Clara Löh - L2-invariants of groups

Abstract

There is a rich interaction between group-theoretic properties and homological invariants of groups and spaces. Taking twisted coefficients in L2-functions on a group (or related constructs such as the group Von Neumann algebra), leads to homological invariants of groups that, e.g., contain information  on homology growth of groups along finite index subgroups and on dynamical properties of groups. This mini-course will give a brief introduction to L2-Betti numbers of groups, their computation, and applications.

Link to the pdf. The pdf contains exercises as well. I might want to do them for the intended experience.

Talk

Short version: Like $\mathbb{Q}$-Betti numbers but multiplicative.

Lecture I: Definition

Betti numbers

Definition Betti Numbers

For a given space, and coefficient, the Betti Zahlen are the dimesions of the homology.

• What about projective spaces? There is not a dimension there or, isit?

But this is not multiplicative. For multiplicativity we need coefficients that know about the underlying groups. For instance, the Group rings $\mathbb{C}\Gamma$, $\mathbb{Z}\Gamma$ are such an example. A nice example is $l^2\Gamma$ which is a completion of $\mathbb{C}\Gamma$, obtained be completion of the scalar product on the group ring. A second nice example is $N\Gamma$ which is the group from Von Neumann algebra.

Theorem

We can define dimensions of $U\subseteq V$ by the trace of a projection $di{m}_{K}U=t{r}_{K}p$ where $p:V\to V$ with $pp=p$ and the image of $p$ is $U$.

This somehow works out.

1. The group von Neumann algebra

Definition The group von Neumann algebra

$N\Gamma =B(l^2\Gamma {)}^{\Gamma }$ Meaning, it is the bounded functions of $l^2\Gamma$ which are equivariant under $\Gamma$.

We define a trace on the Group von Neumann algebra and show that it fulfills the trace property.

• How exactly is the trace defined? It is some scalar product with a basis vector.

Finite groups

If $\Gamma$ is finite then $l^2\Gamma \cong N\Gamma \cong \mathbb{C}\Gamma$.

Whole numbers

If $\Gamma \cong \mathbb{Z}$, then a Fouriertransformation (Analysis) show us that $l^2\cong L^2(S^1,\mathbb{C})\text{ and }N\Gamma \cong L^{\infty }(S^1,\mathbb{C})$ The trace $t{r}_{N\mathbb{Z}}$ then corresponds to taking the integral of the function.

The right apparently explains why this is called “commutating measure theory”.

2. The von Neumann dimension

Definition (von Neumann dimension)

We use the definition from before to define a dimension for Neumann algebras. The value will not depend on the projection chosen, giving us a valid notion of dimension $pdi{m}_{N\Gamma }(P)$ for $P$ a subspace of $N{\Gamma }^n$

• What is a trace?

For an $N\Gamma$ module $M$ we now define $di{m}_{N\Gamma }M$ as a supremum over all $pdim$s. And this works!

Theorem (Lück)

1. If $P$ is a fin.gen. projective $N\Gamma$ module then all $pdims$ are the same and equal to $dim$.
2. Additivity: Given a Kurze Exakte Sequenz $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$ of $N\Gamma$-modules then the dimension of the middle term is the sum of the dimension of the outer terms.
3. Multiplicativity: If $\Lambda \subseteq \Gamma$ is a finite index subgroup then $di{m}_{N\Gamma }$ is $[\Gamma ,\Lambda ]$ times $di{m}_{N\Lambda }$ of some subspace. “res”.

The von Neumann algebra is kind of like a PID. The multiplicativity approximately states that $N\Gamma \cong {⨁}_{[\Gamma :\Lambda ]}N\Lambda$

Studying the $\mathbb{Z}$-example we find that the dimension can be real as well.

3. $L^2$ Betti numbers

Definition ( $L^2$-Betti numbers)

We define the homology as follows $H_k(N\Gamma {\otimes }_{\mathbb{Z}\Gamma }{C}_{\ast }(X,\mathbb{Z}))$ The Betti numbers are the dimensions of the homologies.

