \\ g \mapsto t^{\phi(g)}\gamma(g) \end{array} \right.$
with which the twisted $$L^2$$-complex $$\ell^2(G) \otimes_{\kappa(\gamma, \phi, t)} C_*(\widetilde N)$$. Let $$\tau^{(2)}(N; \gamma, \phi)(t)$$ be the $$L^2$$-torsion of this complex in the case where it is well-defined (when the complex is weakly acyclic and all its differentials of determinant class) and 0 otherwise. Consider this construction as associating to the admissible triple $$(G, \gamma, \phi)$$ a function $$tau^{(2)}(N; \gamma, \phi): ]0, +\infty [ \to [0, +\infty[$$.

Rank, combinatorial cost and homology growth of higher-rank lattices (Miklós Abért)

For any finitely generated group $$H$$ let $$d(H)$$ be its rank, the minimal number of elements needed to generate $$H$$. If $$H$$ is a finite-index subgroup in a finitely generated group $$\Gamma$$ then we have
$d(H) \le |\Gamma / H| (d(\Gamma) – 1)$
and it is thus natural to define:
$r(\Gamma, H) = \frac{d(H) – 1}{|\Gamma / H|}.$
If $$\Gamma = \Gamma_0 \supset \Gamma_1 \supset \cdots$$ is a chain of finite index subgroup then the limit:
$\mathrm{RG}(\Gamma, \Gamma_n) = \lim_{n\to+\infty} r(\Gamma, \Gamma_n)$
exists, and is called the rank gradient of $$(\Gamma, (\Gamma_n))$$.

If $$(\Gamma_n), (\Delta_n)$$ are two residual chains in the same group $$\Gamma$$ (chains with $$\Gamma_n, \Delta_n$$ normal in $$\Gamma$$ and $$\bigcup_n \Gamma_n = \{ 1 \} = \bigcup_n \Delta_n$$), then are $$\mathrm{RG}(\Gamma, (\Gamma_n)$$ and $$\mathrm{RG}(\Gamma, (\Delta_n))$$ equal?

Growth of torsion homology in amenable groups (Nikolay Nikolov)

Let $$G$$ be a discrete group and $$\widetilde X$$ a simply-connected CW-complex with a free $$G$$-action, and $$X$$ the quotient $$G \backslash \widetilde X$$. A particular case is when $$\widetilde X$$ is a classifying space for $$G$$, i.e. contarctible.

The aim is to study the homology groups $$H_n(\cdot; {\mathbb Z})$$ for finite covers of $$X$$. For this suppose that the $$n + 1$$-skeleton of $$X$$ is finite, and take a residual chain $$G_0 = G \supset G_1 \supset \cdots$$ of normal, finite-index subgroups of $$G$$ such that $$\bigcap_i G_i = \{ 1 \}$$. Denote $$X_i = G_i \backslash \widetilde X$$. The Lûck Approximation Theorem states that:
$\lim_{i \to +\infty} \frac{\mathrm{rank}_{\mathbb Z} H_n(X_i ; {\mathbb Z})} {|G / G_i|} = b_n^{(2)}(\widetilde X \to X).$
The question motivating the rest of the talk will be to estimate the growth of $$t(H_n(X_i ; {\mathbb Z}))$$ (where $$t(A)$$ is the size of the torsion subgroup of a finitely generated Abelian group $$A$$). In full generality it is possible to say that $$\log(t(H_n(X_i ; {\mathbb Z})) / |G/G_i|$$ is bounded.

Theorem (Kar–Kropholler–Nikolov): Suppose that $$G$$ is amenable and that $$H_n(\widetilde X; {\mathbb Z}) = 0$$ (for example $$\widetilde X$$ is contractible). Then
$\lim_{i \to +\infty} \frac{t(H_n(X_i; {\mathbb Z}))}{|G/G_i|} = 0.$

Alexander and Thurston norms, and the Bieri–Neumann–Strebel invariants for free-by-cyclic groups (Dawid Kielak, notes by Steffen Kionke)

Hyperbolic manifolds

Let $$\Sigma$$ be a surface and $$f \in \mathrm{Homeo}^+(\Sigma)$$. Let $$M$$ be the 3–manifold obtained from $$\Sigma \times [0, 1]$$ by identifying $$\Sigma \times \{0\}$$ with $$\Sigma \times\{1\}$$ via $$f$$. If $$f$$ is a pseudo-Anosov diffeomorphism then $$M$$ is hyperbolic. If a 3–manifold $$M$$ is obtained from this construction say that it is fibered. A theorem of Agol states that every hyperbolic manifold has a finite cover which is fibered.

