# Some problems and conjectures about $L^{2}$-invariants

Dominik Kirstein<EMAIL_ADDRESS>https://dkirstein.github.io ,
Christian Kremer<EMAIL_ADDRESS>https://sites.google.com/view/christian-kremer-math and  Wolfgang Lück
<EMAIL_ADDRESS>http://www.him.uni-bonn.de/lueck Mathematical
Institute of the University of Bonn
Endenicher Allee 60
53115 Bonn, Germany

###### Abstract.

In this paper we discuss open problems and conjectures concerning
$L^{2}$-invariants.

###### Key words and phrases:

$L^{2}$-invariants, condensed sets

###### 2020 Mathematics Subject Classification:

Primary 57Q99, Secondary 46L99, 20F99

##  1\. Introduction

In this article we give a survey on open problems and conjectures concerning
$L^{2}$-invariants. We cover the whole portfolio and not only certain aspects
as they are considered in the previous more specialized (and within their
scope more detailed) survey articles [88, 90]. Moreover, we include some new
results and problems, which have occurred after these two survey articles were
written. The reader may select a specific topic by looking at the table of
contents below.

### Acknowledgements

This paper is funded by the ERC Advanced Grant “KL2MG-interactions” (no.
662400) of the third author granted by the European Research Council and by
the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy - GZ
2047/1, Projekt-ID 390685813, Hausdorff Center for Mathematics at Bonn. All
three authors are supported by the Max Planck Institute for Mathematics in
Bonn.

We thank Grigori Avramidi for comments on an earlier version of this article
and Emma Brink for helpful discussions concerning Section 14.

The paper is organized as follows:

###### Contents

  1. 1 Introduction
  2. 2 Basics about $L^{2}$-invariants
    1. 2.1 Group von Neumann algebras and their dimension function
    2. 2.2 $L^{2}$-Betti numbers
    3. 2.3 Fuglede-Kadison determinants
    4. 2.4 $L^{2}$-torsion
    5. 2.5 Basic properties of $L^{2}$-Betti numbers and $L^{2}$-torsion
  3. 3 The Atiyah Conjecture
    1. 3.1 Statement of the Atiyah Conjecture
    2. 3.2 Status of the Atiyah Conjecture
    3. 3.3 Embedding the group ring of a torsionfree group into a skewfield
    4. 3.4 The Algebraic Atiyah Conjecture
  4. 4 The Singer Conjecture
    1. 4.1 Statement of the Singer Conjecture
    2. 4.2 Hopf Conjectures
    3. 4.3 Status of the Singer Conjecture
    4. 4.4 The proper Singer Conjecture and the action dimension
    5. 4.5 The ${\mathbb{F}}_{p}$-Singer Conjecture
  5. 5 The Determinant Conjecture
  6. 6 Approximation for finite index normal chains
    1. 6.1 Basic setup for approximation for finite index normal chains
    2. 6.2 The Euler characteristic
    3. 6.3 The Signature
    4. 6.4 Approximation of $L^{2}$-Betti numbers in characteristic zero
    5. 6.5 Approximation of $L^{2}$-Betti numbers in prime characteristic
    6. 6.6 Torsion
    7. 6.7 Homological torsion growth and $L^{2}$-torsion
    8. 6.8 Further examples
  7. 7 Approximation for normal chains
    1. 7.1 The Approximation Conjecture for $L^{2}$-Betti numbers
    2. 7.2 Approximation of Fuglede-Kadison determinants
    3. 7.3 Approximation of $L^{2}$-torsion
    4. 7.4 More general setups
  8. 8 $L^{2}$-invariants and simplicial volume
  9. 9 $L^{2}$-invariants of groups
    1. 9.1 Vanishing results
    2. 9.2 The first $L^{2}$-Betti number and applications to group theory
    3. 9.3 The $L^{2}$-torsion and applications to group theory
  10. 10 Measure equivalence
  11. 11 Zero-in-the-Spectrum-Conjecture
  12. 12 Twisting with finite-dimensional representations
  13. 13 Group von Neumann algebras over ${\mathbb{F}}_{p}$
  14. 14 $L^{2}$-invariants and condensed mathematics

##  2\. Basics about $L^{2}$-invariants

We present some basic definitions, constructions, and properties of
$L^{2}$-invariants for the reader’s convenience. The technical aspects of this
section will not be needed in the other sections.

###  2.1. Group von Neumann algebras and their dimension function

The _group von Neumann algebra_

(2.1) ${\mathcal{N}}(G):=\mathcal{B}(L^{2}(G))^{G}$

of the (discrete) group $G$ is defined to be the ${\mathbb{C}}$-algebra of
$G$-equivariant bounded operators from $L^{2}(G)$ to $L^{2}(G)$, where
$L^{2}(G)$ is the complex Hilbert space of square integrable functions from
$G$ to ${\mathbb{C}}$. It can also be viewed as a completion of the complex
group ring ${\mathbb{C}}G$ with respect to the weak (or equivalently strong)
operator topology. We will view it in the sequel just as a ring with unit and
essentially forget all the functional analytic aspects. It has much nicer
properties than ${\mathbb{C}}G$, for example because it is _semi-hereditary_ ,
i.e., any finitely generated submodule of a projective
${\mathcal{N}}(G)$-module is projective again. More information about von
Neumann algebras and their basic properties can be found for instance in [58,
Section 2.2] and [85, Section 1.1 and Section 9.1].

The main feature is the _dimension function_ that assigns to every
${\mathcal{N}}(G)$-module $M$ an element

(2.2) $\dim_{{\mathcal{N}}(G)}(M)\in{\mathbb{R}}^{\geq 0}\amalg\\{\infty\\}.$

for ${\mathbb{R}}^{\geq 0}=\\{r\in{\mathbb{R}}\mid r\geq 0\\}$. We summarize
its main properties:

  1. (1)

It is additive on short exact sequences of ${\mathcal{N}}(G)$-modules, also in
the extreme cases following the convention that for $x\in{\mathbb{R}}^{\geq
0}\amalg\\{\infty\\}$ we set $x+\infty=\infty$.

  2. (2)

The dimension function extends the classical Murray-von Neumann dimension for
finitely generated projective ${\mathcal{N}}(G)$-modules defined by the
Hattori-Stalling rank associated to the standard trace of ${\mathcal{N}}(G)$.
In particular, we have $\dim_{{\mathcal{N}}(G)}({\mathcal{N}}(G))=1$, which
together with additivity implies that the dimension of a finitely generated
${\mathcal{N}}(G)$-module is finite.

  3. (3)

It is cofinal in the sense that for any directed union $M=\bigcup_{i\in
I}M_{i}$ of ${\mathcal{N}}(G)$-modules the equality
$\dim_{{\mathcal{N}}(G)}(M)=\sup\\{\dim_{{\mathcal{N}}(G)}(M_{i})\mid i\in
I\\}$ holds.

  4. (4)

It satisfies the continuity property. A special case says that for any
finitely generated ${\mathcal{N}}(G)$-module $M$ one can find a finitely
generated projective ${\mathcal{N}}(G)$-module $PM$ together with two exact
sequences $0\to TM\to M\to PM\to 0$ and
$0\to{\mathcal{N}}(G)^{n}\to{\mathcal{N}}(G)^{n}\to TM\to 0$ for which
$\dim_{{\mathcal{N}}(G)}(TM)=0$ and
$\dim_{{\mathcal{N}}(G)}(M)=\dim_{{\mathcal{N}}(G)}(PM)$ hold. The general
version can be found in [85, Theorem 6.7].

  5. (5)

We have $\dim_{{\mathcal{N}}(G)}(M)=0$ if and only if every projective
submodule of $M$ is trivial. Moreover, a finitely generated projective
${\mathcal{N}}(G)$-module $M$ is trivial if and only if
$\dim_{{\mathcal{N}}(G)}(M)=0$ holds.

It turns out that the dimension function is uniquely determined by the
properties (1) - (4). In the special case where $G$ is finite and $M$ is an
${\mathcal{N}}(G)$-module one has ${\mathcal{N}}(G)={\mathbb{C}}G$, and
$|G|\cdot\dim_{{\mathcal{N}}(G)}(M)$ is the (classical) dimension of the
underlying complex vector space of $M$. If $G$ contains an element of infinite
order, then every element $r\in{\mathbb{R}}^{\geq 0}$ can be realized as
$r=\dim_{{\mathcal{N}}(G)}(M)$ for some finitely generated projective
${\mathcal{N}}(G)$-module $M$. More information about the dimension function
can be found for instance in [58, Sections 2.3 and 4.2], [83], [84], [85,
Chapter 6].

###  2.2. $L^{2}$-Betti numbers

Let $Y$ be a $G$-space. Its singular chain complex $C^{s}_{*}(Y)$ is acted
upon by $G$, so it acquires the structure of a chain complex of
${\mathbb{Z}}G$-modules. Note that ${\mathcal{N}}(G)$ can be viewed as an
${\mathcal{N}}(G)$-${\mathbb{Z}}G$-bimodule. Then we obtain an
${\mathcal{N}}(G)$-chain complex
${\mathcal{N}}(G)\otimes_{{\mathbb{Z}}G}C^{s}_{*}(X)$. Its $n$th homology
$H_{n}({\mathcal{N}}(G)\otimes_{{\mathbb{Z}}G}C^{s}_{*}(X))$ is a
${\mathcal{N}}(G)$-module. We define the _$n$ th-$L^{2}$-Betti number_ of the
$G$-space $Y$ to be

(2.3)
$b_{n}^{(2)}(Y;{\mathcal{N}}(G)):=\dim_{{\mathcal{N}}(G)}\bigl{(}H_{n}({\mathcal{N}}(G)\otimes_{{\mathbb{Z}}G}C^{s}_{*}(Y))\bigr{)}\in{\mathbb{R}}^{\geq
0}\amalg\\{\infty\\}.$

If $Y$ is a $G$-$CW$-complex, we can replace $C^{s}_{*}(Y)$ by the cellular
${\mathbb{Z}}G$-chain complex $C_{*}(Y)$ in (2.3). If $Y$ is a
$G$-$CW$-complex such that $Y/G$ is a finite $d$-dimensional $CW$-complex,
then $b_{n}^{(2)}(Y;{\mathcal{N}}(G))$ is finite for every $n\geq 0$ and
vanishes for $n>d$.

We will mainly be interested in the case where $X$ is a path connected space
possessing a universal covering $p\colon\widetilde{X}\to X$. Recall that
$\widetilde{X}$ then comes with a free $\pi$-action for $\pi$ the fundamental
group of $X$, and we abbreviate

(2.4)
$b_{n}^{(2)}(\widetilde{X})=b_{n}^{(2)}(\widetilde{X};{\mathcal{N}}(\pi))\in{\mathbb{R}}^{\geq
0}\amalg\\{\infty\\}.$

If $X$ is not path connected, one defines $b_{n}^{(2)}(\widetilde{X})$ to be
$\sum_{C\in\pi_{0}(X)}b_{n}^{(2)}(\widetilde{C})$. If $X$ is a $CW$-complex of
finite type, then $b_{n}^{(2)}(\widetilde{X})$ is finite. If $M$ is a closed
Riemannian manifold, then $b_{n}^{(2)}(\widetilde{M})$ agrees with the
original definition of the $L^{2}$-Betti number due to Atiyah [9] in terms of
heat kernel on the universal covering, namely, we have

(2.5) $\displaystyle b_{n}^{(2)}(\widetilde{M})$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\int_{\mathcal{F}}\operatorname{tr}(e^{-t\widetilde{\Delta}_{n}}(\widetilde{x},\widetilde{x}))\operatorname{dvol}$

where $\mathcal{F}\subseteq\widetilde{X}$ is a fundamental domain for the
$\pi$-action on $\widetilde{M}$ and
$\operatorname{tr}(e^{-t\widetilde{\Delta}_{n}}(\widetilde{x},\widetilde{x}))$
is the trace of the heat kernel on $\widetilde{M}$ at
$(\widetilde{x},\widetilde{x})$ for $\widetilde{x}\in\widetilde{X}$ at the
time $t$ in degree $n$.

If $G$ is a (discrete) group, we define its _$n$ th-$L^{2}$-Betti number_

(2.6) $b_{n}^{(2)}(G):=b_{n}^{(2)}(EG;{\mathcal{N}}(G))\in{\mathbb{R}}^{\geq
0}\amalg\\{\infty\\}.$

where $EG$ is a the total space of the universal principal $G$-bundle $EG\to
BG$. If $\underline{E}G$ is the classifying space for proper $G$-actions, see
for instance [86], then we conclude
$b_{n}^{(2)}(G):=b_{n}^{(2)}(\underline{E}G;{\mathcal{N}}(G))$ from [85,
Theorem 6.54 (2) on page 265]. This is useful since there are often small
models of $\underline{E}G$ available when $EG$ may have no such nice models.
For example, when $G$ has torsion every $CW$-model for $EG$ has cells in
infinitely many dimension, whereas one might construct a finite dimensional
model for $\underline{E}G$ to conclude vanishing of $L^{2}$-Betti numbers in
high degrees.

Roughly speaking, the usefulness of the $L^{2}$-Betti numbers stems often from
the fact that they vanish quite often and pose interesting obstructions to the
solution of geometric problems if they don’t. For instance, for a connected
$CW$-complex $X$ we have $b_{0}(X)=1$, whereas $b_{0}^{(2)}(\widetilde{X})$
vanishes if $\pi$ is infinite and is equal to $|\pi|^{-1}$ if $\pi$ is finite.
We mention as an appetizer two rather surprising examples.

######  Example 2.1.

If the group $G$ contains an amenable infinite normal subgroup $H\subseteq G$,
then $b_{n}^{(2)}(G)$ vanishes for every $n\geq 0$. This implies that the
Euler characteristic $\chi(BG)$ vanishes, if we additionally assume that $BG$
has a finite $CW$-model. Note that we do not require anything about $BH$. This
is a special case of [85, Theorem 7.2].