• I still want to better understand tensor products

Theorem

Let $\Gamma$ be a countable group and let $X$ be a $\Gamma$-equivariant space.

1. For two spaces connected by a $\Gamma$-equivarient homeo, the two spaces have the same Betti numbers
2. If $X$ is path connected then $b_0^{(2)}(\Gamma \circlearrowright X)=1/\#\Gamma$ (the $\Gamma$-inv points in $l^2\Gamma$) It tells us something on the dynamics
3. Künneth Formel: The betti numbers of a product space can be computed from the betti numbers of the two spaces.
4. Poincare duality: If $M$ is an ICC $n$-mfd with fundamental group $\Gamma$ then $b_k^{(2)}(\Gamma \circlearrowright \tilde{M})=b_{n-k}^{(2)}(\Gamma \circlearrowright \tilde{M})$ So somthing on spaces and cospaces?
5. The Euler characteristic can be calculated from Betti numbers.
6. Multiplicativity: The betti numbers are multiplicative with respect to finite index groups.

Next we apply the theory only to groups

Definition

For a countable group $\Gamma$ we define $b_k^{(2)}(\Gamma )=b_k^{(2)}(\Gamma \circlearrowright E\Gamma )={\dim }_{N\Gamma }{H}_k(\Gamma :N\Gamma )$ where $E\Gamma$ is the Classifying space.

In the end, we describe how the betti number behave under $\Gamma$, $\Lambda$, products and Freies Produkt.

• What is the relationship to MCGs?
• Why did you start to study Betti numbers?

Example

As examples we give

• Betti numbers of $\mathbb{Z}$
• Betti numbers of the free group
• Betti numbers of closed surfaces

• Should this be considered a tool or a object of study.

Lecture II: Multiplicativity

Recall: If $\Lambda \subseteq \Gamma$ finite index then $b_k^{(2)}(\Gamma )=\frac{1}{[\Gamma :\Lambda ]}{b}_k^{(2)}(\Lambda )$

1. Commensurability

Definition

Two groups are Commensurable group if there are exists a finite index subgroup in the two groups that are isomorphic as group.

Corrolary

If two groups are commensurable, then all Betti numbers differ by a factor.

Example

Let $\Gamma =F_2\ast (F_2\times F_2)$, $\Lambda =F_3\ast (F_3\times F_3)$

To calculate the betti numbers, we first use, that the betti numbers are additive under the free group, we use the Künneth formula and we use the Euler characteristic.

We notice that the relations of the betti numbers are not equal. It follows that the two groups are not commensurable.

But, the two groups are quasi-isometric (Whyte).

This means, that the Betti numbers are not Quasi-Isometry-invariant. (We dont want this)

• Why do we want them to be QI-invariants?

2. The approximation theorem

Theorem (Lück)

Let $\Gamma$ be a Residuell Endliche Gruppe of type $F$ and let ${\Gamma }_i$ be a Residual chain. (Meaning, every group is finite index of the former and the groups get smaller and smaller such that in total they deplete the group). Then $b{k}^2(\Gamma )={\lim }_{i\to \infty }\frac{b_k({\Gamma }_i,\mathbb{Q})}{[\Gamma :{\Gamma }_i]}$

The above theorem is really something. It tells us, that the right side does not depend on the residual chain chosen and that it always converges.

Exercise: Compute $b_{\ast }^{(2)}$ for $\mathbb{Z},F_n$ and surface groups by using the approximation theorem.

Remark: There is a whole industry trying to find out which numbers can be betti numbers.

Sketch of proof: We first (a) write the betti numbers as the dimension of the kernel of matrices and then (b) as spectral measures. a) We use some Laplacian and De Rham Kohomologiegruppe to write the homology as the kernel of a matrix. The lim then appears. b) Let ${\mu }_A$ be the Spectral measure of $A$. Then

• ${\mu }_A$ is supported on $[0,,\Vert A\Vert ]$
• Since the kernel is just the Eigenspace with factor zero, we can replace the kernel by the spectral measure

3. Amenable groups

Theorem (Cherger & Gromov)

Let $\Gamma$ be a residually finite, infinite, Amenable Group of type $F$. Then for all $k$ the betti numbers $b_k^{(2)}(\Gamma )=0$.

