Notation

• The group \( K_1({\mathbb Z} G) \) is the abelian group generated by all invertible matrices \( A \in M_n({\mathbb Z} G), n \ge 0 \) with relations \( [AB] = [A] + [B] \) and \( \left[\begin{pmatrix} A & C \\ 0 & B \end{pmatrix} \right] = [A] + [B] \).
• The Whitehead group \( \mathrm{Wh}(G) \) is the quotient of \( K_1({\mathbb Z} G) \) by the elements \( [(\pm g)], g \in G \).
• The weak \( K_1 \)-group \( K_1^w(G) \) is defined as \( K_1(G) \) but by taking the larger set of matrices which define « weak isomorphisms » (have no kernel in \( L^2(G) \)).

Universal \( L^2 \)-torsion

Let \( C_* \) be a finite complex of free, finitely generated \( {\mathbb Z} G \)-complexes, each \( C_i \) having a preffered basis. If \( C_* \) is acyclic, that is \( H_*(C_*) = 0 \), then the Reidemeister torsion \( \rho(C_*) \in K_1({\mathbb Z} G) \) is well-defined (and independent of the choice of basis). If it is only \( L^2 \)-acyclic then there is a well-defined invariant \( \rho_u^{(2)}(C_*) \in K_1^w({\mathbb Z} G) \) which has a universal property.

For the rest of the talk use the following convention: \( X, Y, \ldots \) will be finite CW-complexes, and \( \overline X, \overline Y, \ldots \) will be \( G \)-covers of each of them. If \( H_*^{(2)}(\overline X; G ) = 0 \) then denote by \( \rho_u^{(2)}(\overline X) = \rho_u^{(2)}(C_*(\overline X)) \).

Properties of the universal \( L^2 \)-torsion

• Homotopy invariance: Let \( f: \overline X \to \overline Y \) be a \( G \)-equivariant homotopy equivalence, and let \( \rho(f) \in K_1({\mathbb Z} G) \) be its Whitehead torsion. Then \( \rho(f) \) maps to \( \rho_u^{(2)}(\overline X) – \rho_u^{(2)}(\overline X) \) under the natural map \( K_1({\mathbb Z} G) \to K_1^w({\mathbb Z} G) \).
• Additivity: The universal torsion is additive in exact sequences; if the inclusion maps between CW-complexes are \( \pi_1 \)-injective then everything is functorial with respect to maps from the classical \( K_1 \).
• Fibrations: If \( F \overset{i_*}{\to} E \to B \) is a fibration such that \( i_* \) is \( pi_1 \)-injective then \( \rho_u^{(2)}(E) = \chi(B)i_*\rho_u^{(2)}(\overline F) \).
• Poincaré duality: If \( X \) is a \( n \)-dimensional manifold then \( \rho_u^{(2)}(X) = (-1)^{n+1}\rho^{(2)}(X)^* \).
• Realisability: If there exists an \( L^2 \)-acyclic \( G \)-universal cover then for all \( \eta \in K_1^w({\mathbb Z} G) \) there exists an \( L^2 \)-acyclic covering space \( \overline X \) with \( \rho_u^{(2)}(\overline X) = \eta \).
• 3–manifolds: The universal \( L^2 \)-torsion can be computed from the geometric pieces, and for manifolds with toric boundary it is equal to \( [A_i] – [s_i-1] \) where \( A \) is the Fox matrix minus its i-th column (and \( s_i \) the corresponding generator).

An example

Suppose \( G = {\mathbb Z} \); then \( \mathrm{Wh}(G) = \{ 0 \} \) and \( \mathrm{Wh}^w(G) = {\mathbb Q}(x) \). The universal \( L^2 \)-torsion is the Reidemeister torsion.

\( L^2 \)-torsion and the \( L^2 \)-Alexander invariant

The Fuglede–Kadison determinant gives a morphism \( \mathrm{Wh}^w(G) \to {\mathbb R}_{\ge 0} \). The image of \( \rho_u^{(2)}(\overline M) \) under this map is the \( L^2 \)-torsion \( \rho^{(2)}(\overline M) \).

Note that if \( M \) is a compact, even-dimensional manifold the \( L^2 \)-torsion is always trivial by Poincaré duality, while the universal \( L^2 \)-torsion might not be.

If \( G \) is resudually finite, then for every cohomology class \( \phi \in H^1(M) \) there is associated a map \( {\mathbb R}_{>0} \to {\mathbb R}_{\ge 0} \), the \( L^2 \)-Alexander torsion \( \rho^{(2)}(\overline M, \phi) \). It can be constructed through the universal _( L^2 \)-torsion; a more general hope is that there exists a pairing
\[
\mathrm{Rep}(G) \times \mathrm{Wh}^w(G) \to {\mathbb R}.
\]

There is also a map from \( \mathrm{Wh}^w(G) \to \mathcal P(H_1(G)_{\mathrm{free}}) \), the space of polytopes in the first homology space with integral vertices. This associates to a \( G \)-covering space a polytope \( \mathrm P(\overline M) \).

Theorem: For a closed 3–manifold \( M \), \( \mathrm P(\widetilde M) \) is roughly the unit polytope fr the Thurston norm of \( M \).

\( L^2 \)-Euler characteristic and twisted version

If \( C_* \) is any \( {\mathbb Z} G \) chain complex, consider the \( \mathcal NG \)-cchain complex \( C_* \otimes_{{\mathbb Z} G} \mathcal NG \). It has well-defined (possibly infinite) \( L^2 \)-Betti numbers defined using the following result.

Theorem (Lück, XX^th century): There exists a dimension function for arbitrary \( \mathcal NG \)-modules, which extends the von Neumann dimension.

Suppose that the series \( \sum_{n \in {\mathbb Z}} b_n^{(2)}(C_*) \) converges. Let \( h^{(2)}(C_*) \) be its sum, and define the \( L^2 \)-Euler characteristic by \( \chi^{(2)}(C_*) = \sum_{n\in{\mathbb Z}} (-1)^n b_n^{(2)}(C_*) \).

Let \( \Phi: G \to {\mathbb Z} \). Define
\[
C_*^\Phi(\overline X) = C_*(\overline X) \otimes_{\mathbb Z} \phi^*{\mathbb Z}[{\mathbb Z}]
\]
and let
\[
h^{(2)}(\overline X ; \Phi) = h^{(2)}(C_*^\Phi(\overline X)), \: \chi^{(2)}(\overline X; \Phi) = \chi^{(2)}(C_*^\Phi(\overline X)).
\]
Note that if \( K = \ker(\Phi) \) then \( b_n^{(2)}(C_*^\Phi(\overline X)) = b_n^{(2)}(\mathcal NK \otimes_{{\mathbb Z} K} C_*|_{{\mathbb Z} K}) \).

Theorem: Suppose that \( G \) is torsion-free and satisfies the Atiyah conjecture. Then \( h^{(2)}(\overline X; \Phi) \) is finite and \( \chi^{(2)}(\overline X; \Phi) \) is an integer.

Example

Let \( E \) be the mapping torus of an homeomorphism \( f : F \to F \) and \( p: E \to \mathbb S^1 \) the natural projection. Let \( \Phi = p_* \) and \( \overline E \) the associated infinite cyclic covering space, which is homotopy equivalent to \( F \). Then:
\[
\chi^{(2)}(\widetilde E; \Phi) = \chi^{(2)}(\mathcal NK) \otimes_{{\mathbb Z} K} C_*(\widetilde E)|_K) = \chi^{(2)}(\mathcal NK \otimes C_*(F))
\]
so in the end \( \chi^{(2)}(\widetilde E; \Phi) = \chi(F) \).

In general, the following result is valid.

