Cost

Setting

Let \( X, \mathcal B, \mu \) be a measured space (often we’ll omit \( \mathcal B \) from the notation). Usually \( \mu \) will be a probability measure (i.e. \( \mu(X) = 1 \)). Let \( E \subset X \times X \) be an equivalence relation on \( X \), we will use the abbreviation \( xEy \) for \( (x, y) \in E \). We will suppose that:

• \( E \) is measurable, meaning that it is a measurable subset of \( X \times X \); equivalently, if \( A \in \mathcal B \) then also \( \{ x : \exists y \in A : xEy \} \in \mathcal B \).
• All equivalence classes for \( E \) are countable.
• \( \mu \) is \( E \)-invariant, that is for any partially defined Borel isomorphism \( \phi : A \to B \) (where \( A, B \in \mathcal B \)) such that \( \forall x \in A: xE\phi(x) \) we have \( \phi_*\mu|_A = \mu|_B \).

The basic example is when a finitely generated group \( \Gamma \) acts on a space \( X, \mu \) by Borel automorphisms preserving the measure \( \mu \)(if in addition \( \mu \) is a probability measure we’ll say that this is a p.m.p.—for probability measure preserving—action), and \( E \) is the orbit relation: \( xEy \Leftrightarrow \exists \gamma \in \Gamma: y =\gamma \cdot x \).

The group of Borel automorphisms whose graph is contained in \( E \) (the full group of \( E \)) is denoted by \( [E] \). The pseudo group of partially defined Borel automorphisms with graph contained in \( E \) is denoted by \( [[E]] \).

Graphings and cost of relations

A graphing of \( E \) is an undirected graph \( \mathcal G \) on \( X \) such that:

• \( \mathcal G \) is contained in \( E \) : that is, if \( x, y \) are neighbours in \( \mathcal G \) then \( xEy \) (connected components of \( \mathcal G \) are contained in \( E \)-classes);
• \( \mathcal G \) generates \( E \): that is, if \( xEy \) then \( x, y \) are in the same connected component of \( \mathcal G \).
• \( \mathcal G \) is measurable in the sense that the subset of \( X \times X \) given by those \( (x, y) \) which are neighbours in \( \mathcal G \) is measurable.

Equivalently, a graphing is defined by a collection \( \Phi = (\phi_i : A_i \to B_i)_{i \in I} \) of partially defined Borel automorphisms, such that:
\[
xEy \Leftrightarrow \exists i_1, \ldots, i_r \in I : y = \phi_{i_r} \circ \cdots \circ \phi_{i_1} x.
\]
The cost of a graphing \( \mathcal G \) (with respect to the measure \( \mu\)) is half the average vertex degree:
\[
\mathrm{cost}(\mathcal G, \mu) = \frac 1 2 \int_X d_{\mathcal G}(x) d\mu(x)
\]
(where \( d_{\mathcal G}(x) \) is the number of neighbours of \( x \) in \( \mathcal G \)). If the graphing is defined by \( \Phi \) as above then the cost is equal to:
\[
\mathrm{cost}(\Phi, \mu) = \sum_{i \in I} \mu(A_i).
\]
The cost of a measured equivalence relation is the smallest possible cost for a graphing:
\[
\mathrm{cost}(E, \mu) = \inf_{\mathcal G} \mathrm{cost}(\mathcal G, \mu).
\]

Cost of groups and fixed price

The cost of a discrete group \( \Gamma \), denoted by \( \mathrm{cost}(\Gamma) \), is the infimum over all of its essentially free, p.m.p. actions of their cost. A group is said to have fixed price if the cost of any action is equal to the cost of the group. There are no known exemples of a group which does not have fixed price.

Computing cost

Basic properties

Let \( X, \mu \) a probability space and \( E \) a measured equivalence relation on \( X \) preserving \( \mu \), with infinite countable classes. Then we have the following facts due to Levitt:

1. \( \mathrm{cost}(E, \mu) \ge 1 \).
2. If there exists a graphing \( \mathcal G \) such that \( \mathrm{cost}(E, \mu) = \mathrm{cost}(\mathcal G, \mu) \) then almost all components of \( \mathcal G \) are trees; it is then referred to as a treeing.
3. If \( B \subset X \) is a positive subset (measurable with positive measure) and \( E \) has finite cost then
   \[
   \mathrm{cost}(E|_B, \mu|_B/ \mu(B)) – 1 = \mu(B)^{-1} (\mathrm{cost}(E, \mu) – 1)
   \]
   Moreover \( E|_B \) admits a treeing if and only if \( E \) does.

We give two proofs of 1.: the one by Levitt, and one by Gaboriau which deduces it immediately from 3 (they are essentially the same).