######  Example 2.2.

If $1\to K\to G\to Q\to 1$ is an extension of infinite groups such that $K$ is
finitely generated and $G$ is finitely presented, then $b_{1}^{(2)}(G)$
vanishes and hence the deficiency of $G$ is bounded from above by $1$. Recall
that the deficiency is the maximum of the set of integers $g(P)-r(P)$, where
$P$ runs through all finite presentations $P$ of $G$ and $g(P)$ is the number
of generators and $r(P)$ the number of relations. [81, Theorem 6.1]

###  2.3. Fuglede-Kadison determinants

Given an ${\mathcal{N}}(G)$-map $f\colon P\to Q$ of finitely generated
projective ${\mathcal{N}}(G)$-modules, one can define its _Fuglede-Kadison
determinant_

(2.7) ${\det}_{{\mathcal{N}}(G)}(f)\in{\mathbb{R}}^{\geq
0}:=\\{r\in{\mathbb{R}}\mid r\geq 0\\}.$

Let us briefly explain its definition, even though it will be of no concern to
us later. One can interprete $f$ as a bounded $G$-equivariant operator
$f\colon V\to W$ of _finitely generated Hilbert ${\mathcal{N}}(G)$-modules_,
i.e., $V$ is a Hilbert space $V$ together with a linear isometric $G$-action
such that there exists an isometric linear $G$-embedding of $V$ into
$L^{2}(G)^{r}$ for some natural number $r$, and analogously for $W$. Denote by
$\\{E_{\lambda}^{f^{*}f}\mid\lambda\in{\mathbb{R}}\\}$ the (right-continuous)
family of spectral projections of the positive operator $f^{*}f\colon V\to V$.
Define the _spectral density function of_ $f$ by

$F_{f}\colon{\mathbb{R}}\to{\mathbb{R}}^{\geq
0}\quad\lambda\mapsto\dim_{{\mathcal{N}}(G)}\bigl{(}\operatorname{im}(E_{\lambda^{2}}^{f^{*}f})\bigr{)}.$

The spectral density function is monotone, non-decreasing, right-continuous
and satisfies $F(0)=\dim_{{\mathcal{N}}(G)}(\ker(f))$. Let $dF$ be the unique
measure on the Borel $\sigma$-algebra on ${\mathbb{R}}$ that satisfies
$dF((a,b])=F(b)-F(a)$ for $a<b$. Define Fuglede-Kadison determinant
$\det_{{\mathcal{N}}(G)}(f)\in{\mathbb{R}}^{\geq 0}$ to be

${\det}_{{\mathcal{N}}(G)}(f)=\exp\left(\int_{0+}^{\infty}\ln(\lambda)\;dF\right)$

if the Lebesgue integral $\int_{0+}^{\infty}\ln(\lambda)\;dF$ is a real
number, and to be $0$ if we have $\int_{0+}^{\infty}\ln(\lambda)\;dF=-\infty$.
We say that $f$ is of _determinant class_ if $\det(f)\not=0$. With our
conventions we have $\det(0)=1$ for the zero map $0\colon P\to Q$

######  Example 2.3 (Fuglede-Kadison determinant for finite groups).

To illustrate this definition, we look at the example where $G$ is finite.
Then ${\mathcal{N}}(G)$ is the same as ${\mathbb{C}}G$ and we can think of $f$
as a ${\mathbb{C}}G$-linear map of finite-dimensional unitary
$G$-representations. The spectral density function $F_{f}$ is the right-
continuous step function, whose value at $\lambda$ is the sum of the complex
dimensions of the eigenspaces of $f^{*}f$ for eigenvalues $\mu\leq\lambda^{2}$
divided by the order of $G$. Let $\lambda_{1}$, $\lambda_{2}$, $\ldots$,
$\lambda_{r}$ be the non-zero eigenvalues of $f^{\ast}f$ with multiplicity
$\mu_{i}$. One computes

${\det}_{{\mathcal{N}}(G)}(f)=\exp\left(\sum_{i=1}^{r}\frac{\mu_{i}}{|G|}\cdot\ln(\sqrt{\lambda_{i}})\right)=\prod_{i=1}^{r}\lambda_{i}^{\frac{\mu_{i}}{2\cdot|G|}}={\det}_{{\mathbb{C}}}\bigl{(}\overline{f^{\ast}f}\big{)}^{\frac{1}{2\cdot|G|}}.$

Here, $\overline{f^{\ast}f}$ denotes the automorphism of the orthogonal
complement of the kernel of $f^{\ast}f$ induced by $f^{\ast}f$ and
${\det}_{{\mathbb{C}}}\bigl{(}\overline{f^{\ast}f})$ is put to be $1$ if $f$
is the zero operator. If $f\colon{\mathbb{C}}G^{m}\to{\mathbb{C}}G^{m}$ is an
automorphism, we get

${\det}_{{\mathcal{N}}(G)}(f)=\left|{\det}_{{\mathbb{C}}}(f)\right|^{\frac{1}{|G|}}.$

More information about the Fuglede-Kadison determinant can be found for
instance in [85, Section 3.2].

###  2.4. $L^{2}$-torsion

Let $X$ be a connected finite $CW$-complex. Let $C_{*}(\widetilde{X})$ be the
cellular ${\mathbb{Z}}\pi$-chain complex of its universal covering for $\pi$
the fundamental group of $X$. Then
${\mathcal{N}}(\pi)\otimes_{{\mathbb{Z}}\pi}C_{*}(\widetilde{X})$ is an
${\mathcal{N}}(\pi)$-chain complex, which is finite dimensional and consists
of finitely generated free ${\mathcal{N}}(\pi)$-modules. If $c_{n}\colon
C_{n}(\widetilde{X})\to C_{n-1}(\widetilde{X})$ is the $n$th differential of
$C_{*}(\widetilde{X})$, then

$\operatorname{id}_{{\mathcal{N}}(\pi)}\otimes_{{\mathbb{Z}}\pi}c_{n}\colon{\mathcal{N}}(\pi)\otimes_{{\mathbb{Z}}\pi}C_{n}(\widetilde{X})\to{\mathcal{N}}(\pi)\otimes_{{\mathbb{Z}}\pi}C_{n-1}(\widetilde{X})$

is an ${\mathcal{N}}(\pi)$-map of finitely generated free
${\mathcal{N}}(\pi)$-modules and its Fuglede-Kadison determinant
$\det_{{\mathcal{N}}(\pi)}(\operatorname{id}_{{\mathcal{N}}(\pi)}\otimes_{{\mathbb{Z}}\pi}c_{n})\in{\mathbb{R}}^{\geq
0}$ is defined. We say that $X$ is _of determinant class_ if
$\det_{{\mathcal{N}}(\pi)}\bigl{(}\operatorname{id}_{{\mathcal{N}}(\pi)}\otimes_{{\mathbb{Z}}\pi}c_{n}\bigr{)}\not=0$
holds for every $n\geq 1$, and in this case we define the _$L^{2}$ -torsion_
of $X$ to be

(2.8) $\rho^{(2)}(\widetilde{X})=-\sum_{n\geq
1}(-1)^{n}\cdot\ln\bigl{(}{\det}_{{\mathcal{N}}(\pi)}(\operatorname{id}_{{\mathcal{N}}(\pi)}\otimes_{{\mathbb{Z}}\pi}c_{n})\bigr{)}\in{\mathbb{R}}.$

We say that $X$ is _$L^{2}$ -acyclic_ if $b_{n}^{(2)}(\widetilde{X})=0$ for
$n\geq 0$. We say that $X$ is _$\det$ -$L^{2}$-acyclic_ if it is
$L^{2}$-acyclic and of determinant class. If $X$ is not path connected, we
define
$\rho^{(2)}(\widetilde{X})=\sum_{C\in\pi_{0}(X)}\rho^{(2)}(\widetilde{C})$.

The definition of $L^{2}$-torsion in the analytic setting goes back to Lott
[78] and Mathai [98], and in the topological setting to Lück-Rothenberg [93].
The equality of these two versions has been proved in [20].

###  2.5. Basic properties of $L^{2}$-Betti numbers and $L^{2}$-torsion

Here is a list of basic properties of $L^{2}$-Betti numbers and
$L^{2}$-torsion.

######  Theorem 2.4.

  1. (1)

_(Simple) Homotopy invariance_ , see [85, Theorem 1.35 (1) on page 37 and
Theorem 3.96 (i) on page 163],[89, Theorem 6.7 (2)].

    1. (a)

If $X$ and $Y$ are homotopy equivalent spaces, then we get for $n\geq 0$

$b_{n}^{(2)}(\widetilde{X})=b_{n}^{(2)}(\widetilde{Y});$

    2. (b)

Suppose that $X$ and $Y$ are homotopy equivalent finite $CW$-complexes and
$\widetilde{X}$ or $\widetilde{Y}$ is $\det$-$L^{2}$-acyclic. Then both
$\widetilde{X}$ and $\widetilde{Y}$ are $\det$-$L^{2}$-acyclic;

    3. (c)

Let $f\colon X\to Y$ be a homotopy equivalence of finite $CW$-complexes.
Assume that $\widetilde{X}$ and $\widetilde{Y}$ are $\det$-$L^{2}$-acyclic.
Suppose that $f$ is simple, or that $\pi_{1}(X)$ satisfies the Determinant
Conjecture 5.1, or that the $K$-theoretic Farrell-Jones Conjecture for
${\mathbb{Z}}\pi_{1}(X)$ holds. Then

$\rho^{(2)}(\widetilde{Y})=\rho^{(2)}(\widetilde{X});$

  2. (2)

_Euler-Poincaré formula_ , see [85, Theorem 1.35 (2) on page 37].
We get for the Euler characteristic $\chi(X)$ of a finite $CW$-complex $X$

$\chi(X)=\sum_{n\geq 0}(-1)^{n}\cdot b_{n}^{(2)}(\widetilde{X});$

  3. (3)

_Sum formula_ , see [85, Theorem 3.96 (2) on page 164].
Consider a pushout of finite $CW$-complexes such that $j_{1}$ is an inclusion
of $CW$-complexes, $j_{2}$ is cellular, and $X$ inherits its $CW$-complex
structure from $X_{0}$, $X_{1}$ and $X_{2}$

$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\scriptstyle{j_{2}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{X.}$

Assume that for $k=0,1,2$ the map
$\pi_{1}(i_{k},x)\colon\pi_{1}(X_{k},x)\to\pi_{1}(X,j_{k}(x))$ induced by the
obvious map $i_{k}\colon X_{k}\to X$ is injective for all base points $x$ in
$X_{k}$.

    1. (a)

If $\widetilde{X_{0}}$, $\widetilde{X_{1}}$, and $\widetilde{X_{2}}$ are
$L^{2}$-acyclic, then $\widetilde{X}$ is $L^{2}$-acyclic;

    2. (b)

If $\widetilde{X_{0}}$, $\widetilde{X_{1}}$, and $\widetilde{X_{2}}$ are
$\det$-$L^{2}$-acyclic, then $\widetilde{X}$ is $\det$-$L^{2}$-acyclic and we
get

$\rho^{(2)}(\widetilde{X})=\rho^{(2)}(\widetilde{X_{1}})+\rho^{(2)}(\widetilde{X_{2}})-\rho^{(2)}(\widetilde{X_{0}});$

  4. (4)

_Poincaré duality_ , see [85, Theorem 1.35 (3) on page 37 and Theorem 3.96 (3)
on page 164].
Let $M$ be a closed manifold of dimension $d$

    1. (a)

Then

$b_{n}^{(2)}(\widetilde{M})=b_{d-n}^{(2)}(\widetilde{M});$

    2. (b)

Suppose that $n$ is even and $\widetilde{M}$ is $\det$-$L^{2}$-acyclic. Then

$\rho^{(2)}(\widetilde{M})=0;$

  5. (5)

_Product formula_ , see [85, Theorem 1.35 (4) on page 37 and Theorem 3.96 (4)
on page 164].
Let $X$ and $Y$ be finite $CW$-complexes.

    1. (a)

Then

$b_{n}^{(2)}(\widetilde{X\times
Y})=\sum_{\begin{subarray}{c}i,j\in{\mathbb{N}}\\\
n=i+j\end{subarray}}b_{i}^{(2)}(\widetilde{X})\cdot
b_{j}^{(2)}(\widetilde{Y});$

    2. (b)

Suppose that $\widetilde{X}$ is $\det$-$L^{2}$-acyclic. Then
$\widetilde{X\times Y}$ is $\det$-$L^{2}$-acyclic and

$\rho^{(2)}(\widetilde{X\times Y})=\chi(Y)\cdot\rho^{(2)}(\widetilde{X});$

  6. (6)

_Multiplicativity_ , see [85, Theorem 1.35 (9) on page 38 and Theorem 3.96 (5)
on page 164].
Let $X\to Y$ be a finite covering of finite $CW$-complexes with $d$ sheets.

    1. (a)

Then we get for $n\geq 0$

$b_{n}^{(2)}(\widetilde{X})=d\cdot b_{n}^{(2)}(\widetilde{Y});$

    2. (b)

Then $\widetilde{X}$ is $\det$-$L^{2}$-acyclic if and only if $\widetilde{Y}$
is $\det$-$L^{2}$-acyclic, and in this case

$\rho^{(2)}(\widetilde{X})=d\cdot\rho^{(2)}(\widetilde{Y});$

  7. (7)

_Determinant class_.
If $\pi_{1}(C)$ satisfies the Determinant Conjecture 5.1 for each component
$C$ of the finite $CW$-complex $X$, then $\widetilde{X}$ is of determinant
class;

  8. (8)

_$0$ th $L^{2}$-Betti number_, see [85, Theorem 1.35 (8) on page 38].
If $X$ is a connected finite $CW$-complex with fundamental group $\pi$, then

$b_{0}^{(2)}(\widetilde{X})=\begin{cases}\frac{1}{|\pi|}&\text{if}\;\pi\;\text{is
finite};\\\ 0&\text{otherwise;}\end{cases}$

  9. (9)

_Fibration formula_ , see [85, Lemma 1.41 on page 45 and Corollary 3.103 on
page 166].

    1. (a)

Let $p\colon E\to B$ a fibration whose base space $B$ is a connected finite
$CW$-complex and whose fiber is homotopy equivalent to a finite $CW$-complex
$Z$. Suppose that for every $b\in B$ and $x\in F_{b}:=p^{-1}(b)$ the inclusion
$p^{-1}(b)\to E$ induces an injection on the fundamental groups
$\pi_{1}(F_{b},x)\to\pi_{1}(E,x)$, and that $Z$ is $L^{2}$-acyclic.

Then $E$ is homotopy equivalent to a finite $CW$-complex $X$ which is
$L^{2}$-acyclic;

    2. (b)

Let $F\xrightarrow{i}E\xrightarrow{p}B$ be locally trivial fiber bundle of
finite $CW$-complexes. Suppose that $B$ is connected, the map
$\pi_{1}(F,x)\to\pi_{1}(E,i(x))$ is bijective for every base point $x\in F$,
and $\widetilde{F}$ is $\det$-$L^{2}$-acyclic.

Then $\widetilde{E}$ is $\det$-$L^{2}$-acyclic and

$\rho^{(2)}(\widetilde{E})=\chi(B)\cdot\rho^{(2)}(\widetilde{F});$

  10. (10)

_$S^{1}$ -actions_, see [85, Theorem 1.40 on page 43 and Theorem 3.105 on page
168].
Let $X$ be a connected compact $S^{1}$-$CW$-complex, for instance a closed
smooth manifold with smooth $S^{1}$-action. Suppose that for one orbit
$S^{1}/H$ (and hence for all orbits) the inclusion into $X$ induces a map on
$\pi_{1}$ with infinite image (so in particular the $S^{1}$-action has no
fixed points). Then $\widetilde{X}$ is $\det$-$L^{2}$-acyclic and
$\rho^{(2)}(\widetilde{M})$ vanishes;

  11. (11)

_Aspherical spaces_ , see [85, Theorem 3.111 on page 171 and Theorem 3.113 on
page 172].

    1. (a)

Let $M$ be an aspherical closed smooth manifold with a smooth $S^{1}$-action.
Then the conditions appearing in assertion (10) are satisfied and hence
$\widetilde{M}$ is $\det$-$L^{2}$-acyclic and $\rho^{(2)}(\widetilde{X})$
vanishes;

    2. (b)

If $X$ is an aspherical finite $CW$-complex whose fundamental group contains
an elementary amenable infinite normal subgroup, then $\widetilde{X}$ is
$\det$-$L^{2}$-acyclic and $\rho^{(2)}(\widetilde{X})$ vanishes;

  12. (12)

_Mapping tori_ , see [85, Theorem 1.39 on page 42].
Let $f\colon X\to X$ be a self homotopy equivalence of a finite $CW$-complex.
Denote by $T_{f}$ its mapping torus.

    1. (a)

Then $\widetilde{T_{f}}$ is $L^{2}$-acyclic;

    2. (b)

If $\widetilde{X}$ is $\det$-$L^{2}$-acyclic, then
$\rho^{(2)}(\widetilde{T_{f}})$ vanishes;

  13. (13)

_Hyperbolic manifolds_ , see [50], [85, Theorem 1.39 on page 42].
Let $M$ be a hyperbolic closed manifold of dimension $d$.

    1. (a)

If $d$ is odd, $\widetilde{M}$ is $\det$-$L^{2}$-acyclic;

    2. (b)

Suppose that $d=2m$ is even. Then $b_{n}^{(2)}(\widetilde{M})$ vanishes for
$n\not=m$, and we have $(-1)^{m}\cdot\chi(M)=b_{m}^{(2)}(\widetilde{M})>0$;

    3. (c)

For every number $m$ there exists an explicit constant $C_{m}>0$ with the
following property: If $M$ is a hyperbolic closed manifold of dimension
$(2m+1)$ with volume $\operatorname{vol}(M)$, then

$\rho^{(2)}(\widetilde{M})=(-1)^{m}\cdot C_{m}\cdot\operatorname{vol}(M).$

We have $C_{1}=\frac{1}{6\pi}$. The number $\pi^{m}\cdot C_{m}$ is always
rational;

  14. (14)

_Approximation of $L^{2}$-Betti numbers by classical Betti numbers_, see
[80],[85, Chapter 13].
Let $X$ be a connected finite $CW$-complex with fundamental group
$G=\pi_{1}(X)$. Suppose that $G$ comes with a descending chain of subgroups

$G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq\cdots$

such that $G_{i}$ is normal in $G$, the index $[G:G_{i}]$ is finite, and we
have $\bigcap_{i\geq 0}G_{i}=\\{1\\}$.