Theorem (Friedl–Lück): If \( M \) is any closed 3–manifold then \( \chi^{(2)}(\widetilde M; \Phi) = \|\Phi\|_T \).

An application to knot theory

Question (Simon): Let \( \mathbb S^3 \setminus K, \mathbb S^3 \setminus K’ \) be two knots in teh 3–sphere and \( M = \mathbb S^3 \setminus K, M’ = Y’ \). Suppose that there exists a continuous map \( f: M \to M’ \) such that \( \pi_1(f) \) is surjective and \( f_* \) is an isomorphism \( H_1(M; {\mathbb Q}) \to H_1(M'; {\mathbb Q}) \). Does it follow that \( g(K) \ge g(K’) \)?

It can be shown that if \( G \) is residually elementary amenable locally indicable then \( \chi^{(2)}(\widetilde M; f^*\phi) \ge \chi^{(2)}(\widetilde{M’}; \phi) \), which implies a positive answer by the theorem above.

• 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).

If \( E_{\mathcal F}G \) has a cocompact model the we say that \( G \) is of type \( F^{\mathcal F} \). In case \( \mathcal F \) is the family of finite subgroups we denote \( E_{\mathcal F}G = \underline E G, F^{\mathcal F} = \underline F \) and if \( F \) is the family of virtually cyclic subgroups \( E_{\mathcal F}G = \underline{\underline E} G,\)\( F^{\mathcal F} = \underline{\underline F} \).

We also define:

• The geometric dimension \( \mathrm{gd}_{\mathcal F}(G) \in {\mathbb Z}_{\ge 0} \cup\{\infty\} \) is the minimal dimension for a model of \( E_{\mathcal F}G \);
• If there exists a model for \( E_{\mathcal F}G \) whose \( k \)-skeleton is cocompact then we say that \( G \) is of type \( F_k^{\mathcal F} \), and if this holds for all \( k \) of type \( F_{\mathcal F}^\infty \).

Examples for \( \mathcal F \) = finite subgroups

• If \( G = D_\infty \) (the infinite dihedral group) the the real line is a model for \( \underline E G \).
• More generally, if \( G \) is a crystallographic group then the corresponding tesselation of Euclidean space is a model for \( \underline E G \).
• If \( G \) is hyperbolic then its Rips complex is a model for \( \underline E G \) (Meintrupp–Schick).
• If \( G = \mathrm{Out}(F_n) \) (resp. \( \mathrm{MCG}(S) \) then the spine of outer space (resp. of Teichmüller space) is a model for \( \underline E G \) (Vogtmann, Mislin).
• More generally, any group acting cocompactly on a CAT(0)-space has it as a model for \( \underline E G \).

Note that for crystallographic groups the geometric dimension \( \underline{\mathrm{gd}}(G) \) is equal to the Hirsch length of \( G \) (the sum \( \sum_i (G_i/G_{i+1})\otimes{\mathbb Q} \) is \( G_i \) is the derived series of \( G \)). This is true more generally for solvable groups, and even for elementary amenable groups (with the definition of Hirsch length suitably extended) by a result of Kropholler–Martínez-Pérez–Nucinkis. This is also equal to the virtual cohomological dimension \( \mathrm{vcd}(G) \).

For the mapping class group it also holds that \( \underline{\mathrm{gd}}(G) = \mathrm{vcd}(G) \) (Aramayona–Martínez-Pérez).

On the other hand, in general both \( \underline{\mathrm{gd}}(G) \) and the class \( \underline F_\infty \) do not behave well under finite extensions. Note that if \( N \to G \to Q \) is an extension with \( Q \) finite and \( E \) torsion-free, and \( X \) is a model for \( EN \) then \( E^Q \) is a model for \( \underline EG \). However it is not cocompact.

Leary–Nucinkis have examples where \( N \) is of type \( F \) but \( G \) is not of type \( \underline F \), and where \( \underline{\mathrm{gd}}(G) > \mathrm{gd}(N) \) with \( Q = \mathfrak A_5 \) (Martínez-Pérez has examples with \( Q = {\mathbb Z} / p \).

There are also examples where \( G \) is of type \( \underline F \) but still \( \underline{\mathrm{gd}}(G) > \mathrm{gd}(N) \) (Petrosyan–Leary, Degrisje–Souto).

The case where \( \mathcal F = \) virtually cyclic subgroups

In case \( G = {\mathbb Z}^2 \) a model for \( \underline{\underline E}G \) is given by the join of an infinite set of edges indexed by the cyclic subgroups (an edge is stabilised by its indexing subgroup). This is a non-cocompact model, but it shows that \( \underline{\underline{\mathrm{gd}}} (G) = 3 \).

Juan-Pineole and Leary construct a 2-dimensional model when \( G \) has a unique maximal virtually cyclic subgroup. They conjecture the following.

Conjecture: G has a cocompact \( \underline{\underline E}G \) if and only if it is virtually cyclic.

An easy result is the following.

Lemma: If \( G \) is \( \underline{\underline F}{}_\infty \) then all normalisers of virtually cyclic subgroups in \( G \) are as well.

Kochloukova–Martínez-Pérez–Nucinkis prove that for solvable \( G \) the conclusion of the lemma implies that \( G \) is polycyclic.

Groves–Wilson prove that an elementary amenable group which is \( \underline{\underline F}{}_0 \) has to be virtually cyclic.

The Lück–Weiermann construction

This is a construction used for almost all known classifying spaces.

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

The coefficients of such a representation are the function on \( G \) of the form
\[
c_{v, w}(g) = \langle \pi(g)v, w \rangle_{\mathcal H}
\]
for \( v, w \in \mathcal H \).

The basic problem is to classify those (irreducible) representations for a given group \( G \). It is not solved in general, for example there is no complete classification of unitary representation of the group \( \mathrm{SL}_2(\mathbb Q_p) \).

Classes of groups for which this is of interest include the following:

1. Semisimple Lie groups such as \( \mathrm{Sl}_m({\mathbb R}) \);
2. Semisimple algebraic groups over non-archimedean local fields, such as \( \mathrm{SL}_m(\mathbb Q_p) \);
3. Closed subgroups of the group \( \mathrm{Aut}(T_d) \) of automorphisms of a d-regular tree \( T_d \).

Unitary representations can also be an interesting object of study for some non-locally compact groups such as groups of infinitely-supported permutations.

Before attempting a classification it is useful to look at the behaviour of the matrix coefficients of unitary representations. In particular, representations with the following property are of particular interest: say that a representation is \( C_0 \) if all its coefficients vanish at infinity, that is
\[
\forall v, w \in \mathcal H, \lim_{g\to+\infty} \langle \pi(g)v, w \rangle = 0
\]
where \( g \to +\infty \) means that \( g \) escapes every compact subset of \( G \).

Definition:

• Say that a group has the Howe–Moore property if every unitary representation which does not contain (nonzero) invariant vectors is \( C_0 \).
• Say that a group \( G \) is type I if for every unitary representation \( \pi \) of \( G \) the von Neumann algebra generated by \( \pi(G) \) is of type I (every factor is the algebra of bounded operators on some Hilbert space).

For example:

• Simple Lie groups and algebraic groups over local fields have the Howe–Moore property and are type I (note that in the general semisimple case the property must be defined with more care to take into account representations factoring through a factor);
• If \( H \le \mathrm{Aut}(T_d) \) is a closed, non-ccompact subgroups which acts 2-transitively on the boundary \( \partial T_d \) then it is HM and Type I.