Levitt’s proof goes as follows: for a partial graphing \( \Phi \) of \( E \) (a Borel graph contained in $E$ but which does not necessarily generate it) let
\[
e(\Phi) = \int_X \frac 1 {|\langle \Phi \rangle x|} d\mu(x).
\]
This is 0 if \( \Phi \) is a graphing (all orbits are infinite), and in general it is equal to the infimum over the measures of all Borel sets meeting every orbit of \( \langle \Phi \rangle \). The claim is then that for all partial graphings we have
\[
(\ast) \qquad e(\Phi) + \mathrm{cost}(\Phi) \ge 1.
\]
This is proven by induction: for the empty graphing we have \( e = 1, \mathrm{cost} = 0 \). Now suppose that \( \Phi \) satisfies \( (\ast) \) and \( \psi = \Phi \cup \{\psi : U \to V \} \). Let \( A \) be any Borel subset meeting every orbit of \( \langle \Psi \rangle \): we want to find a subset \( B \) such that \( A \cup B \) meets every orbit of \( \langle \Phi \rangle \) and \( \mu(B) \le \mu(A) + \mu(U) \) (then we get \( \mathrm{cost}(\Phi) + \mu(U) \ge \mathrm{cost}(\Psi) \) and if, fixing an aritrarily small \( \varepsilon > 0 \) we choose \( B \) such that \( \mu(B) \le e(\Psi) + \varepsilon \) also
\[
e(\Phi) \le \mu(A) \le \mu(B) + \mu(U) \le e(\Psi) + \varepsilon + \mathrm{cost}(\Psi) – \mathrm{cost} \Phi
\]
which proves that \( \mathrm{cost}(\Psi) + e(\Psi) \ge \mathrm{cost}(\Phi) + e(\Phi) \)).

For \( x \in X \) there exists \( a \in A \) and \( \phi_{i_k} \in \Phi, m_k \in \mathbb Z \), \( k = 1, \ldots, n \) such that \( x = \prod_{k=1}^n \phi_{i_k} \psi^{m_k} \). Let \( s(x) = \inf \sum_k |m_k| \) and define:
\[
B_1 = \{ x \in U : s(\psi x) < s(x) \}, \quad B_2 = \{ x \in V : s(\psi^{-1}x) < s(x) \}.
\]
Then \( \psi^{-1}B_2 \subset U \) and \( B_1 \cap \psi^{-1} B_2 = \emptyset \) so that \( \mu(B_1) + \mu(B_2) \le \mu(U) \). In addition it is clear that \( X = \langle \Phi \rangle B \), which finishes our proof in case \( \Phi \) is a finite graphing. If \( \Phi = ( \phi_i, i \in \mathbb N ), \Phi_n = (\phi_k, 1\le k \le n) \) then we can conclude bu using \( \mathrm{cost}(\Phi_n) + e(\Phi_n) \ge 1 \) and \( \mathrm{cost}(\Phi) = \lim\mathrm{cost}(\Phi_n) \) and \( \lim e(\Phi_n) = 0 \).

Gaboriau’s proof runs as follows: if \( \Phi \) is a graphing then we know that \( e(\Phi) = 0 \) and thus for every \( \varepsilon \) there is a subset \( X_\varepsilon \) which meets every orbit and has \( \mu(X_\varepsilon) = \varepsilon \). By 3. we have
\[
\mathrm{cost}(\Phi) – 1 = \mu(X_\varepsilon)(\mathrm{cost}(\Phi|_{X_\varepsilon}) – 1) \ge -\mu(X_\varepsilon) = -\varepsilon
\]
since cost is positive, and by taking \( \varepsilon \to 0 \) we conclude that \( \mathrm{cost}(\Phi) – 1 \ge 0 \).

The proof of 2. is as follows. Suppose that \( \Phi \) is a graphing of \( E \) which is not a treeing. Then there exists \( i_k, \varepsilon_k \in \{ \pm 1\}, k=1, \ldots, n \) and a nonzero set \( A \) such that \( \prod_{i=1}^n \phi_{i_k} x = x \) for all \( x \in A \). We may in addition assume that \( \prod_{i=1}^p \phi_{i_k}^{\varepsilon_k} x \not \in A \) for all \( p \le n-1 \) and that \( \varepsilon_k = 1 \). Then the graphing which has \( \phi_{i_k} : A_{i_k} \to B_{i_k} \) by \( \phi_{i_k} : A_{i_k}\setminus A \to B_{i_k}\setminus \phi_{i_k}A \) has the same orbits as \( \Phi \) and cost equal to \( \mathrm{cost}(\Phi) – \mu(A) \), so that \( \mathrm{cost}(\Phi) > \mathrm{cost}(E) \).

When is cost realised by a single graphing?

The main theorem in Gaboriau’s paper is the converse of 2. An immediate corollary is that any essentially free action of a free group on \( r \) generators has cost \( r \), hence the group has fixed price \( r \).

Theorem : If \( \mathcal G \) is a treeing of \( E \) then \( \mathrm{cost}(E, \mu) = \mathrm{cost}(\mathcal G, \mu) \).

A more precise special case is the following, essentially due to Ornstein–Weiss in the hard direction and appearing in Levitt’s paper.

Theorem : Let \( E \) be the relation induced by an essentially free p.m.p. discrete group action. Then there exists a graphing \( \mathcal G \) with \( \mathrm{cost}(E) = \mathrm{cost}(\Phi) = 1 \) if and only if \( \Gamma \) is amenable.

Modulo an easy argument (to go from « heving a graphing with cost 1 » to « being generated by a single automorphism ») this means that:

1. If the orbits of \( \Gamma \) on \( X \) are equal to the orbits of a single Borel automorphism \( T : X \to X \) then \( \Gamma \) is amenable;
2. Conversely, if \( \Gamma \) is amenable there exists a \( T \) whose orbits are exactly that of \( \Gamma \).

