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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