Groups acting on regular tress

Recall that the topology on \( \mathrm{Aut}(T_d) \) is defined by taking pointwise stabilisers of finite subtrees as a basis of neighbourhoods for the identity. In this topology the stabiliser of a vertex is a maximal compact subgroup.

here are some explanations:

• A subgroup \( H \subset \mathrm{Aut}(T_d) \) satisfies Tits’ independence condition if for any edge \( e \) separating \( T_d \) into two half-trees \( T^+, T^- \) the stabiliser \( H_e \) splits as the direct product \( H_{T^+} \times H_{T^-} \) (where \( H_{T^\pm} \) is the stabiliser of the half-tree).
• If \( G \) is a locally compact group and \( K \) a compact subgroup then \( (G, K) \) is called a Gelfand pair if for every irreducible unitary representation \( \pi \) of \( G \) the space \( \mathcal H_\pi^K \) of \( K \)-foxed vectors is of dimension at most 1. This is equivalent to the Hecke algebra of bi-\( K \)-invariant functions being commutative.

It is open whether the Howe–Moore property implies 2-transitivity on the boundary. On the other hand if \( H \) is Type I then it is 2-transitive and has Tits’ independence property (Houdayer–Raum), and if \( (H, H_x) \) is a Gelfand pair then H is 2-transitive (this is due to Caprace–Ciobotaru in the larger context of Euclidean buildings).

Burger–Mozes groups

Let \( F \subset \mathfrak S(d) \) be a non-2-transitive subgroup. Then the Burger–Mozes group \( U(F)^+ \) (finite-index subgroup in the subgroup of automorphisms acting locally by \( F \) is not 2-transitive on the boundary.

Question: Doe \( U(F)^+ \) have the Howe–Moore property?

This group is not type I since it has Tits’ independence property; also, since \( (U(F)^+, U(F)_x^+ \) is not a Gelfand pair there exists an irreducible representation of \( H \) where \( U(F)_x^+ \) has a 2-dimensional subspace of fixed vectors.

Problem: Is it possible to construct this explicitely this representation? Is it \( C_0 \)?

Theorem: If \( F \) is primitive then all its representations induced from « parabolic subgroups » (closed subgroups stabilising a vertex at infinity) are \( C_0 \).

If in addition \( F \) is not 2-transitive then teh Hecke algebra is infinitely generated and infinitely presented.

Remarks:

1. Changing the CW-structure on \( N \) changes the function \( tau^{(2)}(N; \gamma, \phi) \) by a factor \( t^r \) for some \( r \in {\mathbb R} \). It is possible to eleminate this indeterminacy by adding Euler structures to the mix.
2. If \( A \in M_{p\times p}({\mathbb Z}\pi) \) and \( C_* \) is the complex:
   \[
   (\ast) \qquad 0 \to ({\mathbb Z} \pi )^p \overset{A}{\rightarrow} ({\mathbb Z} \pi)^p \to 0
   \]
   then the \( L^2 \)-torsion of the twisted complex \( \ell^2(G) \otimes_{\kappa(\gamma, \phi, t)} C_* \) is equal to the « regularised determinant »
   \[
   \det{}^{\mathrm r}(A) = \left\{\begin{array}{ll}
   \det_{\mathcal NG}(\kappa(\gamma, \phi, t)(A)) \text{ if } \ker(\kappa(\gamma, \phi, t)(A)) = 0 ; \\
   0 \text{ otherwise.} \end{array} \right.
   \]

Examples

1. If \( G = \pi_1(N) \) and \( \gamma \) is the identity map, then for \( \phi \in H^1(N; {\mathbb R}) \) the \( L^2 \)-Alexander torsion is called the full \( L^2 \)-Alexander torsion and denoted \( \tau^{(2)}(N; \phi) \).
2. In the case where \( G = {\mathbb Z}^m \) is a free Abelian group then the Fuglede-Kadison determinant can be computed to be the Mahler measure of the classical determinant. In particular, when \( G = {\mathbb Z} \) and for the complex in \( (\ast) \) above the \( L^2 \)-Alexander torsion is given by:
   \[
   \tau^{(2)}(N; \phi, \phi) = Ct^d \prod_{i=1}^d \max(1, t^{-1}|z_i|)
   \]
   where the \( z_i \) are roots of the polynomial \( \det(\phi(A)) \) (see Stefan Friedl’s second lecture).

Multiplicative convexity

Return to the case where \( G \) is virtually cyclic. The, putting \( t = \log(u) \), in the case of an acyclic complex concentrated in one dimension we have that
\[
\log \tau^{(2)}(N; \phi, \phi)(e^u) = du + \sum_{i=1}^d \max(0, \log|z_i| – u) + \log (C)
\]
which is a piecewise affine function whose derivative is nondecreasing. In particular it is a convex function of \( u \), and this means that \( \tau^{(2)}(N; \phi, \phi) \) is a multiplicatively convex function. For a function \( f: ]0, +\infty[ \to [0, +\infty[ \) this means that for all \( t_0, t_1 > 0 \) we have
\[
\sqrt{f(t_0) f(t_1)} \ge f(\sqrt{t_0 t_1}).
\]
It implies that \( f \) is continuous, and that if it is zero at one point then it is identically zero. In the remainder we will explain how to prove multiplicative convexity for the full Alexander \( L^2 \)-torsion, using approximation by the Abelian case (using a theorem of Boyd on approximating the multivariate Mahler measure by univariate ones it is possible to pass from virtually cyclic to virtually f.g. abelian).

For this let \( 1 = \Gamma_0 \leftarrow \Gamma_1 \leftarrow \cdots \leftarrow \pi \) be a tower of finite quotients of \( \pi \) whose kernels have trivial intersection, and let \( G_n \) be a virtually free abelian quotient of \( \ker(\pi \to \Gamma_n) \), \( \gamma_n \) the morphism from this to \( G_n \) and \( \phi_n \) the lift of . Let \( A_n(t) = \kappa(\gamma_n, \phi, t)(A) \) and \( V_n(t) = \det^{\mathrm r} A_n(t) \). Let \( A_\infty(t) = \kappa(\mathrm{Id}, \phi, t)(A) \) and \( V_\infty(t) = \det^{\mathrm r} A_\infty(t) \). Fix \( t_0, t_1 > 0 \); we want to prove that
\[
\sqrt{V_\infty(t_0) V_\infty(t_1)} \ge V_\infty(\sqrt{t_0 t_1}).
\]
We know by the previous paragraph that
\[
\sqrt{V_n(t_0) V_n(t_1)} \ge V_n(\sqrt{t_0 t_1}).
\]
If we knew approximation for the residual chain we would be done, but this is not the case. We will use instead the two following properties of the regularised determinant:

1. \( \limsup_{n\to+\infty} V_n(t) \le V_\infty(t) \)
2. If \( \varepsilon > 0 \) and \( A \) is positive then \( \lim_{n\to+\infty} \det^{\mathrm r}(A_n(t) + \varepsilon) = \det^{\mathrm r}(A_\infty(t) + \varepsilon) \).
3. If \( A \) is positive then \( \lim_{\varepsilon \to 0} \det^{\mathrm r}(A_\infty(t) + \varepsilon) = \det^{\mathrm r}(A_\infty(t))

Using the first one we get that
\[
\sqrt{V_\infty(t_0) V_\infty(t_1)} \ge \limsup_{n\to+\infty}(V_n(\sqrt{t_0 t_1})).
\]
By adding \( \varepsilon \) to the matrices in the determinants on the right-hand and taking it to 0 in an intelligent way it is then possible to conclude.

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?

One of the motivations for this is the following problem in 3–dimensional topology. Let \( M \) be a closed hyperbolic 3–manifold with Heegard genus \( g(M) \). Then \( d(\pi_1(M)) \le g(M) \). The « rank vs. gradient » problem was to determine whether equality always holds; Tao Li proved that for any \( C > 0 \) there exists an hyperbolic 3–manifold with \( g(m) – d(\pi_1(M)) \ge C \), answering this in the negative. On the other hand the following problem is still open:

Is the quotient \( d(\pi_1(M)) / g(M) \) bounded on the set of all closed hyperbolic 3–manifolds?