The proof of the reciprocal direction 1 goes as follows: the orbit equivalence between \( \Gamma \) and \( \langle T \rangle \cong \mathbb Z\) gives a measure equivalence between these two groups, i.e. a measure space \( \widehat X, \widehat \mu \) with an action of \( \Gamma \times \mathbb Z \) such that both factors admit a finite-volume fundamental set. (This can be obtained as \( \widehat X = X \times \mathbb Z \) with \( \mathbb Z \) acting on the right by translations in the \( \mathbb Z \) factor and \( \Gamma \) acting by \( \gamma\cdot(x, n) = (\gamma x, n+m ) \) where \( \gamma x = T^m x \)).

Now this measure equivalence gives a map \( \pi \mapsto \tilde \pi \) from unitary representations of \( \mathbb Z \) to those of \( \Gamma \). If \( \pi \) is defined on a Hilbert space \( \mathcal H \) then the representation \( \tilde \pi \) is defined on the space:
\[
\tilde{\mathcal H} := \left\{ f: \widehat X \to \mathcal H: \forall m \in \mathbb Z, x \in \widehat X : f(x\cdot m) = \pi(m)f(x) \right\}
\]
on which it acts by \( \tilde \pi(\gamma)f(x) = f(\gamma^{-1} \cdot x) \). Then we have the following facts:

1. If \( \pi \) is the left-regular representation on \( L^2(\Lambda) \) then \( \tilde \pi \) is equivalent to the representation on \( L^2(\widehat X) \) (which is equivalent to the Hilbert sum \( w_n \in \overline \bigoplus_{\mathbb Z} L^2(\Gamma) \) of countably many copies of the left-regular representation on \( L^2(\Gamma) \)).
2. If \( \pi \) has almost-invariant vectors then \( \tilde \pi \) also does.

Thus if \( \Lambda \) is amenable, we get a sequence of quasi-\( \Gamma \)-invariant vectors \( w_n \in \overline \bigoplus_{\mathbb Z} L^2(\Gamma) \). Projecting them onto the factors gives a sequence of quasi-invariant vectors in \( L^2(\Gamma) \), hence it is amenable.

Proof of (a) and (b): For (a) we have the map from \( \tilde H \) to \( L^2(\widehat X) \) given by \( f \mapsto \langle f, \delta_{1_\Lambda} \rangle_{L^2(\Lambda)} \) which is easily checked to be an isometry and \( \Gamma \)-equivariant. For (b), we choose a mesurable fundamental domain \( U \subset \widehat X \) for \( \Lambda \). Given a sequence \( v_n \in \mathcal H, \| v_n \|_{\mathcal H} = 1 \) of quasi -invariant vectors (meaning that \( \langle \pi(g)v_n, v_n \rangle_{\mathcal H} \to 1 \) for every \( g \in \Lambda \)) we define \( f_n \in \tilde H \) by \( f_n(x) = v_n \) for \( x \in U \). Then \( f_n \) is a sequence of quasi-invariant vectors in \( L^2(\Gamma) \).

Rokhlin lemma and cost of amenable groups

Now we want to prove that if \( \Gamma \) is amenable then the orbit relation of every p.m.p. \( \Gamma \)-action is generated by a single Borel automorphism. For this we will use the following characterisation: an equivalence relation is said to be hyperfinite if there exists a nested sequence of finite, measurable subrelations \( F_1 \subset \cdots \subset F_n \subset E \) such that \( E = \bigcup_n F_n \).

Lemma : A measurable equivalence relation is hyperfinite if and only if it is induced by a single Borel automorphism.

The construction of an automorphism from a sequence of nested finite relations is illustarted by the following pictures.

Thus we want to prove that an action of an amenable group gives rise to an hyperfinite equivalence relation. The result that will enable us to do so is the following theorem of Ornstein and Weiss, often called « Rokhlin’s lemma » (Rokhlin proved the case \( \Gamma = \mathbb Z \)). Say that a finite subset \( T \subset \Gamma \) is a tile if there exists \( C \subset \Gamma \) such that \( \Gamma = CT \) and \( cT \cap c’T = \emptyset \) if \( c \not= c’ \in C \)?

Theorem : Suppose that \( \Gamma \) is an amenable group and that \( T \subset \Gamma \) is a tile. Let \( \Gamma \) acting on (X, \mu) \) be an essentially free p.m.p. action. Then for any \( \varepsilon > 0 \) there exists a measurable subset \( A \subset X \) such that \( \mu(T \cdot A) \ge 1 – \varepsilon \) and for almost all \( a \in A \) we have \( (T\cdot) a \cap (T\cdot b) = \emptyset \) for all \( b \in A\setminus\{a\} \).

To deduce hyperfiniteness from this result suppose that there exists a sequence \( T_1, \ldots, T_n, \ldots \) of tiles such that \( T_n \) is \( (T_{n-1} \cup T_{n-1}^{-1}, 1/n) \)-invariant. Let \( T_n’ = T_n \setminus (T_n \Delta T_{n-1}T_n) \) so that \( |T_n’|/|T_n| \ge 1 – 1/n \).

Rokhlin’s lemma gives us subsets \( A_n \subset X \) such that \( \mu(T_nA_n) \ge 1-1/n \) and the \( T_na, a \in A_n \) are paiwise disjoint. Let \( F_n \) be the finite relation which is defined by
\[
xF_n y \Leftrightarrow \exists a \in A: x, y \in T_na
\]
on \( T_nA \) and which is the identity on \( X \setminus T_nA \). Clearly \( E = \bigcup_n F_n \) (because \( \Gamma = \bigcup_n T_n \)); the problem with \( F_n \) is that we don’t have \( F_{n-1} \subset F_n \).