Then $G_{i}\backslash\widetilde{X}\to X$ is a finite $[G:G_{i}]$-sheeted
covering, and we get for $n\geq 0$

$b_{n}^{(2)}(\widetilde{X})=\lim_{i\to\infty}\frac{b_{n}(G_{i}\backslash\widetilde{X})}{[G:G_{i}]},$

where $b_{n}(G_{i}\backslash\widetilde{X})$ is the classical $n$th Betti
number of the finite $CW$-complex $G_{i}\backslash\widetilde{X}$.

We refer for more information about $L^{2}$-invariants and their applications
to algebra, geometry, group theory, index theory, operator algebras, topology,
and $K$-theory for instance to [58] and [85]. We will discuss some of them
later in this article.

##  3\. The Atiyah Conjecture

In this section we discuss one of the most prominent conjectures in the field
of $L^{2}$-invariants, the Atiyah Conjecture. It predicts the possible values
of $L^{2}$-betti numbers and has many interesting implications. For example
Kaplansky’s Zero Divisor Conjecture for rational group rings follows from it.

###  3.1. Statement of the Atiyah Conjecture

######  Conjecture 3.1 (Atiyah Conjecture).

Consider a field $F$ with ${\mathbb{Q}}\subseteq F\subseteq{\mathbb{C}}$. Let
$G$ be a group possessing an upper bound on the orders of its finite
subgroups. Let $\operatorname{lcm}(G)$ be the natural number given by the
least common multiple of the orders of its finite subgroups.

We say that $G$ satisfies the _Atiyah Conjecture with coefficients in $F$_, if
for any finitely presented $FG$-module $M$ the von Neumann dimension satisfies

$\operatorname{lcm}(G)\cdot\dim_{{\mathcal{N}}(G)}({\mathcal{N}}(G)\otimes_{FG}M)\in{\mathbb{Z}}.$

Note that $G$ is torsionfree if and only if $\operatorname{lcm}(G)=1$, so the
Atiyah conjecture in this case predicts integrality of $L^{2}$-Betti numbers.
For a field $F$, let us denote by $\mathcal{A}_{F}$ the collection of all
groups which admit a bound on the order of their finite subgroups and satisfy
the Atiyah conjecture with coefficients in $F$. Obviously
$\mathcal{A}_{{\mathbb{C}}}\subseteq\mathcal{A}_{F}\subseteq\mathcal{A}_{{\mathbb{Q}}}$.

######  Remark 3.2 (Equivalent formulations of the Atiyah Conjecture 3.1).

Consider a field $F$ with ${\mathbb{Q}}\subseteq F\subseteq{\mathbb{C}}$. Let
$G$ be a group that possesses an upper bound on the orders of is finite
subgroups. Then the following assertions are equivalent:

  * (1)

We have $G\in\mathcal{A}_{F}$;

  * (2)

For any matrix $A\in M_{m,n}(FG)$ the dimension
$\dim_{{\mathcal{N}}(G)}(\ker(r_{A}))$ of the kernel of the
${\mathcal{N}}(G)$-homomorphism
$r_{A}\colon{\mathcal{N}}(G)^{m}\to{\mathcal{N}}(G)^{n}$ given by right
multiplication with $A$ satisfies
$\operatorname{lcm}(G)\cdot\dim_{{\mathcal{N}}(G)}(\ker(r_{A}))\in{\mathbb{Z}}$;

  * (3)

For any $FG$-chain complex $C_{*}$ of finitely generated free $FG$-modules and
all $n\in{\mathbb{Z}}$ we have

$\operatorname{lcm}(G)\cdot\dim_{{\mathcal{N}}(G)}(H_{n}({\mathcal{N}}(G)\otimes_{FG}C_{*}))\in{\mathbb{Z}}.$

If we additionally assume $F={\mathbb{Q}}$, then the three assertions above
and the following assertion are equivalent:

  * (4)

For any ${\mathbb{Z}}G$-chain complex $C_{*}$ of finitely generated free
${\mathbb{Z}}G$-modules and all $n\in{\mathbb{Z}}$ we have

$\operatorname{lcm}(G)\cdot\dim_{{\mathcal{N}}(G)}(H_{n}({\mathcal{N}}(G)\otimes_{{\mathbb{Z}}G}C_{*}))\in{\mathbb{Z}}.$

If we additionally assume $F={\mathbb{Q}}$ and that $G$ is finitely presented,
then the four assertions above and the following assertion are equivalent:

  * (5)

For any closed manifold $M$ with $\pi_{1}(M)\cong G$ and any $n\geq 0$ we have

$\operatorname{lcm}(G)\cdot b_{n}^{(2)}(\widetilde{M})\in{\mathbb{Z}}.$

All of these claims follow from from [85, Lemma 10.5 on page 371 and Lemma
10.7 on 372], the equality $\dim_{{\mathcal{N}}(G)}({\mathcal{N}}(G))=1$, and
the Additivity of the dimension function $\dim_{{\mathcal{N}}(G)}$.

The Atiyah Conjecture 3.1 is rather surprizing in view of (2.5) and assertion
(5) appearing in Remark 3.2, since there seems to be no reason why the
expression appearing on the right hand side of (2.5) should be an integer if
$\pi_{1}(M)$ is torsionfree.

###  3.2. Status of the Atiyah Conjecture

The notions of elementary amenable groups and amenable groups are explained
for instance in [85, Subsection 6.4.1].

######  Definition 3.3 (Class of groups ${\mathcal{C}}$ ).

Let ${\mathcal{C}}$ be the smallest class of groups satisfying the following
conditions:

  1. (1)

${\mathcal{C}}$ contains all free groups;

  2. (2)

If $\\{G_{i}\mid i\in I\\}$ is a directed system of subgroups directed by
inclusion such that each $G_{i}$ belongs to ${\mathcal{C}}$, then
$G=\bigcup_{i\in I}G_{i}$ belongs to ${\mathcal{C}}$;

  3. (3)

Let $1\to K\to G\to Q\to 1$ be an extension of groups such that $H$ belongs to
${\mathcal{C}}$ and $Q$ is elementary amenable, then $G$ belongs to
${\mathcal{C}}$.

######  Definition 3.4 (Class of groups $\mathcal{D}$ ).

Let $\mathcal{D}$ be the smallest class of groups satisfying the following
conditions:

  1. (1)

The trivial group belongs to $\mathcal{D}$;

  2. (2)

If $\\{G_{i}:i\in I\\}$ is a filtered system of groups in $\mathcal{D}$ (with
arbitrary structure maps), then its colimit again belongs to $\mathcal{D}$;

  3. (3)

If $\\{G_{i}:i\in I\\}$ is a cofiltered system of groups in $\mathcal{D}$
(with arbitrary structure maps), then its limit again belongs to
$\mathcal{D}$;

  4. (4)

If $G$ belongs to $\mathcal{D}$ and $H\subseteq G$ is a subgroup, then
$H\in\mathcal{D}$;

  5. (5)

If $p\colon G\to A$ is an epimorphism of a torsionfree group $G$ onto an
elementary amenable group $A$ and if $p^{-1}(B)\in\mathcal{A}_{F}$ for every
finite group $B\subset A$, then $G\in\mathcal{A}_{F}$.

Note that each element in $\mathcal{D}$ is a torsionfree group, and the class
$\mathcal{D}$ contains all residually (torsionfree elementary amenable)
groups.

A group is called _locally indicable_ , if every non-trivial finitely
generated subgroup admits an epimorphism onto ${\mathbb{Z}}$. Locally
indicable groups are torsionfree. Examples for locally indicable groups are
one-relator groups.

######  Theorem 3.5 (Status of the Atiyah Conjecture 3.1 with coefficients in
$F$).

Consider a field $F$ with ${\mathbb{Q}}\subseteq F\subseteq{\mathbb{C}}$.

  1. (1)

If $G$ belongs to ${\mathcal{C}}$ and possesses an upper bound on the orders
of is finite subgroups, then $G\in\mathcal{A}_{F}$;

  2. (2)

If $G$ belongs to $\mathcal{D}$, then $G$ is torsionfree and
$G\in\mathcal{A}_{F}$;

  3. (3)

If $1\to H\to G\to Q\to 1$ is an extension of groups, $H$ is torsionfree and
belongs to $\mathcal{A}_{F}$, and $Q$ is locally indicable, then $G$ is
torsionfree and belongs to $\mathcal{A}_{F}$;

  4. (4)

If $G\in\mathcal{A}_{F}$ and $H\subseteq G$ is a subgroup, then
$H\in\mathcal{A}_{F}$;

  5. (5)

If G is the directed union $\bigcup_{i\in I}G_{i}$ of subgroups $G_{i}$
directed by inclusion and each $G_{i}$ belongs to $\mathcal{A}_{F}$, then $G$
belongs to $\mathcal{A}_{F}$;

  6. (6)

The group $G$ belongs to $\mathcal{A}_{F}$ if and only if all its finitely
generated subgroups belong to $\mathcal{A}_{F}$;

  7. (7)

If $1\to K\to G\to Q\to 1$ is an extension of groups such that $K$ is finite
and $G$ belongs to $\mathcal{A}_{F}$, then $Q$ belongs to $\mathcal{A}_{F}$;

  8. (8)

Let $M$ be a connected (not necessarily compact) $d$-dimensional manifold
(possibly with non-empty boundary) such that $d\leq 3$ and its fundamental
group $\pi_{1}(M)$ is torsionfree, then $\pi_{1}(M)\in{\mathcal{C}}$ and hence
$\pi_{1}(M)\in\mathcal{A}_{F}$;

  9. (9)

If the group $G$ possesses an upper bound on the orders of its finite
subgroups and belongs to one of the following classes below, then $G$ belongs
to $\mathcal{A}_{F}$:

    1. (a)

Residually {torsionfree elementary amenable} groups;

    2. (b)

Free by elementary amenable groups;

    3. (c)

Braid groups;

    4. (d)

Right-angled Artin and Coxeter groups;

    5. (e)

Torsionfree $p$-adic analytic pro-$p$-groups;

    6. (f)

Locally indicable groups;

    7. (g)

One-relator groups.

###### Proof.

(1) This is due to Linnell, see for instance [75] or [85, Theorem 10.19 on
page 378].
(2) This follows from [53, Corollary 1.2], which is based on on [26, Theorem
1.4].
(3) This follows from [56, Proposition 6.5].
(4) This follows from [85, Theorem 6.29 (2) on page 253].
(5) See [85, Lemma 10.4 on page 371].
(6) This follows from assertions (4) and (5).
(7) This follows from[85, Lemma 13.45 on page 473].
(8) This follows from [62, Theorem 1.1] for $d=3$. The case $d=2$ can be
reduced to the case $d=3$ by crossing with $S^{1}$ and assertion (4).
(9) This follows from other assertions or from [53, Theorem 1.1 and Corollary
1.2] using [73, Theorem 2] and [32, Theorem 1.1]. ∎

######  Remark 3.6.

The class $\mathcal{A}_{F}$ is very large by aforementioned results.
Nevertheless, we do not know whether the Atiyah Conjecture 3.1 holds for all
hyperbolic groups or for all amenable groups.

There are partial results on the difficult question, whether the Atiyah
Conjectures 3.1 holds for a group $G$ if it holds for a subgroup of finite
index, see for instance [74].

######  Remark 3.7.

Dropping the condition on an upper bound on the orders of finite subgroups,
one might still ask if the $L^{2}$-Betti numbers are always rational. This
goes back to Atiyah’s original question [9, page 72], who asked for
rationality of $L^{2}$-Betti numbers of closed manifolds. Austin [10,
Corollary 1.2] gave the first example of a finitely generated group $G$, where
for some matrix $A\in M_{m,n}({\mathbb{Q}}G)$ the dimension
$\dim_{{\mathcal{N}}(G)}(\ker(r_{A}))$ of the kernel of the
${\mathcal{N}}(G)$-homomorphism
$r_{A}\colon{\mathcal{N}}(G)^{m}\to{\mathcal{N}}(G)^{n}$ is irrational.
Grabowski [41, Theorem 1.3] proved, using Turing machines, that any non-
negative real number can occur in this way for some finitely generated group
$G$ and some matrix $A$. Löh and Uschold [77] investigate the computability
degree of real numbers arising as $L^{2}$-Betti numbers or $L^{2}$-torsion of
groups, parametrised over the Turing degree of the word problem. Roughly
speaking, the complexity of the computation of $L^{2}$-invariants of a group
is the same as the complexity of the word problem. This is due to the
combinatorial computation in terms of characteristic sequences mentioned in
Remark 9.1.

###  3.3. Embedding the group ring of a torsionfree group into a skewfield

Associated to the von Neumann algebra ${\mathcal{N}}(G)$ is the algebra of
affiliated operators ${\mathcal{U}}(G)$ which contains ${\mathcal{N}}(G)$. It
can be defined analytically or just as the Ore localization of
${\mathcal{N}}(G)$ with respect to the multiplicative subset of non-zero
divisors. Now one can consider the so called division closure $\mathcal{D}(G)$
of ${\mathbb{C}}G$ in ${\mathcal{U}}(G)$, i.e., the smallest ring
${\mathbb{C}}G\subset\mathcal{D}(G)\subset{\mathcal{U}}(G)$ such that if
$x\in\mathcal{D}(G)$ is invertible in ${\mathcal{U}}(G)$, its inverse is
already contained in $\mathcal{D}(G)$.

The proof of the following is based on ideas of Peter Linnell from [75] which
have been elaborated and expanded on in [85, Theorem 8.29 on page 330 and
Lemma 10.39 on page 388] and [108], see also [35, Theorem 3.8 (1) and 2]. The
proofs given in the references above for $F={\mathbb{C}}$ and $F={\mathbb{Q}}$
carry directly over to arbitrary $F$.

######  Theorem 3.8 (Main properties of $\mathcal{D}(G)$).

Let $G$ be a torsionfree group. Consider a field $F$ with
${\mathbb{Q}}\subseteq F\subseteq{\mathbb{C}}$.

  1. (1)

The group $G$ satisfies the Atiyah Conjecture with coefficients in $F$ if and
only if $\mathcal{D}(G)$ is a skew field;

  2. (2)

Suppose that $G$ satisfies the Atiyah Conjecture. Let $C_{*}$ be a projective
$FG$-chain complex. Then we get for all $n\geq 0$

$b_{n}^{(2)}\bigl{(}{\mathcal{N}}(G)\otimes_{FG}C_{*}\bigr{)}=\dim_{\mathcal{D}(G)}\bigl{(}H_{n}(\mathcal{D}(G)\otimes_{{\mathbb{Q}}G}C_{*})\bigr{)}.$

In particular $b_{n}^{(2)}\bigl{(}{\mathcal{N}}(G)\otimes_{FG}C_{*}\bigr{)}$
is an integer or $\infty$.

Theorem 3.8 shows that the Atiyah Conjecture 3.1 is related to the question
whether for a torsionfree group $G$ the group ring $FG$ can be embedded into a
skew field, see for instance [49]. Note that the existence of an embedding of
$FG$ into a skewfield implies that $FG$ has no non-trivial zero-divisors which
is predicted by the Zero-Divisor-Conjecture of Kaplansky.

###  3.4. The Algebraic Atiyah Conjecture

Recall that Linnell’s program to approach the Atiyah Conjecture consists of a
$K$-theoretic part and a ring theoretic part, namely that the composite

$\operatorname{colim}_{H\subseteq G,|H|<\infty}K_{0}(FH)\to K_{0}(FG)\to
K_{0}(\mathcal{D}(G))$

is surjective and that $\mathcal{D}(G)$ is semisimple, see [85, Lemma 10.28 on
page 382 and Theorem 10.38 on page 387]. There is the following so called
algebraic Atiyah Conjecture, which is a purely $K$-theoretic statement and
taken from [54, Conjecture 7.2], c,f. [85, Lemma 10.26 on page 382]. It
implies the Atiyah Conjecture 3.1, see [85, Lemma 10.26 on page 382].