Combinatorial cost

Let \( \Gamma \) act via \( \phi \) on a set \( X \) preserving a probability measure \( \mu \). The associated measured groupoid \( \mathcal G \) is defined as follows: it is the set \(\Gamma \times X \) with the composition law \( (g, y) \cdot (h, x) = (gh, x) \) if \( \phi(h)x = y \). Let \( \tilde \mu \) be the product measure on \( \Gamma \times X \) (where \( \Gamma \) is endowed with the counting measure). Then the cost of the action is defined by:
\[
\mathrm{Cost}(\phi) = \inf(\tilde \mu(B): \langle B \rangle = \mathcal G).
\]
If \( \Gamma_n \) is any chain of finite-index subgroups then there is a natural action on the space \( \underset{\leftarrow}{\lim}(\Gamma/\Gamma_n) \) with the probability measure induced from the uniform measures on the finite sets \( \Gamma / \Gamma_n \) (this can be seen as the boundary of the rooted tree induced by the containment relation among cosets of the \( \Gamma_n \)).

Theorem (Abért–Nikolov): \( \mathrm{RG}(\Gamma, (\Gamma_n)) = \mathrm{Cost}( \underset{\leftarrow}{\lim}(\Gamma/\Gamma_n)) – 1 \).

This follows from the easily seen fact that the cost of the action on \( \Gamma/\Gamma_n \) equals \( r(\Gamma, \Gamma_n) + 1\) and « continuity » of the cost, which is the main step in the proof.

In case \( (\Gamma_n) \) is a residual chain the action is essentially free. For those actions Gaboriau asked the following question:

Fixed price conjecture: Is it true that the cost of an essentially free action depends only on the group acting?

Note that since it is possible to construct residual chains in a hyperbolic 3–manifold group which have rank sublinear in the index, a positive answer to this implies a negative answer to the question on rank vs. genus above.

Distortion functions

If \( \mathcal G \) is a groupoid on a set \( X \) generated by a subset \( B \) then we can form the Cayley graph of \( \mathcal G, B \). If we have two generating sets \( B, B’ \) then the bilipschitz constant of the identity map \( X \to X \) between the two Cayley graphs is denoted by \( d_b(B, B’) \in [1, +\infty] \).

If \( \mathcal G \) is the groupoid induced by the action of a group \( \Gamma \) on a probability space \( (X, \mu) \) then an obvious generating set for \( \mathcal G \) is \( S \times X \). Let \( c \) be the cost of the action and define:
\[
f(\varepsilon) = \inf( d_b(X\times S, B): \langle B \rangle = \mathcal G, \tilde \mu(B) \le c + \varepsilon)
\]
the distortion function of the action. Informally the decay of this as \( \varepsilon \to 0 \) measures how complicated a near-optimal generating set must be.

Theorem (Abért–Gelander–Nikolov): Let \( \Gamma_0 = \Gamma \supset \Gamma_1 \supset \cdots \) be a chain of subgroups. Let \( f \) be the distortion function of the action of \( \Gamma \) on \( \underset{\leftarrow}{\lim}(\Gamma / \Gamma_n) \). If \( f \) decays subexponentially at \( 0 \) (meaning that \( \lim_{\varepsilon \to 0} \log f(\varepsilon)/\varepsilon = +\infty \)) then
\[
\lim_{n \to +\infty} \frac{\log|H_1(\Gamma_n; {\mathbb Z})_{\mathrm{tors}}|} {|\Gamma / \Gamma_n|} = 0.
\]

A similar result can also be proved for sequences of subgroups which are not necessarily chains. For this the notion of combinatorial cost of a sequence of finite-index subgroups is needed. This is defined for a sequence \( (G_n) \) of graphs with uniformly bounded degree; then ait can be applied to a sequence of subgroups \( \Gamma_n \subset \Gamma \) via their Schreier graphs. Define:
\[
e((G_n)) = \lim_{n\to +\infty} \frac{|E(G_n)|}{|V(G_n)|}.
\]
A rewiring of the sequence \( (G_n) \) is a sequence \( (H_n) \) such that \( V(H_n) = V(G_n) \) and the bilipschitz distance between \( G_n \) and \( H_n \) is uniformly bounded; this is denoted by \( (H_n) \sim (G_n) \). The combinatorial cost is then defined by:
\[
\mathrm{cc}((G_n)) = \inf_{(H_n) \sim (G_n)} e((H_n)).
\]

A fundamental example of rewiring is as follows. Let \( G_n \) be the \( n \times n \) square grid and let \( K \ge 1 \). Let \( H_n \) be obtained as follows:

• keep all vertical edges;
• in each even column of \( G_n \) delete every horizontal edge but those on the lines indexed by integers \( = 0 \pmod K \);
• on odd columns delete every horizontal edge but those indexed by integers \( = \lfloor K/2 \rfloor \pmod K \)

as illustrated by the following figure:

Then \( (H_n) \) is a rewiring of \( (G_n) \) (the bi-Lipscitz constants are bounded by \( K ) \) and it follows that \( \mathrm{cc}((G_n)) = 1 \).

Right-angled groups

A group \( \Gamma \) is called right-angled if there exists a finite generating set \( g_1, \ldots, g_d \) such that \( g_i g_{i+1} = g_{i+1}g_i \) and each \( g_i \) is of infinite order (to avoid the stupid case \( g_{2i+1} = 1 \)). There are many interesting examples:

• Many lattices in higher rank Lie groups are virtually right-angled;
• Right-angled Artin groups with connected graph are right-angles;
• On the other hand free groups are not right-angled;
• Most lattices in rank 1 Lie groups are not right-angled.

Theorem (Abért–Gelander–Nikolov): If \( \Gamma \) is right-angled, then for any residual chain \( \mathrm{RG}(\Gamma, (\Gamma_n)) = 0 \) and the growth of the torsion subgroup of \( H_1(\Gamma_n; {\mathbb Z}) \) is subexponential.

As a final remark note that there is no known example of a finitely presented, residually finite group \( \Gamma \) with a residual chain \( \Gamma_n \) satisfying
\[
\limsup_{n\to+\infty} \frac{\log|H_1(\Gamma_n; {\mathbb Z})_{\mathrm{tors}}|}{|\Gamma / \Gamma_n|} > 0
\]
although a conjecture of Bergeron–Venkatesh asserts that this should be the case for lattices in \( \mathrm{PSL}_2({\mathbb C}) \).

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

Lück has proven that the same conclusion holds in the case where \( G \) contains an infinite elementary amenable normal subgroup. A natural question is then:

Does the same conclusion holds if \( G \) is only supposed to contain an inifinite amenable normal subgroup?

This was proven by Sauer in the case where \( \widetilde X \) is a simply-connected manifold with a proper, free, cocompact \( G \)-action.

Proof of the theorem

The main idea (already used in previous work by Abért–Jaikin-Zapirain–Nikolov), is to use the following result.

Theorem (Benjamin Weiss): Let \( G \) be a finitely generated amenable group with a residual chain \( G_i \). Then there exists a sequence \( F_i \) such that

• \( G = \bigsqcup_{g \in G_i} gF_i \);
• \( (F_i ) \) is a Følner sequence.

Let \( S \) be any finite subset of \( G \). Weiss’ theorem yields for each \( i \) a finite subcomplex \( \widetilde D_i \subset \widetilde X \) which contains a subcomplex \( \widetilde J_i \) such that \( \partial_S \widetilde J_i \subset \widetilde D_i \) and \( |J_i| / |\widetilde D_i| \underset{i \to +\infty}{\rightarrow} 1 \).