If $$M$$ is fibered with fiber $$\Sigma$$ and monodromy $$f$$ then its fundamental group has a splitting:
$1 \to \pi_1(\Sigma) \to \pi_1(M) \to {\mathbb Z} \to 1$
coming from the presentation
$\pi_1(M) = \langle \pi_1(\Sigma), t | \forall x \in \pi_1(\Sigma) t^{-1}xt = f_*(x) \rangle.$
More generally, if $$H$$ is a group and $$f : H \to H$$ is an injective morphism then the group obtained by:
$G = \langle H, t | \forall x \in H, t^{-1}xt = f(x) \rangle$
is called an ascending HNN-extension and denoted by $$H *_f$$. Then:

• Any semi-direct product $$H \times {\mathbb Z}$$ is an ascending HNN-extension;
• If $$G = H *_f$$ let $$\phi: G \to {\mathbb Z}$$ be the mosphism defined by $$\phi|_G \equiv 0$$ and $$\phi(t) = 1$$; it will be called the induces character of the extension $$H *_f$$.

Definition: Let $$G$$ be a group with a finite generating set $$S$$. The Bieri–Neumann–Strebel invariant is the subset $$\Sigma(G) \subset H^1(G, {\mathbb R}) \setminus \{0\}$$ containing all classes $$\phi$$ such that the subgraph of the Cayley graph of $$G$$ induced by the subset $$\{ g \in G: \phi(g) \ge 0 \}$$ is connected.

The Atiyah problem for k-homology gradients (Łukasz Grabowski)

Lück approximation theorem

Let $$K$$ be a finite CW-complex with residually finite fundamental group $$\Gamma = \pi_1(K)$$. Let $$\Gamma=\Gamma_0 \supset \Gamma_1 \supset \cdots )$$ be a residual chain in $$\Gamma$$, meaning that the $$\Gamma_n$$ are finite-index, normal subgroups and $$\bigcap_n \Gamma_n = \{ 1 \}$$. Denote by $$widetilde K \to K$$ the universal cover and $$K_n = \Gamma_n \backslash \widetilde K$$. Then the Lück approximation theorem states that the $$L^2$$-Betti numbers of the covering $$\widetilde K \to K$$ are given by :
$\beta_q^{(2)}(\widetilde K \to K) = \lim_{n\to+\infty} \frac {b_n(K_n; {\mathbb Q})} {|\Gamma / \Gamma_n|}.$
The following question is then very natural, and was apparently first asked by Farber around 1998:

Question: Can one prove a statement similar to Lück approximation for Betti numbers with coefficients in a field of positive characteristic?

The center-valued Atiyah conjecture (Thomas Schick)

The main theme of this talk is the interplay between the algebra of group rings and the analysis behind $$L^2$$-Betti numbers.

L2-invariants and 3–manifolds, II (Stefan Friedl)

Geometrisation

Let $$N$$ be a compact, orientable 3–manifold. Say that $$N$$ is prime if it cannot be decomposed as the connected sum of two 3–manifolds both not homeomorphic to $$\mathbb S^3$$. (By the Sphere Theorem this is equivalent to $$N$$ not containing any sphere not bounding a 3-ball, and $$M \not= \mathbb S^2 \times \mathbb S^2$$.)

An embedded torus $$T \subset N$$ is said to be essential if the induced map $$\pi_1(T) \to \pi_1(N)$$ is injective. By Papakryakopoulos’ Loop Theorem, if $$N$$ is prime this is equivalent to $$T$$ not being the boundary of an embeded solid torus.

Geometrisation Theorem (Perelman, conjectured by W. Thurston): Let $$N$$ be a prime 3–manifold. Then one of the following holds:

1. $$N$$ is Seifert fibered;
2. $$N$$ is hyperbolic;
3. $$N$$ contains an incompressible torus.