######  Conjecture 3.9 (Algebraic Atiyah Conjecture).

Let $G$ be a group and let $F$ be a field $F$ with ${\mathbb{Q}}\subseteq
F\subseteq{\mathbb{C}}$, which is closed under complex conjugation. Let
$\mathcal{R}_{FG}$ be the $\ast$-regular closure of $FG$ in
${\mathcal{U}}(G)$.

We say that $G$ satisfies the _Algebraic Atiyah Conjecture with coefficients
in $F$_ if the composite of the canonical maps

$\operatorname{colim}_{H\subseteq G,|H|<\infty}K_{0}(FH)\to K_{0}(FG)\to
K_{0}(\mathcal{R}_{FG})$

is surjective.

Note that the Farrell-Jones Conjecture, which is known for a large class of
groups, see for instance [91, Theorem 12.56 and Theorem 15.1], predicts the
bijectivity of the map $\operatorname{colim}_{H\subseteq
G,|H|<\infty}K_{0}(FH)\to K_{0}(FG)$. If the Farrell-Jones Conjecture holds,
then the Algebraic Atiyah Conjecture 3.9 is equivalent to the claim that the
map $K_{0}(FG)\to K_{0}(\mathcal{R}_{FG})$ is surjective. For a group $G$, for
which there is a bound on its finite subgroups, the algebraic Atiyah
Conjecture 3.9 is equivalent to the so called center-valued Atiyah Conjecture
with coefficients in $F$, which implies the Atiyah Conjecture 3.1 with
coefficients in $F$, see [48, Definition 2.5.1, Proposition 2.5.2, Theorem
3.1.4] and also [65, Definition 1.2, Proposition 1.3].

For more information about the Atiyah Conjecture we refer for instance to [85,
Chapter 10].

##  4\. The Singer Conjecture

We now turn attention to a series of conjectures about $L^{2}$-invariants of
aspherical manifolds. They deal with the phenomenon that $L^{2}$-invariants
often vanish outside of the middle dimension.

###  4.1. Statement of the Singer Conjecture

######  Conjecture 4.1 (Singer Conjecture).

If $M$ is an aspherical closed topological manifold, then we get for $n\geq 0$

$b_{n}^{(2)}(\widetilde{M})=0\quad\text{if}\;2n\not=\dim(M).$

If $M$ is an aspherical closed topological manifold of even dimension $2m$ ,
then

$(-1)^{m}\cdot\chi(M)=b_{m}^{(2)}(\widetilde{M})\geq 0.$

If $M$ is a closed connected smooth manifold of even dimension $2m$ admitting
a Riemannian metric of negative sectional curvature, then

$(-1)^{m}\cdot\chi(M)=b_{m}^{(2)}(\widetilde{M})>0.$

The equality $(-1)^{m}\cdot\chi(M)=b_{m}^{(2)}(\widetilde{M})$ appearing in
the Singer Conjecture 4.1 above follows from the the Euler-Poincaré formula
$\chi(M)=\sum_{p\geq 0}(-1)^{p}\cdot b_{p}^{(2)}(\widetilde{M})$.

The Singer Conjecture 4.1 is consistent with the Atiyah Conjecture in the
sense that it predicts that the $L^{2}$-Betti numbers
$b_{n}^{(2)}(\widetilde{M})$ for an aspherical closed manifold $M$ are all
integers.

In original versions of the Singer Conjecture 4.1 the condition aspherical
closed manifolds was replaced by the condition closed Riemannian manifold with
non-positive sectional curvature. Note that a closed Riemannian manifold with
non-positive sectional curvature is aspherical by Hadamard’s Theorem.

###  4.2. Hopf Conjectures

Obviously the Singer Conjecture 4.1 implies the following conjecture in the
cases, where $M$ is aspherical or has negative sectional curvature.

######  Conjecture 4.2 (Hopf Conjecture).

If $M$ is an aspherical closed topological manifold of even dimension
$\dim(M)=2m$, then

$(-1)^{m}\cdot\chi(M)\geq 0.$

If $M$ is a closed smooth manifold of even dimension $\dim(M)=2m$ with
Riemannian metric and with sectional curvature $\sec(M)$, then

$\begin{array}[]{rlllllll}(-1)^{m}\cdot\chi(M)&>&0&&\text{if}&\sec(M)&<&0;\\\
(-1)^{m}\cdot\chi(M)&\geq&0&&\text{if}&\sec(M)&\leq&0;\\\
\chi(M)&=&0&&\text{if}&\sec(M)&=&0;\\\
\chi(M)&\geq&0&&\text{if}&\sec(M)&\geq&0;\\\
\chi(M)&>&0&&\text{if}&\sec(M)&>&0.\end{array}$

The following version of the Hopf Conjecture for $L^{2}$-torsion appears in
[85, Conjecture 11.3 on page 418]

######  Conjecture 4.3 (Hopf Conjecture for $L^{2}$-torsion).

If $M$ is an aspherical closed topological manifold of odd dimension
$\dim(M)=2m+1$, then $M$ is $\det$-$L^{2}$-acyclic, and its $L^{2}$-torsion
satisfies

$(-1)^{m}\cdot\rho^{(2)}(\widetilde{M})\geq 0.$

If $M$ is a closed smooth manifold of odd dimension $\dim(M)=2m+1$ with
Riemannian metric and with sectional curvature $\sec(M)<0$, then $M$ is
$\det$-$L^{2}$-acyclic, and its $L^{2}$-torsion satisfies

$(-1)^{m}\cdot\rho^{(2)}(\widetilde{M})>0.$

If $M$ is an aspherical closed topological manifold of odd dimension
$\dim(M)=2m+1$ whose fundamental group contains an amenable infinite normal
subgroup, then $M$ is $\det$-$L^{2}$-acyclic and its $L^{2}$-torsion satisfies

$\rho^{(2)}(\widetilde{M})=0.$

The Hopf Conjecture 4.3 for $L^{2}$-torsion is known to be true if one of the
following conditions is satisfied:

  1. (1)

$\dim(M)\leq 3$;

  2. (2)

$M$ is a locally symmetric space;

  3. (3)

$\pi_{1}(M)$ contains an elementary amenable infinite normal subgroup;

  4. (4)

$M$ carries a non-trivial $S^{1}$-action.

Statement (1) follows from combining [95, Theorem 0.7] with Thurston’s
Geometrization conjecture, which is known to be true by [64, 99] following the
spectacular outline of Perelman. Statement (2) can be found in [85, Corollary
5.16 on page 231] and statements (3) and (4) are from [87, Corollary 1.13].
See also Remark 6.12.

###  4.3. Status of the Singer Conjecture

The Singer Conjecture 4.1 is known for an aspherical closed smooth manifold
$M$ in the following cases:

  1. (1)

$\dim(M)\leq 3$;

  2. (2)

$M$ comes with a Riemannian metric whose sectional curvature is negative and
satisfies certain pinching conditions;

  3. (3)

$M$ is a locally symmetric space;

  4. (4)

$M$ possesses a Riemannian metric, whose sectional curvature is negative and
$M$ carries some Kähler structure;

  5. (5)

$M$ is an aspherical closed Kähler manifold, whose fundamental group is word-
hyperbolic in the sense of Gromov [44].

  6. (6)

$\pi_{1}(M)$ contains an amenable infinite normal subgroup.

  7. (7)

$M$ carries a non-trivial $S^{1}$-action;

  8. (8)

$M$ fibers over $S^{1}$.

The precise statements and proofs and the relevant references, e.g., [14, 23,
45, 57, 81, 101], can be found in [85, Theorem 1.39 on page 42, Corollary 1.43
on page 48, Theorem 1.44 on page 48, Section 11.1], again using Thurston’s
Geometrization Conjecture for the three dimensional case. We also mention that
proofs of (1) - (5) use geometric properties of the manifold whereas (6) - (8)
use algebraic topological techniques and do not use the manifold structure.

Partial results about the Singer Conjecture 4.1 for right-angled Coxeter
groups can be found in Davis-Okun [25].

The paper by Albanese, Di Cerbo and Lombardi [6] deals with the Singer
Conjecture for aspherical complex surfaces and proves it for aspherical
complex surfaces with residually finite fundamental groups in [6, Theorem
1.5].

In contrast to the Atiyah Conjecture, evidence for the Singer Conjecture 4.1
comes from computations only and no promising proof strategy is known. In some
sense Poincaré duality together with $L^{2}$-bounds on differential forms on
$\widetilde{M}$ seem to force the $L^{2}$-Betti numbers
$b_{p}^{(2)}(\widetilde{M})$ of an aspherical closed manifold to concentrate
in the middle dimension. One may wonder what happens if we replace $M$ by an
aspherical finite Poincaré complex in the Singer Conjecture 4.1. There are
counterexamples to the Singer Conjecture 4.1 if one weakens aspherical to
rationally aspherical, see [11, Theorem 4]. The reader should also take a look
at Remark 6.10 and Remark 6.12.

###  4.4. The proper Singer Conjecture and the action dimension

One may generalize the Singer Conjecture 4.1 to the following

######  Conjecture 4.4 (Proper Singer Conjecture).

A group $G$ satisfies the _proper Singer Conjecture_ , if for any contractible
topological manifold $M$ without boundary on which $G$ acts properly and
cocompactly, we have for all $n\geq 0$

$b_{n}^{(2)}(M;{\mathcal{N}}(G))=0\quad\text{if}\;2n\not=\dim(M).$

Note that the proper Singer Conjecture is a statement about a group $G$,
whereas the Singer Conjecture 4.1 is a statement about an aspherical closed
manifold. Provided that $G$ is torsionfree, the proper Singer Conjecture holds
for $G$ if and only if the Singer Conjecture 4.1 holds for one (and hence all)
aspherical closed manifold $M$ with $\pi_{1}(M)\cong G$.

The proper Singer Conjecture for a group $G$ is equivalent to the cadim-
Conjecture (cadim standing for compact action dimension) for $G$ for manifolds
with PL-boundary, see [100, Theorem 4.10]. The latter predicts that for a
contractible topological manifold $M$ (possibly with boundary) that admits a
cocompact proper topological action of $G$ such that the boundary $\partial M$
of $M$ carries a $\operatorname{PL}$-structure for which the $G$-action on
$\partial M$ is through $\operatorname{PL}$-automorphisms we have

$2\cdot\sup\\{n\in{\mathbb{Z}}^{\geq 0}\mid
b_{n}^{(2)}(G)\not=0\\}\leq\dim(M).$

For more informations about the notion of the (compact) action dimension and
its relationship to $L^{2}$-Betti numbers we refer for instance to [12, 17,
100].

###  4.5. The ${\mathbb{F}}_{p}$-Singer Conjecture

One may also consider the _${\mathbb{F}}_{p}$ -Singer Conjecture_, which
predicts that for an aspherical closed topological manifold $M$ and any prime
$p$ we get for $n\geq 0$

$b_{n}^{(2)}(\pi_{1}(M);{\mathbb{F}}_{p})=0\quad\text{if}\;2n\not=\dim(M).$

Here $b_{n}^{(2)}(\pi_{1}(M);{\mathbb{F}}_{p})$ is defined only if
$\pi_{1}(M)$ is residually finite. Namely, for a chain of normal subgroups of
finite index
$\pi_{1}(M)=\Gamma_{0}\supseteq\Gamma_{1}\supseteq\Gamma_{2}\supseteq\cdots$
with $\bigcap_{n=0}^{\infty}\Gamma_{n}=\\{1\\}$, one puts
$b_{n}^{(2)}(\pi_{1}(M);{\mathbb{F}}_{p})=\limsup_{n\to\infty}\frac{b_{n}(\Gamma_{k};{\mathbb{F}}_{p})}{[\Gamma:\Gamma_{k}]}$.
It is unknown in general whether this definition depends on the chain of
subgroups $\Gamma_{k}$ and the statement above is to be understood in the
sense that it holds for any such chain. The definition of $L^{2}$-Betti
numbers over ${\mathbb{F}}_{p}$ is motivated by the Approximation Theorem in
characteristic zero 2.4 (14). The ${\mathbb{F}}_{p}$-Singer Conjecture is open
for $\dim(M)=3$. However, for any odd prime $p$, the ${\mathbb{F}}_{p}$-Singer
Conjecture fails in all odd dimensions larger or equal than $7$ and all even
dimensions larger or equal than $14$, see [13, Theorem 4].

##  5\. The Determinant Conjecture

Next we explain the Determinant Conjecture [85, Conjecture 13.2 on page 454].
It is needed for the definition of $L^{2}$-determinant class and implies
homotopy invariance of $L^{2}$-torsion, see Theorem 2.4 (1). Furthermore, it
implies the Approximation Conjecture, see Remark 7.2.

######  Conjecture 5.1 (Determinant Conjecture for a group $G$).

For any matrix $A\in M_{r,s}({\mathbb{Z}}G)$, the Fuglede-Kadison determinant
of the ${\mathcal{N}}(G)$-homomorphism
$r_{A}\colon{\mathcal{N}}(G)^{r}\to{\mathcal{N}}(G)^{s}$ given by right
multiplication with $A$ satisfies

${\det}_{{\mathcal{N}}(G)}\left(r_{A}\right)\geq 1.$

The class $\mathcal{F}$ of groups for which it is true has the following
properties:

  1. (1)

_Trivial group_.
The trivial group belongs to $\mathcal{F}$;

  2. (2)

_Amenable quotient_.
Let $H\subset G$ be a normal subgroup. Suppose that $H\in\mathcal{F}$ and the
quotient $G/H$ is amenable. Then $G\in\mathcal{F}$;

  3. (3)

_Filtered colimits_.
The class $\mathcal{F}$ is closed under countable filtered colimits;

  4. (4)

_Cofiltered limits_.
The class $\mathcal{F}$ is closed under countable cofiltered limits;

  5. (5)

_Subgroups_.
If $H$ is isomorphic to a subgroup of a group $G$ with $G\in\mathcal{F}$, then
$H\in\mathcal{F}$.

  6. (6)

_Quotients with finite kernel_.
Let $1\to K\to G\to Q\to 1$ be an exact sequence of groups. If $K$ is finite
and $G$ belongs to $\mathcal{F}$, then $Q$ belongs to $\mathcal{F}$;

  7. (7)

_Sofic groups_.
Sofic groups belong to $\mathcal{F}$.

For the verification of the Determinant Conjecture for the groups above, see
[28, Theorem 5], [85, Section 13.2 on pages 459 ff], [114, Theorem 1.21].

To sketch how large $\mathcal{F}$ really is, let us mention that already the
class of sofic groups is very large. It is closed under direct and free
products, taking subgroups, taking inverse and direct limits over directed
index sets, and is closed under extensions with amenable groups as quotients
and a sofic group as kernel. In particular it contains all residually amenable
groups. One expects that there exists non-sofic groups but no example is
known. More information about sofic groups can be found for instance in [29]
and [103]. More information about the Determinant Conjecture 5.1 can be found
in [85, Chapter 13] and [114].

##  6\. Approximation for finite index normal chains

Recall that the Approximation Theorem 14 predicts that $L^{2}$-Betti numbers
of spaces can be computed as limits of normalized ordinary Betti numbers of
certain towers of coverings. In this section we discuss this phenomenon for
more general invariants.

###  6.1. Basic setup for approximation for finite index normal chains

Let $G$ be a (discrete) group. A _finite index normal chain_ $\\{G_{i}\\}$ for
$G$ is a descending chain of subgroups

(6.1) $\displaystyle G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq\cdots$

such that $G_{i}$ is normal in $G$, the index $[G:G_{i}]$ is finite and
$\bigcap_{i\geq 0}G_{i}=\\{1\\}$.

Let $p\colon\overline{X}\to X$ be a $G$-covering. Put
$X[i]:=G_{i}\backslash\overline{X}$. We obtain a $[G:G_{i}]$-sheeted covering
$p[i]\colon X[i]\to X$. Its total space $X[i]$ inherits the structure of a
finite $CW$-complex, a closed manifold, or a closed Riemannian manifold
respectively if $X$ has the structure of a finite $CW$-complex, a closed
manifold, or a closed Riemannian manifold respectively.