Let \( p_i : \widetilde X \to X_i \) be the covering map and \( J_i = p_i(\widetilde J_i) \). Then for large \( i \) the following exact sequence holds:
\[
H_n(\widetilde X; {\mathbb Z})|_{\widetilde J_i} \overset{p_i}{\to} H_n(X_i; {\mathbb Z}) \to H_n(X_i, J_i; {\mathbb Z})
\]
where \( H_n(\widetilde X; {\mathbb Z})|_{\widetilde J_i} \) is the submodule genertated by those classes in \( H_n(\widetilde X; {\mathbb Z}) \) that can be represented by chains with support in \( \widetilde J_i \). Indeed, taking \( S \) to be the union of supports of all coefficients of a matrix representing \( \tilde d_{n+1} \), we see that for \( i \) large enough so that \( \partial_S \widetilde J_i \subset \widetilde D_i \) we have \( d_{n+1} \widetilde J_i^{n+1} \subset \widetilde D_i^n \), so that if we have a chain \( c \in C_n(X_i; {\mathbb Z}) \) with support in \( J_i \), by lifting it to \( \widetilde D_i \) we retain a chain.

Now if \( H_n(\widetilde X; {\mathbb Z}) = 0 \) we get that
\[
H_n(X_i; {\mathbb Z}) \hookrightarrow H_n(X_i, J_i; {\mathbb Z}).
\]
Since \( H_n(X_i, J_i; {\mathbb Z}) \hookrightarrow C_n( D_i \setminus J_i ) \) the torsion on the left-hand side ic \( o( \|d_{n+1}|^{|D_i \setminus J_i|} ) \) and thus
\[
\frac{\log t(H_n(X_i; {\mathbb Z}))} {|G/G_i|} \ll \frac{|D_i \setminus J_i|}{|G/G_i|} \underset{i\to+\infty}{\rightarrow} 0.
\]

Weakening the hypotheses

Is the hypothesis that \( X \) has a finite \( n+1 \)-skeleton needed to rech the conclusion of the theorem? In general yes, but in some cases one can prove subexponential growth of torsion without it.

Theorem (Kar–Kropholler–Nikolov): Let \( G \) be a metabelian group and let \( G_i \subset G \) be finite-index subgroups satisfying the condition that:
\[
| [G, G] / (G_i \cap [G, G]) | \to +\infty.
\]
Then
\[
\lim_{i \to +\infty} \frac{t(H_n(X_i; {\mathbb Z}))}{|G/G_i|} = 0.
\]

Theorem: For any function \( f: \mathbb N \to \mathbb N \) there exists a finitely generated solvable group \( G \) with a residual chain \( G_i \) such that \( t(H_1(G_i; {\mathbb Z})) \ge f(i) \).

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.

1. If \( \phi \in H^1(G, {\mathbb Z}) \) then \( \phi \in \Sigma(G) \) if and only if there is a finitely generated subgroup \( H \subset G \) and an injective morphism \( f : H \to H \) such that \( G = H *_f \) and \( \phi \) is the induced character.
2. \( \phi \in \Sigma(G) \) if and only if \( \ker(\phi) \) is finitely generated.

For a 3–manifold group we have the following topological interpretation.

Theorem (Bieri–Neumann–Strebel + Perelman): Let \( M \) be a closed 3–manifold; then \( \phi \in \Sigma(\pi_1(M)) \) if and only if \( M \) is fibered and \( \phi \) is the induced character of the resulting HNN-extension. Moreover \( \phi \in \Sigma(\pi_1(M)) \) if and only if \( -\phi \in \Sigma(\pi_1(M)) \).

The Thurston norm

Let \( M \) be a compact 3–manifold. For a class \( \phi \in H^1(M; _ZZ) \) its Thurston norm \( \| \phi \|_T \) is defined as \( \min_S \chi_-(S) \) where \( S \) runs over all surfaces in \( M \) dual to \( \phi \) and \( \chi_-(S) = \sum_i \max(0, -\chi(S_i)) \) if \( S \) is the disjoint union of the connected surfaces \( S_i \).

Theorem (Thurston):

• \( \| \cdot \|_T \) extends to a semi-norm on \( H^1(M, {\mathbb R}) \).
• If \( M \) is hyperbolic then it is a norm.
• The unit ball is a polytope, and \( \Sigma(\pi_1(M)) \) is the cone over a union of maximal open faces of this polytope.
• If \( M \) is fibered withfiber \( \Sigma \) and induced character \( \phi \) then \( \| \phi \|_T = -\chi(\Sigma) \).

Universal \( L^2 \)-torsion

Let \( G \) have a finite, \( L^2 \)-acyclic \( K(G, 1) \), and in addition satisfy the Atiyah conjecture. Let \( {\mathbb Z} G \subset \mathcal DG \) be the division closure of \( {\mathbb Z} G \) in the algebra of affiliated operators (see Thomas Schick’s talk).

Let \( K_1^w(G) \) be the group generated by weak isomorphisms \( {\mathbb Z} G^n \to {\mathbb Z} G^n \) (maps which are invertible over \( \mathcal DG \)) and the same relations as in the usual \( K_1 \). Then Friedl–Lück define the universal \( L^2 \)-torsion \( \rho_w^{(2)}(G) \in K_1^w(G)\) as the classical Reidemeister torsion. Using the Dieudonné determinant this yields an invariant \( \det(\rho_u^{(2)}) \) in the abelianised group \( (\mathcal DG^\times) \).

Let \( G^{\mathrm{ab-f}} \) be the maximal free abelian quotient if \( G \) and \( K \) the kernel of \( G \to G^{\mathrm{ab-f}} \). Then \( \mathcal DG \) is the Ore localisation of the polynomial ring \( \mathcal DK[G^{\mathrm{ab-f}}] \), so we can write \( \det(\rho_u^{(2)}) = pq^{-1} \) for some \( p, q \in \mathcal DK[G^{\mathrm{ab-f}}] \).

Now for any \( p \in \mathcal DK[G^{\mathrm{ab-f}}] \) its support is a finite subset of \( G^{\mathrm{ab-f}} = H_1(G) \). Let \( \mathcal P(p) \) be the convex hull in \( H_1(G; {\mathbb R}) \) of this support.

For a polytope \( \mathcal P \subset H_1(G; {\mathbb R}) \) define the function \( \mathcal N(\mathcal P) : H^1(G; {\mathbb R}) \to [0, +\infty[ \) by :
\[
\mathcal N(\mathcal P)(\phi) = \sup_{a, b \in \mathcal P}(\phi(a) – \phi(b)).
\]
We get a canonical function on \( H^1(G; {\mathbb R}) \) by taking \( \mathcal N(\mathcal P(p)) – \mathcal N(\mathcal P(q)) \) where \( \det \rho_u^{(2)}(G) = pq^{-1} \).

Theorem (Friedl–Lück): If \( G = \pi_1(M) \) for \( M \) a 3–manifold then this function is equal to the Thurston norm.

Now this can be taken as a definition for the Thurston norm for an arbitrary group \( G \) which has a finiet \( K(G, 1) \) and which satisfies the conditions above (being \( L^2 \)-acyclic, satisfying the Atiyah conjecture and \( \mathrm{Wh}(G) = 0 \)).

Theorem (Linnell, Lück, Waldhausen): Any ascending HNN extension of a finitely generated free group satisfies these conditions.

So for this class of groups there is a well-defined function \( \|\cdot\|_T \) on \( H^1(G; {\mathbb R}) \).

Theorem (Funke–Kielak): In this setting \( \|\cdot\|_T \) is a semi-norm.