Let \( F_n’\) be defined as follows: a class \( [F_n]x \) contains exactly one « large » class for \( F_n’ \) consisting of all \( yF_nx \) such that \( [F_n’]y \subset [F_n]x \), and for all other points we take their \( F_{n-1}’ \)-class. Clearly \( F_{n-1}’ \subset F_n’ \), and also \( T_n’a \subset [F_n’]a \) for all \( a \in A_n \). It follows from the latter that \( E = \bigcup_n F_n’ \).

Note that it is not known whether all amenable groups admit such a sequence of tiles; however, this is the case for all elementary amenable groups (Ornstein–Weiss) and for all residually finite amenable groups (Weiss). In general one has to use almost-tilings, for which the Rokhlin lemma also holds.

Proof of the Rokhlin lemma

The first step is to establish the following lemma.

For any \( \varepsilon > 0 \) and \( n \ge 1 \) there is a finite subset \( H \subset \Gamma \) which is \( T \cup T^{-1}, \varepsilon^2 \)-invariant an a collection of measurable subsets \( U_1, \ldots, U_m \subset X \) such that :

• for all \( i = 1, \ldots, n \) and \( u \not= u’ \in U_i \) the sets \( Hu, Hu’ \) are disjoint;
• \( \mu\left( \bigcup_i U_i \right) \ge 1 – \varepsilon \);
• \( \sum_i \mu(U_i) \le \varepsilon^{-1} \).

To prove this Ornstein and Weiss use the following two statements:

1. There is a cover of \( X \) by pairwise disjoint sets \( U_i, i \in \mathbb N \) satisfying 1) ;
2. Given \( \varepsilon > 0 \) and a cover \( X = \bigcup_{j \ge 1} V_j \) such that every \( x \in X \) belongs to exactly \( N \) of the \( V_j \) there exists a finite subsequence \( V_{j_i}, i=1, \ldots, N \) which satisfies 2) and 3).

The point a) is easily proven once it is known that every subset of positive measure in \( X \) contains a set satisfying 1) (choose a first set \( U_1 \) with positive measure, and then do the same in \( X \setminus U_1 \),…) which is more or less obvious.

Point b) is a general fact in combinatorics, proven by Ornstein–Weiss using the following lemma : let \( X = \bigcup_i V_i \) be a cover by measurable such that every point belongs to exactly N of the \( V_i \)s. Then for any \( V \subset X \) there exists \( V_i \) such that :
\[
(\ast\ast) \qquad \frac{\mu(V_i \cap V)}{\mu(V_i)} \le \mu(V).
\]
Now suppose that the \( U_i \) are ordered by measure (so that \( \mu(U_{i+1}) \le \mu(U_i) \)) and use \( (\ast\ast) \) to construct a maximal sequence \( i_1 = 1, i_2, \ldots \) such that for all \( k \le n-1 \) (\( n \in \mathbb N \cup\{\infty\} \) the length of the sequence) \( \mu(\bigcup_{1\le l \le k} U_{i_l}) < 1 – \varepsilon \), \( \mu(U_{i_k} \cap \bigcup_{1\le l \le k-1} U_{i_l}) \le \mu(U_{i_k})\mu(\bigcup_{1 \le l \le k-1} U_{i_l}) \) and \( i_k \) is the smallest index with these properties. If \( n < +\infty \) this finishes the proof. Otherwise, if we had \( \mu(\bigcup_{l \ge 1} U_{i_l}) < 1 – \varepsilon \), by \( (\ast\ast) \) there would be an index \( j \in \mathbb N \) such that \( \mu(U_j \cap \bigcup_{l\ge 1} U_{i_l}) \le \mu(U_j)\mu(\bigcup_{1 \le l} U_{i_l}) \) and this contardicts the choice of the \( U_{i_k} \).

In addition, the fact that \( \mu(U_{i_k} \cap \bigcup_{1\le l \le k-1} U_{i_l}) \le (1 – \varepsilon) \mu(U_{i_k}) \) easily implies that \( \sum_{l \ge 1} \mu(U_{i_l}) \le \varepsilon^{-1} \).

For the second step let \( C \subset \Gamma \) such that \( | TC \Delta H | \le \varepsilon^2 | H | \) which exists because \( T \) tiles and H is \( TT^{-1}, \varepsilon^2 \)-invariant. Let \( W_i = \bigcup_{j=1}^i HU_j \) and construct inductively sets \( W_i’ \) such that \( W_i’ \) contains all \( Tcu, u \in U_i, c \in C \) such that \( Tcu \cap U_{i+1} = \emptyset \). We get
\[
\mu(W_i \setminus W_i’) \le \varepsilon^2 \sum_{j=1}^i \mu(U_j)
\]
so that in the end \( \mu(W_m’) \ge 1 – \varepsilon – \varepsilon^2 \sum_{j=1}^i \mu(U_j) \ge 1 – 2\varepsilon \). The other conclusion of the Rokhlin lemma clearly holds for the set
\[
A = \bigcup_{i=1}^m \{ cu : u \in U_i, c \in C : (Tcu) \subset X \setminus \bigcup_{j \ge i+1} HU_j
\]
which satisfies \( TA = W_m’ \), and this finishes the proof.