Let $\alpha$ be a classical topological invariant such as the Euler
characteristic, the signature, the $n$th Betti number with coefficients in the
field ${\mathbb{Q}}$ or ${\mathbb{F}}_{p}$, torsion in the sense of
Reidemeister or Ray-Singer, or the logarithm of the cardinality of the torsion
subgroup of the $n$th homology group with integral coefficients. We want to
study the sequence

$\displaystyle\left(\frac{\alpha(X[i])}{[G:G_{i}]}\right)_{i\geq 0}.$

######  Problem 6.1 (Approximation Problem).

  1. (1)

Does the sequence converge?

  2. (2)

If yes, is the limit independent of the chain $\\{G_{i}\\}$?

  3. (3)

If yes, what is the limit?

The hope is that the answer to the first two questions is yes and the limit
turns out to be an $L^{2}$-analogue $\alpha^{(2)}$ of $\alpha$ applied to the
$G$-space $\overline{X}$, i.e., one can prove an equality of the type

(6.2)
$\displaystyle\lim_{i\to\infty}\frac{\alpha(X[i])}{[G:G_{i}]}=\alpha^{(2)}(\overline{X};{\mathcal{N}}(G)).$

Here ${\mathcal{N}}(G)$ stands for the group von Neumann algebra and is a
reminiscence of the fact that the $G$-action on $\overline{X}$ plays a role.
Equation (6.2) is often used to compute the $L^{2}$-invariant
$\alpha^{(2)}(\overline{X};{\mathcal{N}}(G))$ by its finite-dimensional
analogues $\alpha(X[i])$. On the other hand, it implies the existence of
finite coverings with large $\alpha(X[i])$, if
$\alpha^{(2)}(\overline{X};{\mathcal{N}}(G))$ is known to be positive.

###  6.2. The Euler characteristic

The _Euler characteristic_ $\chi(X)$ of a finite $CW$-complex is
multiplicative under finite coverings. Because this implies
$\chi(X)=\frac{\chi(X[i])}{[G:G_{i}]}$, the answer in this case is yes to the
questions appearing in Problem 6.1, and the limit is

(6.3) $\displaystyle\lim_{i\to\infty}\frac{\chi(X[i])}{[G:G_{i}]}$
$\displaystyle=$ $\displaystyle\chi(X).$

###  6.3. The Signature

Next we consider the _signature_ $\operatorname{sign}(M)$ of a closed oriented
topological $4k$-dimensional manifold $M$. It is known that it is
multiplicative under finite coverings, however, the proof is more involved
than the one for the Euler characteristic. It follows for instance from
Hirzebruch’s Signature Theorem, see [51], or Atiyah’s $L^{2}$-index theorem
[9, (1.1)] in the smooth case; for closed topological manifolds see Schafer
[113, Theorem 8]. Since this implies
$\operatorname{sign}(X)=\frac{\operatorname{sign}(X[i])}{[G:G_{i}]}$, each of
the questions appearing in Problem 6.1 has a positive answer, and the limit is

(6.4)
$\displaystyle\lim_{i\to\infty}\frac{\operatorname{sign}(X[i])}{[G:G_{i}]}$
$\displaystyle=$ $\displaystyle\operatorname{sign}(X).$

The next level of generality is to pass from a oriented closed topological
manifold to a oriented finite Poincaré complex, whose definition is due to
Wall [123]. For them the signature is still defined if the dimension is
divisible by $4$. There are Poincaré complexes $X$ for which the signature is
not multiplicative under finite coverings, see [106, Example 22.28], [123,
Corollary 5.4.1]. Hence the situation is more complicated here. Nevertheless,
it turns out in this case each of the questions appearing in Problem 6.1 has a
positive answer, and the limit is

(6.5)
$\displaystyle\lim_{i\to\infty}\frac{\operatorname{sign}(X[i])}{[G:G_{i}]}$
$\displaystyle=$
$\displaystyle\operatorname{sign}^{(2)}(\overline{X};{\mathcal{N}}(G)),$

where $\operatorname{sign}^{(2)}(\overline{X};{\mathcal{N}}(G))$ denotes the
$L^{2}$-signature, which is in general different from $\operatorname{sign}(X)$
for a finite Poincaré complex $X$.

For more information and details we refer to [96, 97].

###  6.4. Approximation of $L^{2}$-Betti numbers in characteristic zero

Fix a field $F$ of characteristic zero. We consider the _$n$ th Betti number
with $F$-coefficients_ $b_{n}(X;F):=\dim_{F}(H_{n}(X;F))$. Note that
$b_{n}(X;F)=b_{n}(X;{\mathbb{Q}})=\operatorname{rk}_{{\mathbb{Z}}}(H_{n}(X;{\mathbb{Z}}))$
holds, where $\operatorname{rk}_{{\mathbb{Z}}}$ denotes the rank of a finitely
generated abelian group. In this case each of the questions appearing in
Problem 6.1 has a positive answer by the main result of Lück’s article [80].

######  Theorem 6.2.

Let $F$ be a field of characteristic zero, and let $X$ be a finite
$CW$-complex. Then for each finite index normal chain $\\{G_{i}\\}$ we have

$\lim_{i\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{i}]}=b^{(2)}_{n}(\overline{X};{\mathcal{N}}(G)),$

where $b^{(2)}(\overline{X};{\mathcal{N}}(G))$ denotes the $n$th $L^{2}$-Betti
number.

Löh and Uschold [77, Proposition 6.6] prove a quantitative version of Theorem
6.2. Essentially they show

$\left|b^{(2)}_{n}(\overline{X};{\mathcal{N}}(G))-\frac{b_{n}(X[i];F)}{[G:G_{i}]}\right|\leq
a\cdot\left(1-\frac{1}{b\cdot i}\right)^{i^{2}}+\frac{a\cdot\log(b)}{\log(i)}$

for certain constants $a$ and $b$ depending on $X$.

###  6.5. Approximation of $L^{2}$-Betti numbers in prime characteristic

The situation is more complicated and unclear in prime characteristic. Fix a
prime $p$. Let $F$ be a field of characteristic $p$. We consider the _$n$ th
Betti number with $F$-coefficients_ $b_{n}(X;F):=\dim_{F}(H_{n}(X;F))$. Note
that $b_{n}(X;F)=b_{n}(X;{\mathbb{F}}_{p})$ holds where ${\mathbb{F}}_{p}$ is
the field of $p$-elements. In this setting a general answer to Problem 6.1 is
only known in special cases. The main problem is that one does not have an
analogue of the von Neumann algebra in characteristic $p$ and the construction
of an appropriate extended dimension function, see [83], is not known in
general.

If $G$ is torsionfree elementary amenable, one gets the full positive answer
by Linnell-Lück-Sauer [72, Theorem 0.2], where more explanations, e.g., about
Ore localizations, are given and actually virtually torsionfree elementary
amenable groups are considered.

######  Theorem 6.3.

Let $F$ be a field (of arbitrary characteristic) and $X$ be a connected finite
$CW$-complex. Let $G$ be a torsionfree elementary amenable group. Then for
each finite index normal chain $\\{G_{i}\\}$:

$\dim_{FG}^{\operatorname{Ore}}\bigl{(}H_{n}(\overline{X};F)\bigr{)}=\lim_{n\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{n}]}.$

Note that Theorem 6.3 is consistent with Theorem 6.2 since for a field $F$ of
characteristic zero and a torsionfree elementary amenable group $G$ we have
$b_{n}^{(2)}(\overline{X};{\mathcal{N}}(G))=\dim_{FG}^{\operatorname{Ore}}\bigl{(}H_{n}(\overline{X};F)\bigr{)}$.
The latter equality follows from [85, Theorem 6.37 on page 259, Theorem 8.29
on page 330, Lemma 10.16 on page 376, and Lemma 10.39 on page 388].

Here is another special case taken from Bergeron-Lück-Linnell-Sauer [15], (see
also Calegari-Emerton [21, 22]), where we know the answer only for special
chains. Let $p$ be a prime, let $n$ be a positive integer, and let $\phi\colon
G\rightarrow\operatorname{GL}_{n}({\mathbb{Z}}_{p})$ be a homomorphism, where
${\mathbb{Z}}_{p}$ denotes the $p$-adic integers. The closure of the image of
$\phi$, which is denoted by $\Lambda$, is a $p$-adic analytic group admitting
an exhausting filtration by open normal subgroups
$\Lambda_{i}=\ker\left(\Lambda\rightarrow\operatorname{GL}_{n}({\mathbb{Z}}/p^{i}{\mathbb{Z}})\right)$.
Put $G_{i}=\phi^{-1}(\Lambda_{i})$.

######  Theorem 6.4.

Let $F$ be a field (of arbitrary characteristic). Put $d=\dim(\Lambda)$. Let
$X$ be a finite $CW$-complex. Then for any integer $n$ and as $i$ tends to
infinity, we have:

$b_{n}(X[i];F)=b_{n}^{(2)}(\overline{X};F)\cdot[G:G_{i}]+O\left([G:G_{i}]^{1-{1/d}}\right),$

where $b_{n}^{(2)}(\overline{X};F)$ is the $n$th mod $p$ $L^{2}$-Betti numbers
occurring in [15, Definition 1.3] which is defined using homology coefficients
in the Iwasawa algebra of $G$. In particular

$\lim_{i\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{i}]}=b_{n}^{(2)}(\overline{X};F).$

Returning to the setting of arbitrary finite index normal chain
$(G_{i})_{i\geq 0}$, we get by the universal coefficient theorem
$b_{n}(X[i];{\mathbb{Q}})\leq b_{n}(X[i];F)$ for any field $F$ and hence by
Theorem 6.2 the inequality

$\liminf_{i\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{i}]}\geq
b_{n}^{(2)}(\overline{X};{\mathcal{N}}(G)).$

If $p$ is a prime and we additionally assume that each index $[G:G_{i}]$ is a
$p$-power, then the sequence $\frac{b_{n}(X[i];F)}{[G:G_{i}]}$ is monotone
decreasing and in particular
$\lim_{i\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{i}]}$ exists, see [15, Theorem
1.6].

######  Question 6.5 (Approximation in prime characteristic).

Does the sequence $\frac{b_{n}(X[i];F)}{[G:G_{i}]}$ converge and is the limit
$\lim_{i\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{i}]}$ independent of the chain
$\\{G_{i}\\}$ for all fields $F$ and $n\geq 0$, provided that $X$ is finite
and $\overline{X}$ is contractible?

######  Remark 6.6.

The third author conjectured in the situation of Question 6.5 that the
$\lim_{i\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{i}]}$ is equal to
$b_{n}^{(2)}(X;{\mathcal{N}}(G))$ for every field $F$ and for all finite index
normal chains $\\{G_{i}\\}$, see [88, Conjecture 3.4 on page 275]. This is
true in characteristic zero by Theorem 6.2 and if $G$ is torsionfree
elementary amenable by Theorem 6.3, but not in prime characteristic by
Avramidi-Okun-Schreve [13, Corollary 2]. Namely, Avramidi-Okun-Schreve [13,
Theorem 1] prove that for a right-angled Artin group $A_{L}$ with defining
flag complex $L$ and any field $F$ the limit
$\lim_{i\to\infty}\frac{b_{n}(X[i];F)}{[G:G_{i}]}$ exists, is independent of
the chain $\\{G_{i}\\}$, and actually agrees with the reduced Betti number
$\overline{b}_{n}(L;F)$ of $L$ with coefficients in $F$. Furthermore, they
construct examples where $\overline{b}_{3}(L;{\mathbb{Q}})=0$ and
$\overline{b}_{3}(L;{\mathbb{F}}_{2})=1$ hold. These counterexamples do not
exist in degree $n=1$.

######  Remark 6.7 ($L^{2}$-Betti numbers in finite characteristic via skew
fields).

Recall from Theorem 3.8 that if a torsionfree group $G$ satisfies the Atiyah
conjecture, then the group ring ${\mathbb{C}}G$ embeds into a skew field
$\mathcal{D}(G)$ and $L^{2}$-Betti numbers are the Betti numbers with
coefficients in this skew field. Motivated by this, one might hope that for a
torsionfree group $G$ and any field $F$ there exists a skew field
$\mathcal{D}_{F}(G)$ together with an embedding
$FG\hookrightarrow\mathcal{D}_{F}(G)$ which can be used to define
$L^{2}$-Betti numbers with coefficients in $F$ as
$b^{(2)}_{n}(\overline{X};F)\coloneqq\dim_{\mathcal{D}_{F}(G)}H_{n}^{G}(\overline{X};\mathcal{D}_{F}(G))$.
An approach to Question 6.5 would then be to show that the sequence
$\frac{b_{n}(X[i];F)}{[G:G_{i}]}$ converges to $b^{(2)}_{n}(\overline{X};F)$
for $i\to\infty$.

Jaikin-Zapirain constructs such embeddings in [55, Corollary 1.3] for large
classes of groups. He shows that such an embedding of $FG$ into a skew field
with very nice properties (it is a Hughes free division ring and the universal
division ring of fractions of $FG$) exists if $G$ is residually (locally
indicable amenable). Combining the approximation results [55, Theorem 1.2] and
[72, Theorem 0.2] it is easy to check that in this situation the equality

(6.6)
$b^{(2)}_{n}(\overline{X};F)=\inf\left\\{\frac{b_{n}(\overline{X}/H;F)}{[G:H]}:H\leq
G,[G:H]<\infty\right\\}$

holds. In the case $F={\mathbb{Q}}$ of the classical approximation theorem the
easier to prove Kazhdan’s inequality predicts that $L^{2}$-Betti numbers are
bounded by

$b^{(2)}_{n}(\overline{X};{\mathcal{N}}(G))\geq\limsup_{i\to\infty}\frac{b_{n}(X[i];{\mathbb{Q}})}{[G:G_{i}]}$

for any finite index normal chain $(G_{i})$ for $G$. It is not clear, however,
whether an analogue of this holds in the present setting. This would imply
that (6.6) still holds when one replaces $\inf$ by the limit over a finite
index normal chain.

###  6.6. Torsion

For a detailed discussion of approximation for finite index normal chains for
torsion invariants such as the Ray-Singer torsion or the integral torsion we
refer for instance to [88, Sections 8 – 10] and also to Subsection 7.3.

###  6.7. Homological torsion growth and $L^{2}$-torsion

The following conjecture is taken from [87, Conjecture 1.12 (2)]. For locally
symmetric spaces it reduces to the conjecture of Bergeron and Venkatesh [16,
Conjecture 1.3].

######  Conjecture 6.8 (Homological torsion growth and $L^{2}$-torsion).

Let $M$ be an aspherical closed manifold.

Then we get for any natural number $n$ with $2n+1\not=\dim(M)$

$\lim_{i\to\infty}\;\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{n}(M[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G:G_{i}]}=0.$

If the dimension $\dim(M)=2m+1$ is odd, then $\widetilde{M}$ is
$\det$-$L^{2}$-acyclic and we get

$\lim_{i\to\infty}\;\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{m}(M[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G:G_{i}]}=(-1)^{m}\cdot\rho^{(2)}(\widetilde{M}).$

Conjecture 6.8 is true, if $\pi_{1}(M)$ contains an elementary amenable
infinite normal subgroup or $M$ carries a non-trivial $S^{1}$-action, see [87,
Corollary 1.13].

######  Remark 6.9 (Criteria for vanishing homology torsion growth).

Recently, there has been some progress in proving vanishing of homology growth
and torsion homology growth for certain groups. These results are based on the
general vanishing result of Abert-Bergeron-Fraczyk-Gaboriau [2, Theorem 10.20]
and have been applied to show vanishing for special linear groups and mapping
class groups [2, Theorem A, D]. Further applications of this theorem can be
found in [7, Theorem A], [8, Theorem A], and [122, Proposition 1.6], where
vanishing of homology growth and torsion homology growth of mapping tori of
finitely generated free groups with respect to a polynomially growing
automorphism and of right-angled Artin groups which are inner amenable is
shown. Combining the first result with [24, Theorem 5.1] confirms Conjecture
6.8 for mapping tori of aspherical closed manifolds which have finitely
generated free fundamental group and the monodromy is polynomially growing.

######  Remark 6.10 (The Singer Conjecture 4.1 and Conjecture 6.8 are not
compatible).

The Singer Conjecture 4.1 and Conjecture 6.8 about homological torsion growth
and $L^{2}$-torsion cannot both be true in general. Namely, if both are true,
then the ${\mathbb{F}}_{p}$-Singer Conjecture of Subsection 4.5 would be true
as pointed out by Avramidi-Okun-Schreve before Theorem 4 appearing in [13].
The argument is as follows.