Comparison with the Alexander norm

For any finitely generated group \( G \) the Alexander polynomial \( \Delta_G \) is an element of the group ring \( {\mathbb Z} G^{\mathrm{ab-f}} \). Define the Alexander norm \( \|\cdot\|_A \) of \( G \) as the function \( \mathcal N(\mathcal P(\Delta_G)) \). Then McMullen proved that:

• If \( M \) is a 3–manifold with \( b_1(M) \ge 2 \) and \( G = \pi_1(M) \) then \( \|\cdot\|_A \le \|\cdot\|_T \);
• If in addition \( M \) is fibered with induced character \( \phi \) then \( \|\phi\|_A = \|\phi\|_T \).

(For non-fibered 3–manifolds the inequality can be strict as shown by Dunfield.)

Theorem (Funke–Kielak):If \( G = F_n * _f \) where either \( n = 2 \) or \( f \) is unipotent and polynomially growing, and \( b_1(G) \ge 2 \) then \( \|\cdot\|_A \le \|\cdot\|_T \) with equality on the BNS-invariant.

Theorem (Funke–Kielak):Under the same hypotheses the universal \( L^2 \)-torsion \( \rho_u^{(2)} \) determines the BNS-invariant.

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?

A major step towards answering this question is to define an analogue for \( L^2 \)-Betti numbers for fields with positive characteristic, which has so far been done only in specific cases. An example is the following result.

Theorem (Elek–Szabó): Let \( k \) be a field. If \( \Gamma \) is a discrete, residually finite amenable group. Then for any finite CW-complex \( K \) with \( \pi_1(K) \cong \Gamma \) and any residual chain \( (\Gamma_n) \) the limit
\[
\lim_{n\to+\infty} \frac {b_n(K_n; k)} {|\Gamma / \Gamma_n|}
\]
exists. Moreover it is independent of the choice of \( (\Gamma_n) \).

Other results in this direction are due to Lackenby, Abért–Nikolov, Linnell–Lück–Sauer,…

Classical Atiyah problem

The classical Atiyah problem can be formulated as follows.

Atiyah problem: Fix a discrete group \( \Gamma \). What are the possible values for the \( \ell^2 \)-Betti numbers \( \beta^{(2)}(\widetilde K \to K) \) of \( \Gamma \)-covers? In particular, are they all rational?

In full generality the answer to the second question is very strongly « no ». The work of various people on this can be summerised in the following statement.

Theorem (Austin, Grabowski, Pichot–Schick–Zuk):

1. For any positive real number \( \alpha \) there exists a group \( G \) and a \( G \)-covering \( \widehat K \to K \) such that \( \beta_3^{(2)}(\widehat K \to K) = \alpha \).
2. If moreover \( \alpha \) is computable then we can take \( \widehat K \) to be simply connected in the previous point.

This statement is in fact really about group ring elements. The proof proceeds by showing that:

1. For any positive real number \( \alpha \) there exists a group \( G \) and \( M \in M_{n \times m}({\mathbb C} G) \) such that \( \dim_{\mathcal NG} \ker(M) = \alpha \).
2. If moreover \( \alpha \) is computable then we can take \( G \) to be finitely presented.

Homology gradients over finite fields

From now on \( k \) will be a finite field of characteristic at least 3 (it being of characteristic 2 causes some technical issues, but most of what follows goes through in general). Let \( G \) be an amenable group and \( T _in kG \). For \( G \) residually finite we can define :
\[
\dim_{kG} \ker(T) = \lim_{n\to+\infty} \frac{\dim_k(\ker(T_n)} {|\Gamma / \Gamma_n|}
\]
where \( (\Gamma_n) \) is a residual chain and \( T_n \in k(\Gamma/\Gamma_n) \) is the reduction of \( T \) (in general it is possible to define the right-hand side by using Følner sequences). For these homology gradients there is a result similar to the one in characteristic 0.

Theorem (Grabowski–Schick):

1. For any positive real number \( \alpha \) there exists an amenable group \( G \) and \( T \in kG) \) such that \( \dim_{kG} \ker(T) = \alpha \).
2. There exists a finitely presented amenable \( G \) and \( T \in kG \) with \( \dim_{kG} \ker(T) \in {\mathbb R} \setminus {\mathbb Q} \).

Later on it will be explained with point 2. above is much weaker than the corresponding statement in characteristic 0.

The same kind of techniques as used in the proof of the previous statement can also be used to give a counter example to a conjecture by A. Thom:

Theorem (Grabowski–Schick): Let \( G = {\mathbb Z}/2 \wr {\mathbb Z} \). There exists a matrix \( T \) with coefficients in the group ring \( {\mathbb Z} G \) such that the set \( \{ \dim_{\mathbb F_pG} \ker(T) \) is infinite (it does not « stabilise » to the von Neumann dimension \( \dim_{\mathcal NG} \ker(T) \)).

Note that for classical Betti numbers (i.e. for a matrix \( A \in M_{n\times m}({\mathbb Z}) \) it is clear that \( \dim_{\mathbb F_p} \ker A = \dim_{\mathbb C} \ker(A) \) for large enough \( p \). On the other hand the limit:
\[
\lim_{p \to +\infty} \dim_{\mathbb F_pG} \ker(T) = \dim_{\mathcal NG} \ker(T)
\]
is provable (?).

In view of this it is perhaps surprising that the following result holds.

Theorem (Grabowski–Schick): For \( G = {\mathbb Z}/2 \wr {\mathbb Z} \), any \( n, m \ge 1, T \in M_{n \times m}({\mathbb Q} G) \), the von Neumann dimension \( \dim_{\mathcal NG} \ker(T) \in {\mathbb Q} \).

On the other hand one can find such \( T \) with their \( \mathbb F_p \)-gradients not rational (Grabowski).

One of the main tools used in the proofs is the following interpretation of the von Neumann dimension of certain particular modules.

Theorem (Lehner–Neuhauser–Woess): Let \( \Gamma = \langle \gamma_1, \ldots, \gamma_r \rangle \) be a finitely generated group. For \( n \ge 1 \) let \( G_n = ({\mathbb Z}_2)^n \wr \Gamma \) and define \( A, T \in {\mathbb Q} G_n \) by:
\[
A = \frac 1 {2^n} \sum_{a \in ({\mathbb Z}/2)^n} a, \quad T = \sum_{i=1}^r (\gamma_i A + A\gamma_i^{-1}).
\]
The for large enough \( n \) it holds that:
\[
\dim_{\mathcal NG_n} \ker(T) = \sum_{\mathcal G} \frac {\mathrm P\mathcal G}{|\mathcal G} \dim_{\mathbb C} \ker(M_{\mathcal G})
\]

In this statement:

• The sum is over all finite graphs \( \mathcal G \);
• \( \mathrm P\mathcal G \) is the probability that the identity component of the identity in teh Cayley graph ofg \( \Gamma \) after percolation of parameter \( 2^{-n} \) is isomorphic to \( \mathcal G \);
• \( M_{\mathcal G} \) is the Markov operator associated to the simple random walk on \( \mathcal G \).

If \( \Gamma \) is amenable then the same formula holds with \( \mathcal NG_n, \mathcal {\mathbb C} \) replaced with \( \mathbb F_pG_n, \mathbb F_p \). It would be possible to obtain an optimal statement in 2. in positive characteristic if it was known to hold for nonamenable groups as well.

Let \( \Gamma \) be a discrete group, \( \widetilde X \) a free \( \Gamma \)-CW-complex with finite quotient \( X = \Gamma \backslash \widetilde X \) and chain complex \( C_*(\widetilde X), d_* \).