Références

• Gilbert Levitt, On the cost of generating an equivalence relation, Ergodic Theory and Dynamical systems.
• Damien Gaboriau, Coût des relations d’équivalence et des groupes, Inventiones Mathematicae.
• Donald Ornstein, Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d’analyse mathématique.
• Alex Furman, A survey of measured group theory, in Geometry, Rigidity, and Group Actions, U. Chicago press.

The coset tree

Let \(\Gamma\) be a countable, discrete group and let \(\Gamma = \Gamma_0 \ge \Gamma_1 \ge \Gamma_2 \ge \cdots\) be a chain of finite index subgroups. For the moment we do not make any further assumptions on the chain \((\Gamma_n)\). So the subgroups \(\Gamma_n\) must neither be normal, nor are they required to have trivial total intersection.

Definition: The (right) coset tree \(T\) of \((\Gamma, (\Gamma_i))\) has vertex set \(\coprod_{n \ge 0} \Gamma_n \backslash \Gamma\) and an edge from \(\Gamma_ng\) to \(\Gamma_{m}h\) if and only if \(m = n+1\) and \(\Gamma_mh \subset \Gamma_ng\).

Not that \(T\) has the canonical root \(\Gamma_0\), so that the \(n\)-th level vertex set is just \(\Gamma_n \backslash \Gamma\), and each node in \(\Gamma_n \backslash \Gamma\) has precisely \([\Gamma_n \colon \Gamma_{n+1}]\) children.

A typical coset tree.

Definition: The boundary \(\partial T\) of the coset tree \(T\) consits of all infinite rays starting at the root \(\Gamma_0\).

Thus in mathematical terms we have \(\partial T = \underset{\leftarrow}{\lim} \Gamma_n \backslash \Gamma\) and this description makes sense not only in the category of sets but also in the category of topological spaces and of measure spaces. Each \(\Gamma_n \backslash \Gamma\) carries the discrete topology and the uniform probability measure. Thus as a space, \(\partial T\) is compact, totally disconneted and Hausdorff. It has a basis of the topology given by shadows where a shadow \(\mathrm{sh}(\Gamma_n g)\) consists of all rays going through the vertex \(\Gamma_n g\). The Borel probability measure \(\mu\) on \(\partial T\) is determined by the values \(\mu (\mathrm{sh}(\Gamma_n g)) = \frac{1}{[\Gamma : \Gamma_n]}\). The group \(\Gamma\) permutes the cosets in \(\Gamma_n \backslash \Gamma\) by right multiplication preserving the child–parent relation. Thus \(\Gamma\) acts on \(T\) from the right by tree automorphisms and we obtain an induced probability measure preserving right action of \(\Gamma\) on \(\partial T\) by homeomorphisms.

Lemma: The action of \( \Gamma\) on \(\partial T \) is ergodic.

Proof: If the chain stabilizes, the boundary is finite and the action is transitive so that the assertion is clear. Otherwise, Kuratowski’s theorem says that \(\partial T\) is Borel-isomorphic to the unit interval with Lebesgue measure so that Lebesgue’s theorem applies: any measurable \(A \subseteq \partial T\) is almost everywhere dense. In particular, if \(\mu(A) > 0\), then for all \(\varepsilon > 0\) there is some \(\Gamma_n g_0\) with
\[ \mu (\mathrm{sh}(\Gamma_n g_0) \cap A) > (1 – \varepsilon) \mathrm{sh}(\Gamma_n g_0). \]
If \(A\) is moreover \(\Gamma\)-invariant, then the same must hold for all \(\Gamma_n g\). Adding up all these inequalities gives
\[ \mu(A) = \sum_{\Gamma_n g \,\in\, \Gamma_n \backslash \Gamma} \mu(\mathrm{sh}(\Gamma_n g) \cap A) > 1 – \varepsilon, \]
which implies \(\mu(A) = 1\).

Lemma : Suppose each \(\Gamma_n\) is normal in \(\Gamma\) and that the total intersection \(\bigcap_{n \ge 0} \Gamma_n\) is trivial. Then the action of \( \Gamma\) on \(\partial T \) is free.

Proof: Given any nontrivial \(g \in \Gamma\) there is \(n > 0\) such that \(g \notin \Gamma_n\). Since \(\Gamma_n\) is a normal subgroup of \(\Gamma\), the element \(g\) permutes the set \(\Gamma_n \backslash \Gamma\) without fixed points. Thus \(g\) moves all rays in \(\partial T\).

In view of this lemma the following concept is a natural generalization of the common assumptions on a chain \((\Gamma_n)\) given in the Lemma.

Definition: A chain \((\Gamma_n)\) of subgroups of \(\Gamma\) is called Farber if the action of \( \Gamma\) on \(\partial T \) is essentially free.

Here, as usual, « essentially free » means the set of points in \(\partial T\) with nontrivial stabilizer has measure zero.

Cost and groupoid cost

Cost

Let \((X, \mu)\) be a standard Borel space and let \( \Gamma \) act on \( X \) preserving a probability measure, with finitely many ergodic components. « Lying in the same orbit » defines an equivalence relation \(E\) on \(X\). So \(E\) is a measurable subset of \(X \times X\) which we can picture as (the edge set of) a directed graph. Reflexivity, symmetry and transitivity say that the connected components of \(E\) are complete as directed gaphs: each vertex carries a loop and any two distinct vertices in the same component are joined by precisely two edges, one in each direction. Since \(\Gamma\) is countable, so are the complete directed graphs given by each component.