Suppose we have an aspherical closed $d$-dimensional manifold $M$ and both the
Singer Conjecture 4.1 and Conjecture 6.8 are true. Consider $n$ with
$2n\not=d$. We have to show

(6.7)
$\displaystyle\lim_{i\to\infty}\frac{b_{n}(M[i];{\mathbb{F}}_{p})}{[G:G_{i}]}=0$

for $G=\pi_{1}(M)$ and a finite index normal chain $\\{G_{i}\\}$. Because of
Poincare duality it suffices to consider the case $2n>d$. Put $M^{3}=M\times
M\times M$, $G^{3}=G\times G\times G$ and $G_{i}^{3}=G_{i}\times G_{i}\times
G_{i}$. Then
$\widetilde{M^{3}}=\widetilde{M}\times\widetilde{M}\times\widetilde{M}$,
$G^{3}=\pi_{1}(M^{3})$, and $M^{3}[i]=M[i]^{3}$. By the Singer Conjecture 4.1
applied to $M^{3}$ we get $b_{3n}^{(2)}(\widetilde{M^{3}})=0$. This implies by
Theorem 6.3

$\lim_{i\to\infty}\frac{\operatorname{rk}_{{\mathbb{Z}}}(H_{3n}(M[i]^{3};{\mathbb{Z}}))}{[G^{3}:G_{i}^{3}]}=\lim_{i\to\infty}\frac{b_{3n}(M[i]^{3};{\mathbb{Q}})}{[G^{3}:G_{i}^{3}]}=b_{3n}^{(2)}(\widetilde{M^{3}};{\mathbb{Q}})=0.$

Since $2(3n)>2(3n-1)>3d$ holds, we conclude from Conjecture 6.8

$\displaystyle\lim_{i\to\infty}\;\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{3n}(M^{3}[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G^{3}:G_{i}^{3}]}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\lim_{i\to\infty}\;\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{3n-1}(M^{3}[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G^{3}:G_{i}^{3}]}$
$\displaystyle=$ $\displaystyle 0.$

We have for any $k\geq 0$

$\displaystyle\dim_{{\mathbb{F}}_{p}}({\mathbb{Z}}^{k}\otimes_{{\mathbb{Z}}}{\mathbb{F}}_{p})$
$\displaystyle=$ $\displaystyle\dim_{{\mathbb{Z}}}({\mathbb{Z}}^{k});$
$\displaystyle\dim_{{\mathbb{F}}_{p}}({\mathbb{Z}}/p^{k}\otimes_{{\mathbb{Z}}}{\mathbb{F}}_{p})$
$\displaystyle=$ $\displaystyle 1;$
$\displaystyle\dim_{{\mathbb{F}}_{p}}\bigl{(}\operatorname{Tor}^{{\mathbb{Z}}}_{1}({\mathbb{Z}}/p^{k},{\mathbb{F}}_{p})\bigr{)}$
$\displaystyle=$ $\displaystyle 1;$
$\displaystyle\ln\bigl{(}|\operatorname{tors}({\mathbb{Z}}/p^{k})|\bigr{)}$
$\displaystyle=$ $\displaystyle k\cdot\ln(p).$

If $q$ is an prime different from $p$, we get
${\mathbb{Z}}/q^{k}\otimes_{{\mathbb{Z}}}{\mathbb{F}}_{p}=0$ and
$\operatorname{Tor}^{{\mathbb{Z}}}_{1}({\mathbb{Z}}/q^{k},{\mathbb{F}}_{p})=0$.
Hence we get for any finitely generated abelian group $M$

$\displaystyle\dim_{{\mathbb{F}}_{p}}(M\otimes_{{\mathbb{Z}}}{\mathbb{F}}_{p})$
$\displaystyle\leq$
$\displaystyle\dim_{{\mathbb{Z}}}(M)+\frac{\ln\bigl{(}|\operatorname{tors}(M)|\bigr{)}}{\ln(2)};$
$\displaystyle\dim_{{\mathbb{F}}_{p}}(\operatorname{Tor}^{{\mathbb{Z}}}_{1}(M,{\mathbb{F}}_{p})\bigr{)}$
$\displaystyle\leq$
$\displaystyle\frac{\ln\bigl{(}|\operatorname{tors}(M)|\bigr{)}}{\ln(2)}.$

We conclude from the Künneth Formula and the Universal Coefficient Theorem

$\displaystyle\frac{b_{n}(M[i];{\mathbb{F}}_{p})^{3}}{[G:G_{i}]^{3}}$
$\displaystyle=$
$\displaystyle\frac{b_{n}(M[i];{\mathbb{F}}_{p})^{3}}{[G^{3}:G_{i}^{3}]}$
$\displaystyle\leq$
$\displaystyle\frac{b_{3n}(M[i]^{3};{\mathbb{F}}_{p})}{[G^{3}:G_{i}^{3}]}$
$\displaystyle=$
$\displaystyle\frac{\dim_{{\mathbb{F}}_{p}}\bigl{(}H_{3n}(M[i]^{3};{\mathbb{Z}})\otimes_{{\mathbb{Z}}}{\mathbb{F}}_{p}\bigr{)}+\dim_{{\mathbb{F}}_{p}}\bigl{(}\operatorname{Tor}^{{\mathbb{Z}}}_{1}(H_{3n-1}(M[i];{\mathbb{Z}}),{\mathbb{F}}_{p})\bigr{)}}{[G^{3}:G_{i}^{3}]}$
$\displaystyle\leq$
$\displaystyle\frac{\dim_{{\mathbb{Z}}}(H_{3n}(M[i]^{3};{\mathbb{Z}}))}{[G^{3}:G_{i}^{3}]}+\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{3n}(M^{3}[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G^{3}:G_{i}^{3}]\cdot\ln(2)}$
$\displaystyle\hskip
85.35826pt+\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{3n-1}(M^{3}[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G^{3}:G_{i}^{3}]\cdot\ln(2)}.$

Hence (6.7) is true.

We have already mentioned in Subsection 4.5 that the ${\mathbb{F}}_{p}$-Singer
Conjecture is not true in general, see [13, Theorem 4]. Hence the Singer
Conjecture 4.1 or Conjecture 6.8 is not true.

In view of Remark 6.10 we suggest to keep the Singer Conjecture 4.1 and weaken
Conjecture 6.8 to the following version.

######  Conjecture 6.11 (Modified Homological torsion growth and
$L^{2}$-torsion).

Let $M$ be an aspherical closed manifold of odd dimension $\dim(M)=2m+1$.

Then $M$ is $\det$-$L^{2}$-acyclic, the limit
$\lim_{i\to\infty}\left(\sum_{n=0}^{2m+1}(-1)^{n}\cdot\frac{\ln(|\operatorname{tors}(H_{n}(M[i];{\mathbb{Z}}))|)}{[G:G_{i}]}\right)$
exists and is given by

$\lim_{i\to\infty}\left(\sum_{n=0}^{2m+1}(-1)^{n}\cdot\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{n}(M[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G:G_{i}]}\right)=\rho^{(2)}(\widetilde{M}).$

######  Remark 6.12 (Discussion of Conjecture 6.11).

It is conceivable that both the Singer Conjecture 4.1 and the modified
Conjecture 6.11 about homological torsion growth and $L^{2}$-torsion are true,
since the argument in Remark 6.10 that the Singer Conjecture 4.1 or Conjecture
6.8 is not true does not apply anymore.

The difference between Conjecture 6.8 and Conjecture 6.11 is that we drop the
claim
$\lim_{i\to\infty}\;\frac{\ln(|\operatorname{tors}(H_{n}(M[i];{\mathbb{Z}}))|)}{[G:G_{i}]}=0$
in Conjecture 6.8 and allow in Conjecture 6.11 contributions from all $n\geq
0$ and not only from $n=m$ as in Conjecture 6.8. We conclude
$\operatorname{tors}\bigl{(}H_{n}(M[i];{\mathbb{Z}})\bigr{)}\cong\operatorname{tors}\bigl{(}H_{2m-n}(M[i];{\mathbb{Z}})\bigr{)}$
for $n\geq 0$ from Poincaré duality and the universal coefficient theorem and
the vanishing of
$\operatorname{tors}\bigl{(}H_{0}(M[i];{\mathbb{Z}})\bigr{)}$. We get the
equality

$\sum_{n=0}^{2m+1}(-1)^{n}\cdot\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{n}(M[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G:G_{i}]}\\\
=(-1)^{m}\cdot\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{m}(M[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G:G_{i}]}+2\cdot\sum_{n=1}^{m-1}(-1)^{n}\cdot\frac{\ln\big{(}\bigl{|}\operatorname{tors}\bigl{(}H_{n}(M[i];{\mathbb{Z}})\bigr{)}\bigr{|}\bigr{)}}{[G:G_{i}]}.$

Hence in dimension $\dim(M)=3$ Conjecture 6.8 and Conjecture 6.11 agree. This
does not lead to any contradiction since no $3$-dimensional counterexample to
the ${\mathbb{F}}_{p}$-Singer Conjecture known.

Note that Conjecture 6.11 does not imply the Hopf Conjecture for
$L^{2}$-torsion 4.3 for aspherical closed topological manifolds. One may
wonder whether the Hopf Conjecture for $L^{2}$-torsion 4.3 is not true in odd
dimensions $\dim(M)\geq 5$ in general. It is known to be true in dimension
$\dim(M)=3$ by [95, Theorem 0.7], yet again since Thurston’s Geometrization
Conjecture is known to be true. The modified Conjecture 6.11 resolves the
problem mentioned above, but may only be true in special cases.

One may replace in Conjecture 6.11 the aspherical closed manifold $M$ of odd
dimension $\dim(M)=2m+1$ by a connected finite $CW$-complex $X$, for which
$\widetilde{X}$ is $\det$-$L^{2}$-acyclic. This corresponds to [88, Conjecture
8.9 on page 290]. This version is true if $\pi_{1}(X)\cong{\mathbb{Z}}$ holds,
see [16, Theorem 7.3], but at the time of writing nothing is known in general
if $\pi_{1}(X)$ is infinite and not equal to ${\mathbb{Z}}$.

For applications of these approximation conjectures for $L^{2}$-torsion to
questions about profinite rigidity for fundamental groups of $3$-manifolds and
lattices in higher rank Lie groups we refer to [58, Section 6.7] and [59].
Estimates for the homological growth in terms of the volume for aspherical
manifolds are established in [111].

###  6.8. Further examples

Other invariants, such as the rank gradient and the cost, truncated Euler
characteristics, and minimal numbers of generators, are discussed in [88,
Sections 4 and 5]. For consideration concerning the speed of convergence we
refer to [88, Section 6].

##  7\. Approximation for normal chains

A _normal chain_ $\\{G_{i}\\}$ for the group $G$ is a descending chain of
subgroups

(7.1) $\displaystyle G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq\cdots$

such that $G_{i}$ is normal in $G$ and $\bigcap_{i\geq 0}G_{i}=\\{1\\}$. Note
that a normal chain is a finite index normal chain, if and only if $[G:G_{i}]$
is finite for each $i$. Next we want to discuss approximation results for
normal chains.

###  7.1. The Approximation Conjecture for $L^{2}$-Betti numbers

Next we deal with the Approximation Conjecture for $L^{2}$-Betti numbers (see
[114, Conjecture 1.10], [85, Conjecture 13.1 on page 453]).

######  Conjecture 7.1 (Approximation Conjecture for $L^{2}$-Betti numbers).

A group $G$ satisfies the _Approximation Conjecture for $L^{2}$-Betti numbers_
if for any normal chain $\\{G_{i}\\}$ one of the following equivalent
conditions holds

  1. (1)

Matrix version
Let $A\in M_{r,s}({\mathbb{Q}}G)$ be a matrix. Then

$\displaystyle\dim_{{\mathcal{N}}(G)}\bigl{(}\ker\bigl{(}r_{A}^{(2)}\colon
L^{2}(G)^{r}\to L^{2}(G)^{s}\bigr{)}\bigr{)}$ $\displaystyle\hskip
39.83385pt=$
$\displaystyle\lim_{i\to\infty}\;\dim_{{\mathcal{N}}(G/G_{i})}\big{(}\ker\big{(}r_{A[i]}^{(2)}\colon
L^{2}(G/G_{i})^{r}\to L^{2}(G/G_{i})^{s}\bigr{)}\bigr{)};$

  2. (2)

$CW$-complex version
Let $X$ be a $G$-$CW$-complex of finite type. Then $X[i]:=G_{i}\backslash X$
is a $G/G_{i}$-$CW$-complex of finite type and we get for $n\geq 0$

$\displaystyle b_{n}^{(2)}(X;{\mathcal{N}}(G))$ $\displaystyle=$
$\displaystyle\lim_{i\to\infty}\;b_{n}^{(2)}(X[i];{\mathcal{N}}(G/G_{i})).$

The two conditions appearing in Conjecture 7.1 are equivalent by [85, Lemma
13.4 on page 455].

######  Remark 7.2 (The Determinant Conjecture and the Approximation
Conjecture).

Let $G$ be a group and $\\{G_{i}\\}$ be a normal chain. Suppose that $G$ and
each $G/G_{i}$ satisfies the Determinant Conjecture 5.1. Then the
Approximation Conjecture 7.1 for $L^{2}$-Betti numbers holds for $G$ for this
normal chain $\\{G_{i}\\}$ by [85, Theorem 13.3 (1) on page 454]. We mention
that in [85, Theorem 13.3 (1) on page 454] there is a misprint, $G_{i}$ has to
be replaced by $G/G_{i}$. Moreover, as pointed out to us by Bin Sun, [85,
Theorem 13.3 (2) on page 454] is not correct as stated. It is true that $G$ is
of $\det\geq 1$-class if the group $G$ belongs to $\mathcal{G}$. But it is not
correct that $G$ satisfies the Approximation Conjecture for any normal chain
$\\{G_{i}\\}$ if $G$ belongs to $\mathcal{G}$, one additionally needs that
each $G/G_{i}$ belongs to $\mathcal{G}$.

Recall that the Determinant Conjecture 5.1 is known for a large class of
groups, for instance it is true for all sofic groups. Suppose that each
quotient $G/G_{i}$ is finite. Then we recover Theorem 6.2 from Remark 7.2.

A typical application of the Approximation Conjecture 7.1 is the following. If
$G$ satisfies the Approximation Conjecture 7.1 and for a given normal chain
$\\{G_{i}\\}$ each $G/G_{i}$ is torsionfree and satisfies the Atiyah
Conjecture 3.1, then $G$ is torsionfree and satisfies the Atiyah Conjecture
3.1, since a limit of a convergent sequence of integers is an integer again.

For more information about the Approximation Conjecture and its applications
we refer to [85, Chapter 13] and [114].

###  7.2. Approximation of Fuglede-Kadison determinants

The following conjecture is taken from [88, Conjecture 14.1 on page 308].

######  Conjecture 7.3 (Approximation Conjecture for Fuglede-Kadison
determinants).

A group $G$ satisfies the _Approximation Conjecture for Fuglede-Kadison
determinants_ if for any normal chain $\\{G_{i}\\}$ and any matrix $A\in
M_{r,s}({\mathbb{Q}}G)$ we get for the Fuglede-Kadison determinant

$\displaystyle{\det}_{{\mathcal{N}}(G)}\bigl{(}r_{A}^{(2)}\colon
L^{2}(G)^{r}\to L^{2}(G)^{s}\bigr{)}$ $\displaystyle\hskip 39.83385pt=$
$\displaystyle\lim_{i\in
I}\;{\det}_{{\mathcal{N}}(G/G_{i})}\big{(}r_{A[i]}^{(2)}\colon
L^{2}(G/G_{i})^{r}\to L^{2}(G/G_{i})^{s}\bigr{)}.$

An equivalent chain complex version can be found in [88, Conjecture 15.3 on
page 309].