Fix lifts of the cells of \( X \) in \( \widetilde X \), so that for each \( p \) we can identify \( C_p \cong {\mathbb Z}\Gamma^{n_p} \) and let \( A_p \in M_{n_p \times n_{p-1}}({\mathbb Z}\Gamma) \) represent the differential \( d_p \), which extends to a continuous operator between the Hilbert spaces \( \ell^2(\Gamma)^{n_p} \to \ell^2(\Gamma)^{n_{p-1}} \). Denote by \( P \) the orthogonal projection from \( \ker(d_p) \) onto the closure of \( \mathrm{Im}(d_{p+1}) \). The \( L^2 \)-Betti numbers are then defined by \( b_p^{(2)}(\widetilde X ; \Gamma) = \mathrm{tr}_{\mathcal N(\Gamma)}(P) \).

1. When \( \widetilde X = E\Gamma \) these numbers depend only on the group and are called the \( L^2 \)-Betti numbers of \( \Gamma \).
2. The image of \( P \) is also the kernel of the operator \( \Delta_p = d_p^*d_p + d_{p-1}d_{p-1}^* \in M_{n_p \times n_p}({\mathbb Z}\Gamma) \).

Let \( \mathrm{Bet}(\Gamma) \) be the set of all real numbers appearing as \( L^2 \)-Betti numbers of \( \Gamma \)-complexes \( \widetilde X \) as above, which is also equal (by point 2. just above) to the set of von Neumann dimensions of matrices in \( M_{n \times n}({\mathbb Z}\Gamma) \) for \( n \ge 1 \).

Strong Atiyah Conjecture: Let \( \mathrm{lcm}(\Gamma) = \mathrm{lcm}(|G| : G \le \Gamma, |G| < +\infty) \). Then \( \mathrm{Bet}(\Gamma) \subset \frac 1 {\mathrm{lcm}(\Gamma)} {\mathbb Z} \).

The same statement with \( {\mathbb Z} \) replaced by any number field is equivalent to the statement for \( {\mathbb Z} \). The conjecture is also made for coefficients in \( K[G] \) where \( K \) is any subfield of the complex numbers.

The conjecture is known to hold in many cases. For free groups Linnell proved that the von Neumann dimension of the kernel any matrix with coefficients in a group ring \( K\Gamma, K\subset {\mathbb C} \) is an integer. His statement is in fact more general and rests on an algebraic reformulation of the conjecture. The von Neumann algebra \( \mathcal N\Gamma \) (the von Neumann algebra of bounded operators on \( \ell^2(\Gamma) \) which commute with the right-\( \Gamma \)-action) fits into the following diagram :
\[
\begin{array}{ccc}
K\Gamma & \subset & \mathcal N\Gamma \\
\cap & & \cap \\
D_K\Gamma & \subset & \mathcal U\Gamma \end{array}
\]
where:

• \( \mathcal U\Gamma \) is the algebra of affiliated operators, the Ore localisation of \( \mathcal N\Gamma \) at the multiplicative subset of non-zero-divisors (in general it is not a field, not even an integral domain);
• \( D_K\Gamma \) is the division closure of \( K\Gamma \) in \( \mathcal U\Gamma \), the subset of all \( y^{-1}x \) for \( x\in K\Gamma \) and \( y\in K\Gamma \) invertible in \( \mathcal U\Gamma \).

As an example, in the case \( \Gamma = \langle z \rangle \cong {\mathbb Z} \) these algebras are explicitely described as follows:

• \( \mathcal U\Gamma \) is the algebra of classes of measurable functions on \( \mathbb T^1 = \{z\in{\mathbb C} : |z|=1 \} \);
• \( K\Gamma = K[z, z^{-1}] \) is the subalgebra of Laurent polynomials;
• \( \mathcal N\Gamma = L^\infty(\mathbb T^1) \) is the subalgebra of essentially bounded functions;
• \( D_K\gamma = K(z) \) is the subalgebra of rational functions.

Theorem (Linnell): Let \( \Gamma \) be a torsion-free group. Then the strong Atiyah conjecture with coefficients in \( K \) holds for \( \Gamma \) if and only if \( D_K\Gamma \) is a skew field.

A remarkable feature of this statement is that if one manages to give an intrinsic characterisation of the algebra \( D_K\Gamma \) (i.e. without referring to the von Neumann algebra) then it gives a completely algebraic formulation for the Atiyah conjecture. In special classes of groups this is in fact possible, for instance:

• If \( \Gamma \) is amenable then \( D_K\Gamma \) is the Ore localisation of \( K\Gamma \) at non-zero-divisor elements ;
• If \( \Gamma \) is bi-orderable then \( D_K\Gamma \) is the Mal’cev–von Neumann completion of \( K\Gamma \).

This gives an approach to the Atiyah conjecture which might be valid beyond characteristic 0 fields, in particular for finite fields.

It is desirable to have an extension of Linnell’s approach to the case where \( \Gamma \) is not necessarily torsion-free.

Theorem (Knebusch–Linnell–Schick): Let \( \Gamma \) be a discrete group with \( \mathrm{lcm}(\Gamma) < +\infty \) and let \( K \subset {\mathbb C} \) be a subfield stable under complex conjugation. Then the Atiyah conjecture for center-valued \( L^2 \)-Betti numbers for \( K\Gamma \) holds if and only if \( D_K\Gamma \) is a direct sum of finite matrix algebras over skew fields.

The theorem in fact gives a precise description of the degrees of the matrix algebras involved; the nature of the skew fields remains mysterious (as it is already in general in the torsion-free case).

A particular case where the statement becomes simpler is when \( \Gamma \) does not contain any non-trivial finite normal subgroup. In this setting, if \( d = \mathrm{lcm}(\Gamma) \) the statement is that the center-valued Atiyah conjecture holds for \( K\Gamma \) if and only if \( D_K\Gamma \cong M_{d\times d}(D) \) for some skew field \( D \).

More precisely, in this case « center-valued von Neumann dimensions » are the same as classical von Neumann dimensions, and the result is that the dimensions of kernels of matrices with coefficients in \( K\Gamma \) are dimensions of associated \( M_{d\times d}(D) \)-modules, which are easily seen to lie in \( d^{-1}{\mathbb Z} \) (they are direct sums of the module \( M_{d\times 1}(D) \), which is of dimension \( 1/d \)).

In general the center-valued dimensions lie a priori in the center \( Z(\mathcal N\Gamma) \). The center-valued Atiyah conjecture predicts that they in fact lie in the subalgebra \( Z({\mathbb C}\Gamma) \), which is equal to the elements supported on finite conjugacy classes in \( \Gamma \), and constant along them.

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.

By induction, this implies that every prime 3–manifold contains a finite collection of incompressible tori which cuts it into « geometric pieces », i.e. manifolds with torus boundary which are either hyperbolic or Seifert fibered. For ou examples of knot exteriors we have:

• As we saw, the exteriors of the trefoil of 5₁ are Seifert fibered;
• The exterior of the figure-eight and of 5₂ are hyperbolic;
• The exterior of a Whitehead double contains an essential torus, namely the boundary of the tubular neighbourhood of the original knot. For example this decomposes the exterior of the double of the trefoil into a Seifert fibered piece (the exterior of the trefoil) and an hyperbolic piece.

\( L^2 \)-invariants

\( L^2 \)-Betti numbers

Given a finite CW-complex \( X \) and a morphism \( \gamma: \pi_1(X) \to \Gamma \) for some discrete group \( \Gamma \) we obtain \( L^2 \)-Betti numbers \( b_n^{(2)}(X ; \gamma) \) from the complex
\[
C_*^{(2)}(X ; \gamma) = \ell^2(\Gamma) \otimes_{{\mathbb Z}[\pi_1(X)]} C_*(\widetilde X).
\]
If \( \Gamma \) is residually finite, i.e. there exists a sequence \( \Gamma = \Gamma_0 \supset \Gamma_1 \supset \cdots \) of finite-index, normal subgroups \( \Gamma_i \le \Gamma \) with \( \bigcup_i \Gamma_i = \{ 1 \} \) then an alternative definition is given by:

Lück Approximation Theorem: With the notation above, let \( X_i = \Gamma_i \backslash \widetilde X \). Then:
\[
b_n^{(2)}(X ; \gamma) = \lim_{i \to +\infty} \frac{b_n(X_i)} {|\Gamma/\Gamma_i|}.
\]

Using only this theorem it is possible to prove the following properties of the \( L^2 \)-Betti numbers:

• If \( \Gamma = \langle x \rangle \cong {\mathbb Z} \) then \( b_n^{(2)}(X; \gamma) = b_n(X ; {\mathbb Q}(x)) \).
• For any aspherical, prime compact 3–manifold \( N \) we have \( b_*(N ; \pi_1(N) = 0 \).