Lecture III: More Multiplicativity

1. The proportionality principle

Theorem (Atiyah, Dodzin)

Let $M,N$ be occ. Riemannsche Mannigfaltigkeit with isometric Riem. Universal cover. Then $\frac{b_k^{(2)}({\pi }_1(M)\circlearrowright \tilde{M})}{vol(M)}=\frac{b_k^{(2)}({\pi }_1(N)\circlearrowright \tilde{N})}{vol(N)}$

Example

Let $M$ be an occ. hyperbolic $n$-manifold, then $b_k^{(2)}({\pi }_1(M)\circlearrowright \tilde{M})=\{\begin{array}{ll}0& 2k\ne n\\ \ne 0& 2k=n\end{array}$

2. Orbit equivalence

Idea: We transform groups into dynamical systems.

Definition

Let $\Gamma$ be a countable group. A standard $\Gamma$-action is an essentially free prob-measure preserving action on a standard Borel prob, space $(X,\mu )$.

Example

Examples of standard actions.

• Irrational rotation actions of $\mathbb{Z}$ on $S^1$
• If $\Gamma$ is countable and residually finite: The translation action of $\Gamma$ ob its profinite completion.
• If $\Gamma$ is countable infinite the Bernoulli shift on the space of $\{0,1\}$ indexed by $\Gamma$.

Definition

A standard equivalence relation is an equivalence relation $R$ on a std. Borel prob space $(X,\mu )$ such that

• $R\subseteq X\times X$ is measurable
• Each equivalence class is countable

Example

If $\Gamma$ on $(X,\mu )$ is a standard action then the Orbit relation (two points are related if they are in the same orbit) is a standard equivalence relation

We can take any subset $A\subseteq X$ and by $A\times A\cap R$ we get a new standard relation.

Definition

Let $\Gamma ,\Lambda$ be a stable action on $(X,\mu ),(Y,\nu )$

• The two actions are orbit equivalent if their standard equivalence relation is isomorphic.
• The two actions are stable orbit equivalent if there exist a nonzero-measure sets such that the standard equivalence relation restricted to those subsets is isomorphic.

It is known that two groups are stably Orbit equivalent groups iff. they are Measure equivalent groups.

Example

Two Riemannian manifolds whose universal civering are isometric have SOE Fundamental group. If $\Gamma$ is ammenable, countable, infinite, then every standard action is SOE to any irrational rotation action of $\mathbb{Z}$ on the circle.

3. More $L^2$-Betti numbers

Definition (Geboriau, Sauer)

If $R$ is a stadard equivalence relation on $(X,\mu )$ we define $b_k^{(2)}(R)={\dim }_{NR}To{r}_k^{\mathbb{C}R}(NR,L^{\infty }(X))\in [0,\infty ]$ where

We said, we wanted to assign groups to dynamics. We start from two simple objects and gradually do an analogy two show how similar they are:

• $\mathbb{C}\mapsto L^{\infty }(X)$
• $\mathbb{C}\Gamma \mapsto \mathbb{C}R$
• $N\Gamma \mapsto NR$
• $t{r}_{N\Gamma }\mapsto t{r}_{NR}$
• $di{m}_{N\Gamma }\mapsto t{r}_{NR}$

Theorem (Gaburiau, Sauer)

If $\Gamma$ is a standard action then $b_k^{(2)}(\Gamma )=b_k^{(2)}(R_{\Gamma \circlearrowright (X,\mu )})$

Proposition

$b_k^{(2)}(R)=\mu (A)b_k^{(2)}(R\_A)$

As a consequence the Betti numbers of two SOE of index $c$ differ by a factor $c$. Furthermore the Betti numbers are zero for all countable, infinite, amenable groups. Two free groups $F_n,F_m$ are Orbit equivalent iff. $n=m$.