Apparently, we can reconstruct \(E\) if we only know which pairs of points in \(X\) are connected by some finite path along edges in \(E\), regardless of their direction. This information, in turn, is captured by any measurable subset \(S \subset E\) with the property that two points \(x,y \in X\) are joined by an edge in \(E\) if and only if there is a path between \(x\) and \(y\) along edges in \(S\).

Definition:

1. A measurable subset \(S \subseteq E\) is called a subgraph of \(E\).
2. The \(k\)-th power \(S^k\) of a subgraph \(S \subseteq E\) is defined by requiring \((x,y) \in S^k\) if and only if there is an undirected path from \(x\) to \(y\) in \(S\) of length \(l\) with \(0 \le l \le k\).
3. We say that a subgraph \(S \subseteq E\) spans \(E\) if \(E = \bigcup_{k \ge 0} S^k\). In this case we write \(E = \langle S \rangle\).

Note that \(S^0\) is the set of loop edges in \(X\), regardless of what \(S\) is: each point \(x \in X\) is joined to itself by the empty path of length zero. The rules of the game are to find spanning subgraphs \(S\) of \(E\) which are as « small » as possible. It makes sense to quantify « small » by the average number of edges starting in a vertex of \(S\). The next definition makes this idea precise.

Definition: The edge measure \(e(S)\) of a subgraph \(S \subseteq E\) is given by
\[ e(S) = \int_X \deg_S(x) \,\mathrm{d} \mu(x) \quad \text{where} \quad \deg_S(x) = |\{y \in X \colon (x,y) \in S \}|. \]

Note that \(e(S)\) can be infinite. The largest lower bound of the edge measures of spanning subgraphs of \(E\) can be thought of measuring how much effort we must invest to generate the graph and thus the equivalence relation \(E\) which the action of \( \Gamma \) on \( X \) defines.

Definition: The cost of the action of \( \Gamma \) on \( X \) is given by
\[ \mathrm{cost}(X, \Gamma) = \mathrm{cost} E = \inf_{\langle S \rangle = E} e(S). \]

As an example, say \(\Gamma\) is finitely generated by \(g_1, \ldots, g_n\). Then
\[ S = \bigcup_{i=1}^n \{ (x, xg_i) \colon x \in X \} \]
clearly defines a spanning subgraph of \(E\). So we always have
\[ \mathrm{cost}(X, \Gamma) \le d(\Gamma) \]
where \(d(\Gamma)\) is the minimal number of generators of \(\Gamma\), also known as the rank of \(\Gamma\). Note that if the action of \( \Gamma \) on \( X \) is essentially free, almost every connected component of this subgraph \(S\) looks like the Cayley graph of \(\Gamma\) with respect to the generators \(g_1, \ldots, g_n\) but with no distinguished base point whatsoever.

The idea comes to mind that one can improve upon a spanning subgraph \(S\) as above by picking a maximal tree in each component. The problem is that depending on \(\Gamma\) there might be no measurable such choice which already hints at cost being quite a delicate invariant of the action. Let us mention that the cost of an action immediately yields an invariant for the group \(\Gamma\).

Definition: We define \(\mathrm{cost} \Gamma\) as the infimum of the numbers \(\mathrm{cost}(X, \Gamma)\) running over all essentially free, ergodic, probability measure preserving actions of \(\Gamma\) on a standard Borel space \((X, \mu)\).

Groupoid cost

In addition to examining (sub-)graphs, meaning measurable subsets of \(X \times X\), it is useful to also study graphings by which we mean any measurable subset \(M \subseteq X \times \Gamma\). We find it convenient to picture an element \((x,g) \in X \times \Gamma\) as an « arrow » in \(X\) pointing from \(x\) to \(xg\). Note that (almost all) these arrows are determined by their initial and final point if and only if the action of \(\Gamma\) on \( X \) is (essentially) free. We can sort the arrows in the subset \(M\) either by initial point or by direction: either by the \(X\)- or by the \(\Gamma\)-coordinate. So interchangeably we think of \(M\) as a family of subsets
\[ M_g = \{ x \in X \colon (x, g) \in M \} \subseteq X\]
parametrized by group elements \(g \in \Gamma\), or as a family of subsets
\[ M_x = \{ g \in \Gamma \colon (x, g) \in M \} \subseteq \Gamma\]
parametrized by points \(x \in X\). Guided by what we did above, we define the \(k\)-th power \(M^k\) of \(M\) by all the arrows we obtain by composing up to \(k\) arrows from \(M\) regardless of their direction. In more mathematical terms this means \((x,g) \in M^k\) if and only if there is \(0 \le l \le k\) and a decomposition \(g = g_1 \cdots g_l\) in \(\Gamma\) such that for all \(0 \le i \le l-1 \) either
\[ (x g_1 \cdots g_i, g_{i+1}) \in M \quad \text{or} \quad (x g_1 \cdots g_{i+1}, g_{i+1}^{-1}) \in M. \]
Note that \(M^0 = X \times \{ 1 \}\), regardless of what \(M\) is.

Definition: We say that a graphing \(M \subset X \times \Gamma\) spans \(X \times \Gamma\) if we have \(X \times \Gamma = \bigcup_{k \ge 0} M^k\). In this case we write \(\langle M \rangle = X \times \Gamma\).

Let \(e\) be the measure on \(X \times \Gamma\) given by the product of \(\mu\) and the counting measure on \(\Gamma\).