The proof of the second point follows if we can prove it for geometric manifolds. For Seifert manifolds the fundamental groups are extensions of Fuchsian groups by \( {\mathbb Z} \) and it is easy to find an explicit sequence of finite-index subgroup where \( b_1 \) grows sublinearly in the index.

For hyperbolic manifolds one can use the following deep theorem of Agol :

Theorem: If \( N \) is an hyperbolic manifold there exists a finite cover \( M \to N \) such that \( M \) is a surface bundle over \( \mathbb S^1 \).

The fundamental group is then an extension of \( {\mathbb Z} \) by a surface group, for which it is easy to find a sequence of finite-index subgroup whose first Betti numbers grows sublinearly.

\( L^2 \)-torsion

Suppose that \( N, \gamma \) is \( L^2 \)-acyclic, meaning that the Betti numbers \( b_n^{(2)}(N ; \Gamma) \) are all zero. Then it is possible to define the \( L^2 \)-torsion \( \tau^{(2)}(N ; \gamma) \), which is a positive real number (at least if \( \Gamma \) is residually finite).

In the case where the cellular complex is concentrated in one degree, say \( C_2^{(2)} \overset{\partial_2}{\to} C_1^{(2)} \), then the Reidemeister torsion is equal to the Fuglede–Kadison determinant: \( \tau^{(2)} = \det^{(2)}(\partial) \).

For 3–manifolds the universal cover is \( L^2 \)-acyclic, and Geometrisation allows to compute exactly the \( L^2 \)-torsion.

Theorem (Lück–Schick): Let \( N \) be a compact, prime 3–manifold and let \( N_1, \ldots, N_r \) be the hyperbolic pieces in the geometric decomposition of \( N \). Then:
\[
log \tau^{(2)}(N ; \pi_1(N)) = \frac 1 {6\pi} \sum_{i=1}^r \mathrm{vol}(N_i).
\]

The \( L^2 \)-Alexander invariant

Definition

Let \( N \) be a 3–manifold with morphisms \( \gamma: \pi_1(N) \to \Gamma \) and \( \phi: \pi_1(N) \to \langle x \rangle \cong {\mathbb Z} \). We say that the triple \( (N , \gamma, \phi) \) is admissible if \( \phi \) factors through \( \gamma \). In this setting, for \( t \in ]0, +\infty[ \) put:
\[
\tau^{(2)}(N ; \gamma, \phi)(t) = \tau^{(2)}(N ; g \mapsto t^{\phi(g)}\gamma(g)).
\]
The \( L^2 \)-Alexander invariant is the function \( t \mapsto \tau^{(2)}(N ; \gamma, \phi)(t) \). Note that for \( t \not\in{\mathbb Q} \) it is not obvious at all that this is positive, since for this we need to prove that a matrix with coefficients in the group ring \( {\mathbb R}[\Gamma] \) (as opposed to \( {\mathbb Z}[\Gamma] \) in the definition of \( L^2 \)-torsion) is of determinant class.

Baby case: \( \Gamma = {\mathbb Z} \)

Let \( \Gamma = \langle x \rangle \cong {\mathbb Z} \), and consider the admissible triple \( N, \gamma=\phi, \phi) \). For a matrix \( A \in M_{n \times n}({\mathbb R}[x, x^{-1}]) \) the Fuglede–Kadison determinant is given by the Mahler measure of the classical determinant. Namely, \( \det(A) \) is a polynomial in \( x, x^{-1} \) and we have:
\[
\begin{array}{cc}
\det^{(2)}(A) = m(\det(A)) & := \int_{|z|=1} \log|\det(A)(z)| \frac{dz} z \\
& = C\prod_{\det(A)(z_i) = 0} \max(1, |z_i|)
\end{array}
\]
where \( C \) is the leading coefficient of \( \det(A) \).

It is then easy to see that, in the simplest case where the complex is \( {\mathbb Z}[x, x^{-1}]^m \overset{A}{\to} {\mathbb Z}[x, x^{-1}]^n \), that:
\[
\tau^{(2)}(N; \phi, \phi) = Ct^n \prod_{\det(A)(z_i) = 0} \max(1, t^{-1}|z|_i).
\]
We see that \( \tau^{(2)}(N ; \phi, \phi) \) is a piecewise polynomial function, in particular \( \tau^{(2)}(N ; \phi, \phi) \equiv C \prod_i |z_i| \) in a neighbourhood of 0 and \( \equiv Ct^n \) in a neighbourhood of \( +\infty\).

In this setting the \( L^2 \)-Alexander invariant has many properties similar to the Reidemeister torsion.

1. If \( N \) is a surface bundle over the circle and \( \phi \) is dual to the fiber then \( \tau^{(2)}(N ; \phi, \phi) \equiv 1 \) in a neighbourhood of \( 0 \) and \( \equiv t^k \) for some \( k \ge 0 \) at infinity.
2. The « degree at infinity » \( k \) is smaller than the Thurson norm of \( \phi \).
3. The \( L^2 \)-Alexander invariant is « symmetric », in the sense that \( \tau^{(2)}(N ; \phi, \phi)(t^{-1}) = t^k\tau^{(2)}(N ; \phi, \phi)(t) \).
4. If \( N \) is Seifert fibered then \( \tau^{(2)}(N ; \phi, \phi)(t) = \max(1, t^{\| \phi \|_T}) \).

\( L^2 \)-Alexander invariant for the universal cover

From now on \( \gamma \) is the identity \( \pi_1(N) \to \pi_1(N) \) and we denote by \( \tau^{(2)}(N ; \phi ) \) the \( L^2 \)-Alexander function \( \tau^{(2)}(N ; \gamma, \phi) \).

Theorem (Dubois–Friedl–Lück): Suppose that \( N \) is a fiber bundle over the circle with fiber a surface \( F \) and monodromy \( f \), and \( \phi \) is the class dual to \( F \). Let \( \lambda \) be the dilatation of \( f \) (the largest modulus of an eigenvalue of \( f_*: H_1(F; {\mathbb C}) \to H_1(F; {\mathbb C}) \)). Then \( \tau^{(2)}(N ; \phi)(t) = 1 \) for \( t \le \lambda^{-1} \) and \( \tau^{(2)}(N ; \phi, \phi)(t) = t^{-\chi(F)} \) for \( t \ge \lambda \).

It is not known whether \( \tau^{(2)}(N; \phi) \) can be constant in an interval strictly larger than \( [0, \lambda^{-1}] \).

Other properties of this \( L^2 \)-Alexander function, similar to the ones we saw in other settings, are:

1. \( \tau^{(2)}(N; \phi) \) is symmetric around \( t = 1 \).
2. The Fox–Milnor theorem extends to the \( L^2 \)-Alexander function, but it is essentially useless in this setting since every nonnegative function can be written in the form \( t \mapsto f(t)f(t^{-1}) \).

Theorem (G. Hermann): If \( N \) is Seifert fibered then \( \tau^{(2)}(N; \phi)(t) = \max(1, t^{\|\phi\|_T}) \).