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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Examples of 3–manifolds

1. Let \( K \subset \mathbb S^3\) be a knot, and \( \nu K \) a tubular neigbourhood. Then the knot complement \( \mathbb S^3 \setminus \nu K \) is a compact three-manifold with boundary a torus.

   Examples of knots are :

   • The unknot;
   • The trefoil :
     
   • The figure-eight :
     

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Torsion (continued from previous lecture)

Locally homogeneous setting

We now consider a compact locally symmetric manifold \( M = \Gamma \backslash X \), with a local system \( \mathscr L \) coming from a representation \( \rho: G \to \mathrm{SL}(V) \) of the whole Lie group (we need \( \Gamma \) to stabilise a lattice in \( V \)).

To compute the analytic torsion it is also convenient to use the heat kernel. For this recall the trace formula for the heat kernel:
\[
\mathrm{tr}(e^{-t\Delta_q}) := \sum_{j \ge 0} e^{-t\lambda_j} = \int_M \mathrm{tr}(e^{-t\Delta_q}(x, x)) dx
\]
where \( \lambda_j \) are the eigenvalues of \( \Delta_q \) in \( L^2C^q(M) \). With this equality, we can use the Mellin transform to get the following expression for the spectral zeta function \( \zeta_q \) of \( \Delta_q \):
\[
\zeta_q(s) = \frac 1 {\Gamma(s)} \int_0^{+\infty} t^s \mathrm{tr}(e^{-t\Delta_q}) \frac {dt} t.
\]
This expression can be adapted to the \( L^2 \)-setting, and we define the \( L^2 \)-torsion of \( X \) with coefficients in \( \rho \) to be:
\[
\tau_X^{(2)}(\rho) = \frac 1 2 \frac d{ds} \left( \frac 1 {\Gamma(s)} \int_0^{+\infty} dt t^{s-1} \int_{\widehat G} d_\pi e^{-t(\rho(\Omega) – \pi(\Omega))} d\mu_{pl}(\pi) \right)_{s=0}
\]
where:

• For a unitary representation \( \pi \), we put \( d_\pi = \sum_q q(-1)^q \dim \left( \wedge^q\mathfrak p^* \otimes \mathcal H \otimes \pi \right)^K \);
• \( \Omega \) is the Casimir operator of \( G \);
• \( \mu_{pl} \) is the Plancherel measure of \( G \).

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Betti numbers of symmetric spaces (continuing previous lecture)

Recall the setup: \( G, K, X=G/K \) are respectively a semisimple Lie group, a maximal compact subgroup and the associated symmetric space; \( \Gamma, M = \Gamma \backslash S \) are a discrete, torsion-free subgroup and the associated \( X \)-manifold. The de Rham complex of \( M \) can be computed via:
\[
C^q(M) = \mathrm{Hom}_K \left( \wedge^q\mathfrak p, C^\infty(\Gamma \backslash G) \right)
\]
and there is a Laplace operator \( \Delta_q \) which is essentially auto-adjoint and positive on the \( L^2 \)-completion of \( C^q \), and an associated heat operator which is given by convolution with the heat kernel
\[
e^{-t\Delta_q} \in \left( \mathrm{End}(\wedge^q \mathfrak p^*) \otimes C^\infty(\Gamma \backslash G \times \Gamma \backslash G) \right)^{K \times K}.
\]
The Betti numbers of \( M \) are then given by the formula:
\[
b_q(M) = \lim_{t \to +\infty} \int_M \mathrm{tr}(e^{-t\Delta_q}(x, x)) dx.
\]

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Betti numbers of locally symmetric spaces

Let \( G \) be a semisimple Lie group, for example:

• \( G = \mathrm{SL}_2({\mathbb R}) \) or \( \mathrm{SL}_m({\mathbb R}), m \ge 3 \);
• \( G = \mathrm{SO}(p, q), p, q \ge 1 \) ;
• \( G = \mathrm{SL}_2({\mathbb C}) \), which is isogenic to \( \mathrm{SO}(3,1) \).

Let \( K \) be a maximal compact subgroup of \( G \), and let \( X = G/K \) be the associated Riemannian symmetric space, whose group of orientation-preserving isometries is \( G \) (at least if the latter is connected). In the examples:

• If \( G = \mathrm{SL}_2({\mathbb R}) \), we can take \( K = \mathrm{SO}(2) \) and \( X \) is then the hyperbolic plane \( {\mathbb H}^2 \);
• If \( G = \mathrm{SL}_2({\mathbb C}) \), \( K = \mathrm{SU}(2) \) then \( X \) is the hyperbolic space \( {\mathbb H}^3 \);
• If \( G = \mathrm{SL}_m({\mathbb R}) \) and \( K = \mathrm{SO}(m) \) then \( X \) is the subset of the projective space of m by m matrices which is the image of the cone of positive definite matrices.

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