Definition: The groupoid cost of the action of \(\Gamma\) on \( X \) is given by
\[ \mathrm{gcost}(X, \Gamma) = \inf_{\langle M \rangle = X \times \Gamma} e(M). \]

To explain the terminology, we observe that the set \(X \times \Gamma\) has a groupoid structure: two arrows \((x_1, g_1), (x_2, g_2) \in X \times \Gamma\) can be composed if and only if \(x_1 g_1 = x_2\), meaning the first arrow points to the initial point of the second, and in that case their composition is \((x_1, g_1 g_2)\). So a graphing spans \(X \times \Gamma\) if and only if it generates \(X \times \Gamma\) as groupoid.

Proposition: We have
\[ \mathrm{gcost}(X, \Gamma) \ge \mathrm{cost}(X, \Gamma) \]
with equality if the action is essentially free.

Proof: A graphing \(M \subseteq X \times \Gamma\) defines a subgraph \(\Phi(M) \subseteq E\) of the orbit relation \(E \subseteq X \times X\) by setting
\[ \Phi(M) = \{ (x, xg) \colon (x,g) \in M \}. \]
Clearly \(\Phi(M^k) = \Phi(M)^k\) so that \(\Phi\) preserves the spanning property. We have \(\deg_{\Phi(M)} (x) \le |M^x|\) with equality almost everywhere if the action of \(\Gamma\) on \( X \) is essentially free. Integrating over \(X\) gives the inequality. To obtain equality for essentially free actions one still has to show that each spanning subgraph can be obtained from a spanning graphing via \(\Phi\); we skip this argument which needs some technical care but no unusual ideas.

Rank gradient and cost of the coset tree boundary

Let us now assume that \(\Gamma\) is finitely generated so that there is an epimorphism \(\varphi \colon F_k \rightarrow \Gamma\) from the free group on a finite number \(k\) of letters. If \(\Lambda \subseteq \Gamma\) is a subgroup of finite index, then by the Schreier theorem \(\varphi^{-1}(\Lambda)\) is likewise free. It has index \([\Gamma \colon \Lambda]\) in \(F_k\) so that an Euler characteristic argument reveals that the rank \(l\) of \(\varphi^{-1}(\Lambda)\) is given by \(l-1 = [\Gamma \colon \Lambda](k-1)\). For the ranks \(d(\Gamma_n)\) of the groups in our chain \((\Gamma_n)\) this means that the numbers
\[ \frac{d(\Gamma_n) -1}{[\Gamma \colon \Gamma_n]} \]
form a monotone non-increasing sequence of positive numbers.

Definition: The rank gradient of \(\Gamma\) with respect to \((\Gamma_n)\) is given by
\[ \mathrm{rg}(\Gamma, (\Gamma_n)) = \lim_{n \rightarrow \infty} \frac{d(\Gamma_n) – 1}{[\Gamma \colon \Gamma_n]}. \]

This notion is due to Lackenby and here is its dynamic interpretation.

Theorem (Abért–Nikolov, 2012): For any chain \((\Gamma_n)\) we have
\[ \mathrm{gcost}(\partial T, \Gamma) = \mathrm{rg}(\Gamma, (\Gamma_n)) + 1. \]

Before we come to the proof let us combine this theorem with the proposition.

Corollary: For any Farber chain \((\Gamma_n)\) we have
\[ \mathrm{cost}(\partial T, \Gamma) = \mathrm{rg}(\Gamma, (\Gamma_n)) + 1. \]

Proof of Theorem: We first show \(\mathrm{gcost}(\partial T, \Gamma) \le \mathrm{rg}(\Gamma, (\Gamma_n)) + 1\). For all \(\varepsilon > 0\) we find some \(n\) with \(\frac{d(\Gamma_n) -1}{[\Gamma \colon \Gamma_n]} \le \mathrm{rg}(\Gamma, (\Gamma_n)) + \varepsilon\). Thus the integer
\[ d = \lfloor(\mathrm{rg}(\Gamma, (\Gamma_n)) + \varepsilon)[\Gamma \colon \Gamma_n]\rfloor + 1 \]
gives an upper bound for \(d(\Gamma_n)\). Say \(\Gamma_n\) is generated by \(g_1, \ldots, g_d\) and let \(1 = \gamma_1, \ldots, \gamma_{[\Gamma \colon \Gamma_n]}\) be a system of representatives for \(\Gamma_n \backslash \Gamma\). We define a graphing \(M \subseteq \partial T \times \Gamma\) by setting
\[
M_g = \mathrm{sh}(\Gamma_n) \text{if } g = g_i \text{ for some } i \ge 1 \text{ or } g = \gamma_i \text{ for some } i > 1
\]
and \( M_g = \emptyset \) otherwise. We claim that \(\langle M \rangle = \partial T \times \Gamma\). Indeed, let \((x, g) \in \partial T \times \Gamma\) and let \(\gamma_a\) and \(\gamma_b\) be the representatives from the list for which \(x \in \mathrm{sh}(\Gamma_n \gamma_a)\) and \(\gamma_a g \gamma_b^{-1} \in \Gamma_n\). Hence we can write \(g\) as a word of the form \(\gamma_a^{-1} g^{\pm 1}_{i_1} \cdots g^{\pm 1}_{i_k} \gamma_b\). With respect to this factorization of \(g\) one easily verifies the criterion to conclude \((x,g) \in M^{k+2}\) proving the claim. By definition \(M_g\) equals \(\mathrm{sh}(\Gamma_n)\) for precisely \(d + [\Gamma \colon \Gamma_n] – 1\) elements in \(\Gamma\) and is empty otherwise. Hence the graphing \(M\) has measure
\[ e(M) = \frac{d + [\Gamma \colon \Gamma_n] – 1}{[\Gamma \colon \Gamma_n]}. \]
It follows that
\[ \mathrm{gcost}(\partial T, \Gamma) \le \frac{d-1}{[\Gamma \colon \Gamma_n]} + 1 \le \mathrm{rg}(\Gamma, (\Gamma_n)) + 1 + \varepsilon. \]