###  7.3. Approximation of $L^{2}$-torsion

Let $\overline{M}$ be a Riemannian manifold without boundary that comes with a
proper free cocompact isometric $G$-action. Denote by $M[i]$ the Riemannian
manifold obtained from $\overline{M}$ by dividing out the $G_{i}$-action. The
Riemannian metric on $M[i]$ is induced by the one on $M$. There is an obvious
proper free cocompact isometric $G/G_{i}$-action on $M[i]$ induced by the
given $G$-action on $\overline{M}$. Notice that $M=\overline{M}/G$ is a closed
Riemannian manifold and we get a $G$-covering $\overline{M}\to M$ and a
$G/G_{i}$-covering $M[i]\to M$ which are compatible with the Riemannian
metrics. Denote by

(7.2)
$\displaystyle\rho^{(2)}_{\operatorname{an}}(\overline{M};{\mathcal{N}}(G))$
$\displaystyle\in$ $\displaystyle{\mathbb{R}};$ (7.3)
$\displaystyle\rho^{(2)}_{\operatorname{an}}(M[i];{\mathcal{N}}(G/G_{i}))$
$\displaystyle\in$ $\displaystyle{\mathbb{R}},$

their _analytic $L^{2}$-torsion_ over ${\mathcal{N}}(G)$ and
${\mathcal{N}}(G/G_{i})$ respectively, see [85, Definition 3.128 on page 178].
Note that we do _not_ require that $\overline{M}$ or $M[i]$ is
$L^{2}$-acyclic. If $[G:G_{i}]$ is finite,
$\rho^{(2)}_{\operatorname{an}}(M[i];{\mathcal{N}}(G/G_{i}))$ is the Ray-
Singer torsion of the closed Riemannian manifold multiplied with
$\frac{1}{[G:G_{i}]}$ defined in [107].

######  Conjecture 7.4 (Approximation Conjecture for analytic
$L^{2}$-torsion).

A group $G$ satisfies the _Approximation Conjecture for analytic
$L^{2}$-torsion_ if for any normal chain $\\{G_{i}\\}$ and Riemannian manifold
$\overline{M}$ without boundary and with a proper free cocompact isometric
$G$-action

$\rho^{(2)}_{\operatorname{an}}(\overline{M};{\mathcal{N}}(G))=\lim_{i\in
I}\;\rho^{(2)}_{\operatorname{an}}(M[i];{\mathcal{N}}(G/G_{i})).$

There are topological counterparts which we will denote by
$\rho^{(2)}_{\operatorname{top}}(X[i];{\mathcal{N}}(G/G_{i}))$ and
$\rho^{(2)}_{\operatorname{top}}(\overline{X};{\mathcal{N}}(G))$ see [85,
Definition 3.120 on page 176]. They agree with their analytic versions by
[20]. So Conjecture 7.4 is equivalent to its topological counterpart.

######  Conjecture 7.5 (Approximation Conjecture for topological torsion).

A group $G$ satisfies the _Approximation Conjecture for topological
$L^{2}$-torsion_ if for any normal chain $\\{G_{i}\\}$ and Riemannian manifold
$\overline{M}$ without boundary and with a proper free cocompact isometric
$G$-action

$\rho^{(2)}_{\operatorname{top}}(\overline{M};{\mathcal{N}}(G))=\lim_{i\to\infty}\rho_{\operatorname{top}}(M[i];{\mathcal{N}}(G/G_{I}))$

The next result is proved in [88, Theorem 15.6 on page 310].

######  Theorem 7.6.

Suppose that $G$ satisfies the Approximation Conjecture 7.3 for Fuglede-
Kadison determinants.

Let $\overline{M}$ be a Riemannian manifold without boundary that comes with a
proper free cocompact isometric $G$-action. Suppose that
$b_{n}^{(2)}(\overline{M};{\mathcal{N}}(G))=0$ holds for all $n\geq 0$.

Then

$\rho^{(2)}_{\operatorname{an}}(\overline{M};{\mathcal{N}}(G))=\lim_{i\in
I}\;\rho^{(2)}_{\operatorname{an}}(M[i];{\mathcal{N}}(G/G_{i})).$

######  Remark 7.7 (Relating the Approximation Conjectures for Fuglede-Kadison
determinant and torsion invariants).

It is conceivable that Theorem 7.6 is still true, if we drop the assumption
that $b_{n}^{(2)}(\overline{M};{\mathcal{N}}(G))$ vanishes for all $n\geq 0$,
but our present proof works only under this assumption, see [88, Remark 16.2
on page 315]. If we can drop this assumption in Theorem 7.6, then Theorem 7.6
just boils down to the statement that $G$ satisfies the equivalent Conjectures
7.4 and 7.5 about $L^{2}$-torsion, provided that $G$ satisfies the
Approximation Conjecture 7.3 for Fuglede-Kadison determinants.

######  Remark 7.8 (Strategy of proof of Conjecture 7.3).

A strategy to prove the Approximation Conjecture 7.3 for Fuglede-Kadison
determinants is described in [88, Theorem 17.1 on page 316]. The key problem
is isolated in the _uniform integrability condition_ appearing in [88, Theorem
17.1 (v)]. Why it does not follow from known facts about spectral density
functions is explained in [88, Lemma 17.2 on page 318], whereas in [88, Remark
17.3 on page 321] it is discussed why there is some hope that it may hold in
many situations. In [88, Theorem 17.6 on page 322] a sufficient condition for
the validity of the uniform integrability condition is described.

######  Remark 7.9 ((Very poor) status of Conjecture 7.3).

Unfortunately, there is no infinite group $G$ besides infinite virtually
cyclic groups for which the Approximation Conjecture 7.3 for Fuglede-Kadison
determinants is known to be true. It holds for finite groups $G$ for trivial
reasons. For $G={\mathbb{Z}}$ (and hence for every infinite virtually cyclic
group) Conjecture 7.3 has been proven by Schmidt [116], see also [85, Lemma
13.53 on page 478] and [89, Lemma 7.25]. Note that the proof of this very
special case already requires some input from the theory of diophantine
equations, namely the estimate [85, 13.65] taken from [121, Corollary B1 on
page 30].

We conclude from [88, Theorem 15.7] that in the situation of the Approximation
Conjecture for Fuglede-Kadison determinants 7.3 we always have the inequality

$\displaystyle\dim_{{\mathcal{N}}(G)}\bigl{(}\ker\bigl{(}r_{A}^{(2)}\colon
L^{2}(G)^{r}\to L^{2}(G)^{s}\bigr{)}\bigr{)}$ $\displaystyle\hskip
19.91692pt\geq$
$\displaystyle\limsup_{i\to\infty}\;\dim_{{\mathcal{N}}(G/G_{i})}\big{(}\ker\big{(}r_{A[i]}^{(2)}\colon
L^{2}(G/G_{i})^{r}\to L^{2}(G/G_{i})^{s}\bigr{)}\bigr{)}$

provided that each quotient $G/G_{i}$ satisfies the Determinant Conjecture
5.1. If $G={\mathbb{Z}}^{n}$ and each $G/G_{i}$ is finite, then the inequality
(7.9) for the limit superior is known to be an equality by Lê [69, Theorem 3],
see also Raimbault [105].

######  Remark 7.10.

The relationship between Conjecture 6.11 about homological torsion growth and
$L^{2}$-torsion and the Approximation Conjecture 7.5 for finite index normal
chains is described in terms of regulators in [88, Sections 8 – 10].

###  7.4. More general setups

Some of the approximation conjectures above can also be formulated in more
generality or a slightly different context. One can replace the normal chain
$\\{G_{i}\\}$ indexed by $i\in\\{0,1,2,\ldots\\}$ by an inverse system
$\\{G_{i}\mid i\in I\\}$ of normal subgroups of $G$ directed by inclusion over
the directed set $I$ such that $\bigcap_{i\in I}G_{i}=\\{1\\}$, see [88,
Section 13]. One may consider so called _Farber sequences_ , where the
subgroups $G_{i}$ are not necessarily normal but normal in an asymptotic
sense, see [2, Section 10]. There is the notion of Benjamini-Schramm
convergence, see [1, Definition 1.1]. Or one can consider convergence in the
space or marked groups, see [56, Section 1.3].

##  8\. $L^{2}$-invariants and simplicial volume

The simplicial volume of a manifold is a topological variant of the
(Riemannian) volume which agrees with it for hyperbolic manifolds up to a
dimension constant and was introduced by Gromov [42].

######  Conjecture 8.1 (Simplicial volume and $L^{2}$-invariants).

Let $M$ be an aspherical closed orientable manifold of dimension $\geq 1$.
Suppose that its simplicial volume $\lVert M\rVert$ vanishes. Then
$\widetilde{M}$ is of determinant class and

$\displaystyle b_{p}^{(2)}(\widetilde{M})$ $\displaystyle=$ $\displaystyle
0\hskip 14.22636pt\mbox{ for }p\geq 0;$
$\displaystyle\rho^{(2)}(\widetilde{M})$ $\displaystyle=$ $\displaystyle 0.$

Gromov first asked in [46, Section 8A on page 232] whether under the
conditions in Conjecture 8.1 the Euler characteristic of $M$ vanishes, and
notes that in all available examples even the $L^{2}$-Betti numbers of $M$
vanish. The part about $L^{2}$-torsion appears in [82, Conjecture 3.2].

Conjecture 8.1 is discussed in detail in [85, Chapter 14]. No essential
breakthrough has been made since then, but there are some interesting papers
connected to this problem, such as [36, 110, 111]. So far, all evidence for
Conjecture 8.1 has been computational, but there is no structural reason known
for it. It is intriguing, since it relates rather different invariants to one
another and is one of the typical conjectures, which only make sense for
closed manifolds when they are aspherical.

For an aspherical closed orientable manifold of dimension $\geq 1$ its
simplicial volume $\lVert M\rVert$ vanishes if $\pi_{1}(M)$ is amenable, see
[42, page 40] and [52, Theorem 4.3]. However, it is not known whether $\lVert
M\rVert$ vanishes if $\pi_{1}(M)$ contains a normal infinite amenable subgroup
or at least an elementary amenable infinite normal subgroup. Recall that
$\rho^{(2)}(\widetilde{M})$ vanishes if $\pi_{1}(M)$ contains a normal
infinite elementary amenable subgroup, see (11) in Subsection 2.5, but it is
not known wether $\rho^{(2)}(\widetilde{M})$ vanishes if $\pi_{1}(M)$ contains
a normal infinite amenable subgroup.

As an approach to Conjecture 8.1, Gromov suggested in [47] to use a new
invariant, the integral foliated simplicial volume, to estimate $L^{2}$-Betti
numbers from above. This was carried out in [117, Corollary 5.28]. However, it
is still open whether vanishing of simplicial volume implies vanishing of
integral foliated simplicial volume (of oriented closed aspherical manifolds).

##  9\. $L^{2}$-invariants of groups

Recall that $L^{2}$-Betti numbers $b_{n}^{(2)}(G)$ of a group $G$ were defined
in (2.6). We call a group $G$ _admissible_ , if there exists a finite
$CW$-model $BG$ for its classifying space, we have
$b_{n}^{(2)}(EG;{\mathcal{N}}(G))=0$ for $n\geq 0$, and $G$ satisfies the
Determinant Conjecture 5.1. We define the _$L^{2}$ -torsion of an admissible
group $G$_

(9.1) $\rho^{(2)}(G)=\rho^{(2)}(EG;{\mathcal{N}}(G))\in{\mathbb{R}}$

by the $L^{2}$-torsion of $EG=\widetilde{BG}$. Since two models for $EG$ are
$G$-homotopy equivalent and $G$ satisfies the Determinant Conjecture 5.1, the
real number $\rho^{(2)}(EG;{\mathcal{N}}(G))$ is well-defined and is
independent of the choice of the model for $EG$. Hence there notion of the
$L^{2}$-torsion of an admissible group in (9.1) makes sense.

###  9.1. Vanishing results

Some vanishing criterions for the $L^{2}$-Betti numbers $b_{n}^{(2)}(G)$ of a
group can be found in [85, Theorem 7.2 on page 294 and Theorem 7.4 on page
295]. For instance, if $G$ contains a normal subgroup $H$ such that
$b_{n}^{(2)}(H)=0$ for all $n\leq d$ for some fixed natural number $d$, then
$b_{n}^{(2)}(G)=0$ for all $n\leq d$. If $G$ contains an amenable infinite
normal subgroup, then $b_{n}^{(2)}(G)=0$ holds for all $n\geq 0$. If $G$
occurs as an extension $1\to H\to G\to Q\to 1$ of infinite groups such that
$H$ is finitely generated or, more generally, satisfies
$b_{1}^{(2)}(H)<\infty$, then $b_{1}^{(2)}(G)=0$. See [109] and [112] for
generalizations of the last assertion. If $G$ is admissible and contains an
elementary amenable infinite normal subgroup, then $\rho^{(2)}(G)=0$ holds,
see (11) in Subsection 2.5. A very interesting interaction between the notions
of $L^{2}$-torsion and entropy is developed in [70] and used to show that for
any admissible group which is amenable, $\rho^{(2)}(G)=0$ holds, see [70,
Theorem 1.3]. It is unknown whether $\rho^{(2)}(G)=0$ vanishes if $G$ contains
an amenable infinite normal subgroup.

######  Remark 9.1.

A combinatorial computation of $L^{2}$-invariants is described in [85, Section
3.7] Given a matrix $A\in M_{m,n}({\mathbb{C}}G)$, one can assign to it its
_characteristic sequence_ $\\{c(A,K)_{n}\\}$ for some large enough number $K$.
The sequence $\\{c(A,K)_{n}\\}$ is monotone decreasing sequence of non-
negative real numbers that converges to the von Neumann dimension
$\dim_{{\mathcal{N}}(G)}(\ker(R_{A}))$ of the kernel of the induced
${\mathcal{N}}(G)$-homomorphism
$R_{A}\colon{\mathcal{N}}(G)^{m}\to{\mathcal{N}}(G)^{n}$. There is some
control over the speed of convergence, which is often very fast, actually
exponentially in $n$. Analogously there is a way of computing the Fuglede-
Kadison determinant of $R_{A}$ in terms of the characteristic sequence. Given
an algorithm to decide the word problem in $G$, one obtains an algorithm to
compute the characteristic sequence. Unfortunately this algorithm seems to be
exponentially running. But at least each $c(A,K)_{n}$ gives an upper bound for
$\dim_{{\mathcal{N}}(G)}(R_{A})$. This may be useful in view of the Atiyah
Conjecture 3.1 to prove the vanishing of $L^{2}$-Betti numbers
$b_{n}^{(2)}(\widetilde{X})$ for a finite $CW$-complex $X$ for which there is
an upper bound on the orders of the finite subgroups of $\pi_{1}(X)$. See also
Remark 3.7.

###  9.2. The first $L^{2}$-Betti number and applications to group theory

The vanishing of $b_{1}^{(2)}(G)$ has some interesting consequences, provided
that $G$ is finitely presented. Namely, it implies that the deficiency of $G$
is bounded from above by $1$ and that for any oriented closed manifold $M$ of
dimension $M$ the inequality $|\operatorname{sign}(M)|\leq\chi(M)$ holds for
its signature $\operatorname{sign}(M)$ and its Euler characteristic $\chi(M)$,
see [81, Theorems 5.1 and 6.1]. The following result is due to Kielak [61,
Theorem 5.3] and generalizes the work of Agol [5]

######  Theorem 9.2.

Let G be an infinite finitely generated group which is virtually RFRS, where
RFRS stands for residually finite rationally solvable.

Then G is virtually fibered, in the sense that it admits a finite index
subgroup mapping onto Z with a finitely generated kernel, if and only if
$b^{(2)}_{1}(G)=0$ holds.

There are relations between the non-vanishing of $b_{1}^{(2)}(G)$ and the
question whether the finitely presented group $G$ is _large_ , i.e, has a
subgroup of finite index which maps surjectively to a non-abelian free group,
see for instance [67, Theorem 1.4], [68, Theorem 1.6].

The following questions are taken from [88, Section 4], where also more
explanations and references to the literature, such as [3, 4, 19, 30, 38, 39,
40, 66, 92, 102, 115], are given.

######  Question 9.3 (Rank gradient, cost, and first $L^{2}$ Betti number).

Let $G$ be an infinite finitely generated residually finite group. Let
$(G_{i})_{i\geq 0}$ be a descending chain of normal subgroups of finite index
of $G$ with $\bigcap_{i\geq 0}G_{i}=\\{1\\}$.

