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.

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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).

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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 $\lambda(g)f(x) = f(g^{-1}x)$.

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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 $tau^{(2)}(N; \gamma, \phi): ]0, +\infty [ \to [0, +\infty[$.

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Rank gradient

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?

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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

$$\lim_{i \to +\infty} \frac{t(H_n(X_i; {\mathbb Z}))}{|G/G_i|} = 0.$$

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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.

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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?

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

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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.

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