The reverse inequality \(\mathrm{gcost}(\partial T, \Gamma) \ge \mathrm{rg}(\Gamma, (\Gamma_n)) + 1\) is somewhat harder. Given \(\varepsilon 0\), there exists a graphing \(M\) which spans \(\partial T \times \Gamma\) and has measure \(e(M) \le \mathrm{gcost}(\partial T, \Gamma) + \frac{\varepsilon}{2}\). The first thing to do now is to construct yet another graphing \(N \subseteq \partial T \times \Gamma\) which is close to \(M\) in the sense that \(e(N \triangle M) \le \frac{\varepsilon}{2}\) and has the convenient property that each \(N_g\) is a finite union of shadows which is nonempty only for finitely many \(g \in \Gamma\). Since the shadows form a countable basis of the topology of \(\partial T\), it is conceivable that such a « finite approximation » to \(M\) exists. So we shall allow ourselves to skip the precise technical construction. Since \(N\) is made up from only finitely many shadows altogether, there exists a large enough \(n\) such that each \(N_g\) is in fact a finite union of level-\(n\) shadows of the form \(\mathrm{sh}(\Gamma_n h)\).

We define a finite, directed, labeled graph \(G\) as follows. The vertex set is \(V = \Gamma_n \backslash \Gamma\) and for each \(g \in \Gamma\) we connect \(w \in V\) with \(wg \in V\) by an edge of label \(g\) if and only if \(\mathrm{sh}(w) \subseteq N_g\). The graph \(G\) has the canonical base point \(v = \Gamma_n \in V\). This data clearly defines a homomorphism of groups
\[
\begin{array}{cc}
\varphi \colon \pi_1(G,v) & \longrightarrow \Gamma \\
l = (e_1, \ldots, e_k) & \longmapsto \mathrm{label}(e_1)^{\pm 1} \cdots \mathrm{label}(e_k)^{\pm 1}
\end{array}
\]
which multiplies the labels along a loop of edges, inverting the label whenever we travel through an edge in reverse direction. We claim that the image of the homomorphism \(\varphi\) is precisely \(\Gamma_n\). Indeed, for each \(l \in \pi_1(G,v)\) we have \(v \varphi(l) = v\) by the construction of the graph \(G\). Thus \(\varphi(l) \in \mathrm{Stab}(v) = \Gamma_n\). Let \(h \in \Gamma_n\) be any element and pick some ray \(x \in \mathrm{sh}(v)\). Since \(N\) spans \(\partial T \times \Gamma\), there is a factorization \(h = g_1 \cdots g_k\) with \(g_i \in \Gamma\) such that for all \(0 \le i \le k-1\) either
\[ (x g_1 \cdots g_i, g_{i+1}) \in N \quad \text{or} \quad (x g_1 \cdots g_{i+1}, g_{i+1}^{-1}) \in N. \]
For \(0 \le i \le k\) let \(w_i \in V = \Gamma_n \backslash \Gamma\) be the level-\(n\) vertex in the coset tree \(T\) through which the ray \(x \gamma_1 \cdots \gamma_i\) passes. Then we have \(w_0 = w_k = v\) and for each \(0 \le i \le k-1\) either \(w_i\) is connected to \(w_{i+1}\) by an edge in \(G\) with label \(g_{i+1}\) or \(w_{i+1}\) is connected to \(w_i\) by an edge of label \(g_{i+1}^{-1}\). Hence these edges form a loop \(l\) with \(\varphi(l) = h\) proving the claim.

Thus \(\Gamma_n\) is a quotient group of \(\pi_1(G,v)\). The latter group is free of rank \(1 – \chi(G)\) where \(\chi(G)\) is the Euler characterisitc of the graph \(G\). Note that
\[ \textstyle e(N) = \sum_{g \in \Gamma} \ \sum_{\mathrm{sh}(\Gamma_n h) \subseteq N_g} \frac{1}{[\Gamma \colon \Gamma_n]} \]
so that \(e(N) [\Gamma \colon \Gamma_n]\) is the number of edges in \(G\) while the number of vertices in \(G\) is of course \([\Gamma \colon \Gamma_n]\). Putting pieces together we obtain
\[ d(\Gamma_n) \le d(\pi_1(G,v)) = 1 + e(N)[\Gamma \colon \Gamma_n] -[\Gamma \colon \Gamma_n]. \]
By subadditivity of the measure \(e\) applied to \(N \subseteq M \cup (N \triangle M)\) we conclude
\[ \textstyle \mathrm{rg}(\Gamma, (\Gamma_i)) \le \frac{d(\Gamma_n) – 1}{[\Gamma \colon \Gamma_n]} = e(N) – 1 < \mathrm{gcost}(\partial T, \Gamma) + \varepsilon – 1. \]

References

• Miklós Abért, Nikolay Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus, J. Eur. Math. Soc.

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.