Do we have

$b_{1}^{(2)}(G)={\rm cost}(G)-1=\operatorname{RG}(G;(G_{i})_{i\geq 0})?$

######  Question 9.4 (Rank gradient, cost, first $L^{2}$-Betti number, and
approximation).

Let $G$ be a finitely presented infinite residually finite group. Let
$(G_{i})$ be a descending chain of normal subgroups of finite index of $G$
with $\bigcap_{i\geq 0}G_{i}=\\{1\\}$. Let $F$ be any field.

Do we have

$\lim_{i\to\infty}\frac{b_{1}(G_{i};F)-1}{[G:G_{i}]}=b_{1}^{(2)}(G)-b_{0}^{(2)}(G)=\operatorname{cost}(G)-1=\operatorname{RG}(G;(G_{i})_{i\geq
0})?$

Note that a positive answer to the questions above also includes the
statement, that $\lim_{i\to\infty}\frac{b_{1}(X[i];F)}{[G:G_{i}]}$ and
$\operatorname{RG}(G;(G_{i})_{i\geq 0})$ are independent of the chain and the
characteristic of $F$. It is possible that the answer is positive also in the
case, where the characteristic of $F$ is not zero, since in Remark 6.6 the
counterexamples of Avramidi-Okun-Schreve [13, Theorem 1] do not work in degree
$n=1$.

######  Question 9.5.

Let $G$ be a finitely generated group. Do we have

$b_{1}^{(2)}(G)=\operatorname{cost}(G)-1$

and is the Fixed Price Conjecture true? (The Fixed Price Conjecture predicts
that the cost of every standard action of $G$, i.e., an essentially free
$G$-action on a standard Borel space with $G$-invariant probability measure,
is equal to the cost of $G$.)

Higher rank versions of the rank gradient are discussed in [88, Section 5].

###  9.3. The $L^{2}$-torsion and applications to group theory

As we have explained above, $L^{2}$-Betti numbers have been exploited for
group theory. We think that there is a lot of potential for the
$L^{2}$-torsion of a group $G$ to have striking applications to group theory,
and we encourage group theorists to work on these types of questions.

A typical question is the following. If $M$ is a closed hyperbolic manifold of
odd dimension, then $\rho^{(2)}(\pi_{1}(M))$ is a up to a constant depending
only on the dimension the volume of $M$, see [50, 95]. Since $\pi_{1}(M)$ is a
word-hyperbolic group, one may ask what $\rho^{(2)}(G)$ measures for a
$\det$-$L^{2}$-acyclic torsionfree word-hyperbolic group.

We want to mention the following invariant of an automorphism $f\colon BG\to
BG$ of a group $G$ for which there exists a finite model for $BG$ and which
satisfies the Determinant Conjecture 5.1. Then $G\rtimes_{f}{\mathbb{Z}}$ is
admissible, and hence we can define the $L^{2}$-torsion of $f$ by

(9.2) $\rho^{(2)}(f):=\rho^{(2)}(G\rtimes_{f}{\mathbb{Z}}).$

This invariant has the following properties, see [85, Theorem 7.27 on page
305].

######  Theorem 9.6.

Suppose that all groups appearing below have finite $CW$-models for their
classifying spaces and satisfy the Determinant Conjecture 5.1.

  1. (1)

Suppose that $G$ is the amalgamated product $G_{1}\ast_{G_{0}}G_{2}$ for
subgroups $G_{i}\subset G$ and the automorphism $f\colon G\to G$ is the
amalgamated product $f_{1}\ast_{f_{0}}f_{2}$ for automorphisms $f_{i}\colon
G_{i}\to G_{i}$. Then

$\rho^{(2)}(f)=\rho^{(2)}(f_{1})+\rho^{(2)}(f_{2})-\rho^{(2)}(f_{0});$

  2. (2)

Let $f\colon G\to H$ and $g\colon H\to G$ be isomorphisms of groups. Then

$\rho^{(2)}(f\circ g)=\rho^{(2)}(g\circ f).$

In particular $\rho^{(2)}(f)$ is invariant under conjugation with
automorphisms;

  3. (3)

Suppose that the following diagram of groups

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{i}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\scriptstyle{p}$$\textstyle{G_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{G_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{G_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

commutes, has exact rows and its vertical arrows are automorphisms. Then

$\rho^{(2)}(f_{2})=\chi(BG_{3})\cdot\rho^{(2)}(f_{1});$

  4. (4)

Let $f\colon G\to G$ be an automorphism of a group. Then for all integers
$n\geq 1$

$\rho^{(2)}(f^{n})=n\cdot\rho^{(2)}(f);$

  5. (5)

We have $\rho^{(2)}(f)=0$, if $G$ satisfies one of the following conditions:

    1. (a)

We have $b_{n}^{(2)}(G)=0$ for every $n\geq 0$;

    2. (b)

$G$ contains an amenable infinite normal subgroup.

Let $S$ be a compact connected orientable $2$-dimensional manifold, possibly
with boundary. Let $f\colon S\to S$ be an orientation preserving
homeomorphism. The mapping torus $T_{f}$ is a compact connected orientable
$3$-manifold, whose boundary is empty or a disjoint union of $2$-dimensional
tori. Then there is a maximal family of embedded incompressible tori, which
are pairwise not isotopic and not boundary parallel, such that it decomposes
$T_{f}$ into pieces that are Seifert or hyperbolic. Let $M_{1}$, $M_{2}$,
$\ldots$, $M_{r}$ be the hyperbolic pieces. They all have finite volume
$\operatorname{vol}(M_{i})$. The following result is taken from [85, Theorem
7.28 on page 307]

######  Theorem 9.7.

If $S$ is $S^{2}$, $D^{2}$, or $T^{2}$, then $\rho^{(2)}(f)=0$. Otherwise we
get

$\rho^{(2)}(\pi_{1}(f)\colon\pi_{1}(S)\to\pi_{1}(S))=\frac{-1}{6\pi}\cdot\sum_{i=1}^{r}\operatorname{vol}(M_{i}).$

The combinatorial approach for the computation of $\rho^{(2)}(f)$ in terms of
characteristic sequences of Remark 9.1 is described in detail in [85,
Subsection 7.34]. The favourite and so far unexplored case is, when $G$ is a
finitely generated free group.

######  Question 9.8.

Let $G$ be finitely generated non-abelian free group.

  1. (1)

Is $\rho^{(2)}(f)\leq 0$ for any automorphism $f$ of $G$;

  2. (2)

What is the structure of the countable set $\\{\rho^{(2)}(f)\\}$, where $f$
runs through the automorphisms of $G$?

  3. (3)

Given a real number $r<0$, is the set of conjugacy classes of fully
irreducible automorphisms $f$ of $G$ with $\rho^{(2)}(f)=r$ finite?

We will discuss measure equivalence in Section 10 and twisted $L^{2}$-torsion
in Section 12.

##  10\. Measure equivalence

Gaboriau [39] introduced _$L^{2}$ -Betti numbers of measured equivalence
relations_ and proved that two measure equivalent countable groups have
proportional $L^{2}$-Betti numbers. This notion turned out to have many
important applications in recent years, most notably through the work of Popa
[104].

The notion of _measure equivalence_ was introduced by Gromov [46, 0.5.E].

######  Definition 10.1.

Two countable groups $G$ and $H$ are called _measure equivalent with index
$c=I(G,H)>0$_ if there exists a non-trivial standard measure space
$(\Omega,\mu)$ on which $G\times H$ acts such that the restricted actions of
$G=G\times\\{1\\}$ and $H=\\{1\\}\times H$ have measurable fundamental domains
$X\subset\Omega$ and $Y\subset\Omega$, with $\mu(X)<\infty$, $\mu(Y)<\infty$,
and $c=\mu(X)/\mu(Y)$. The space $(\Omega,\mu)$ is called a _measure coupling_
between $G$ and $H$ (of index $c$).

The following conjecture is taken from [94, Conjecture 1.2].

######  Conjecture 10.2.

Let $G$ and $H$ be two admissible groups, which are measure equivalent with
index $I(G,H)>0$. Then

$\rho^{(2)}(G)=I(G,H)\cdot\rho^{(2)}(H).$

Due to Gaboriau [39], the vanishing of the $n$th $L^{2}$-Betti number
$b_{n}^{(2)}(G)$ is a invariant of the measure equivalence class of a
countable group $G$. If all $L^{2}$-Betti numbers vanish and $G$ is an
admissible group, then the vanishing of the $L^{2}$-torsion is a secondary
invariant of the measure equivalence class of a countable group $G$ provided
that Conjecture 10.2 holds.

Evidence for Conjecture 10.2 comes from [94, Conjecture 1.10] which says that
Conjecture 10.2 is true if we replace measure equivalence by the stronger
notion of uniform measure equivalence, see [94, Definition 1.3], and assume
that $G$ satisfies the Measure Theoretic Determinant Conjecture, see [94,
Conjecture 1.7].

##  11\. Zero-in-the-Spectrum-Conjecture

The next conjecture appears for the first time in Gromov’s article [43, page
120].

######  Conjecture 11.1 (Zero-in-the-Spectrum Conjecture).

Suppose that $\widetilde{M}$ is the universal covering of the aspherical
closed Riemannian manifold $M$ (with the Riemannian metric coming from $M$).
Then for some $p\geq 0$ zero is in the spectrum of the minimal closure

$(\Delta_{p})_{\min}\colon\operatorname{dom}\left((\Delta_{p})_{\min}\right)\subset
L^{2}\Omega^{p}(\widetilde{M})\to L^{2}\Omega^{p}(\widetilde{M})$

of the Laplacian acting on smooth $p$-forms on $\widetilde{M}$.

The Zero-in-the-Spectrum Conjecture 11.1 is known to be true if one of the
following conditions is satisfied:

  1. (1)

$\dim(M)\leq 3$;

  2. (2)

M is a locally symmetric space;

  3. (3)

$M$ possesses a Riemannian metric whose sectional curvature is non-positive;

  4. (4)

$M$ is an aspherical closed Kähler manifold whose fundamental group is word-
hyperbolic in the sense of [44];

  5. (5)

$\pi_{1}(M)$ satisfies the strong Novikov Conjecture.

If one drops the conditions “aspherical” in the Zero-in-the-Spectrum
Conjecture 11.1, then there are counterexamples.

For the proofs of the claim above and the relevant reference, such as [31, 43,
79], and for more information about the Zero-in-the-Spectrum Conjecture 11.1,
we refer to [85, Chapter 12].

##  12\. Twisting with finite-dimensional representations

A prominent open problem is whether one can twist $L^{2}$-invariants with (not
necessarily unitary) finite-dimensional representations. A basic and
systematical treatment of this problem can be found in [89].

For $L^{2}$-Betti numbers there is the conjecture that this just boils down to
multiplying the untwisted $L^{2}$-Betti number with the dimension of the
representation, see [18, Conjecture 2] and [89, Question 0.1]. This conjecture
is proved for sofic groups by Boschheidgen-Jaikin-Zapirian[18, Theorem 1.1]
and for locally indicable groups by Kielak-Sun [63, Theorem 4.5].

For $L^{2}$-torsion the effect of the twisting is much more interesting and
this leads to new invariants. Especially in dimension $3$ there has been made
a lot of progress, and interesting open problems occur. For instance, there
are interesting relations between the Alexander-torsion function, which is
given by twisting the $L^{2}$-torsion with a family of one-dimensional
representations associated to an element in $H^{1}(M;{\mathbb{Z}})$, and the
Thurston polytope. A prominent open problem is whether the regulare Fuglede-
Kadison determinant is continuous, see [89, Question 9.11]. We refer for more
information and the relevant references, such as [27, 33, 34, 35, 37, 60, 71,
76], to [89, Section 10] and the survey article [90].

##  13\. Group von Neumann algebras over ${\mathbb{F}}_{p}$

Some of the problems discussed above are of the shape that we understand the
case of a field of characteristic zero well using the group von Neumann
algebra ${\mathcal{N}}(G)$, but do not know what happens in prime
characteristic $p$. A prominent example are the questions about approximation
of $L^{2}$-Betti numbers in prime characteristic, see Subsection 6.5. It seems
to be conceivable that the relevant limits exists and are independent of the
chains, but the value of the limits depend on whether we work in
characteristic zero or in prime characteristic. So the original hope that one
always gets as a limit the $L^{2}$-invariants defined in terms of the von
Neumann algebra also in prime characteristic turns out not to be fullfilled.
So we face the new problem what the limit could be in the prime characteristic
case.

This raises the question whether there is an ${\mathbb{F}}_{p}$-analogue of
the group von Neumann algebra? In order to treat $L^{2}$-Betti numbers in
prime characteristic, one would hope for the existence of a
${\mathbb{F}}_{p}$-algebra ${\mathcal{N}}(G;p)$ which contains the group ring
${\mathbb{F}}_{p}G$ and comes with a dimension function for finitely generated
projective ${\mathcal{N}}(G;p)$-modules satisfying [85, Assumption 6.2 on page
238]. Then [85, Definition 6.6 and Theorem 6.7 on page 239] would apply, and
one would get a dimension function for all ${\mathcal{N}}(G;p)$-modules which
has many useful features. Finally, one would define the $n$th $L^{2}$-Betti
number in characteristic $p$ of a $G$-space $X$ to be

(13.1)
$b_{n}^{(2)}(Y;{\mathcal{N}}(G;p)):=\dim_{{\mathcal{N}}(G;p)}\bigl{(}H_{n}({\mathcal{N}}(G;p)\otimes_{{\mathbb{F}}_{p}G}C^{s}_{*}(Y;{\mathbb{F}}_{p}))\bigr{)}\in{\mathbb{R}}^{\geq
0}\amalg\\{\infty\\},$

where $C^{s}_{*}(Y;{\mathbb{F}}_{p})$ is the ${\mathbb{F}}_{p}G$-chain complex
given by the singular chain complex with coefficient in ${\mathbb{F}}_{p}$.
For a group $G$ one would define its $n$th $L^{2}$-Betti number in
characteristic $p$

(13.2) $b_{n}^{(2)}(G;p)=b_{n}^{(2)}(EG;{\mathcal{N}}(G;p)).$

The hope is that then the corresponding sequences of normalized Betti numbers
with coefficients in ${\mathbb{F}}_{p}$ converge for any normal chain to the
$L^{2}$-Betti numbers with coefficients in ${\mathbb{F}}_{p}$, see also Remark
6.7.

##  14\. $L^{2}$-invariants and condensed mathematics

Condensed mathematics, a theory recently developed by Clausen-Scholze [119,
118, 120], is a framework which aims to remedy ill behaviour of the category
of topological spaces with respect to algebraic structure, for example the
fact that the category of topological abelian groups is not an abelian
category. It has proven useful in incorporating geometric and analytic
structures that appear in arithmetic, or complex geometry and we are far from
understanding its full capabilities. In the context of $L^{2}$-invariants,
condensed mathematics poses at least two interesting questions.

  1. (1)

The theory of $L^{2}$-invariants heavily relies on techniques mixing algebra
and topology or functional analysis. Can condensed mathematics help to extend
the current formalism of $L^{2}$-invariants in a way which sheds light on open
problems in the area? For example, are there analogues of the group von
Neumann algebra of a discrete group over ${\mathbb{F}}_{p}$?

  2. (2)

Can $L^{2}$-invariants be applied to a wider class of geometric problems
coming from condensed mathematics?

In this section, we attempt to make a first step towards the second question
by defining $L^{2}$-Betti numbers of condensed sets carrying an action of a
discrete group in three different ways that extend the current definition for
nicely behaved spaces, e.g., CW-complexes:

  1. (1)

homology through solidification;

  2. (2)

condensed cohomology;

  3. (3)

condensed singular homology.

We would hope that more sophisticated setups to associate $L^{2}$-Betti
numbers to objects in analytic geometry lead to new interesting invariants.
The reader should be aware that we are freely using the language of
$\infty$-categories.

Let us begin by giving a brief summary of the main definitions from condensed
mathematics necessary for this purpose. For more details and proofs, we refer
the reader to Peter Scholze’s lecture notes [118, 119, 120]. We also ignore
set theoretic size issues, which are treated with more care in aforementioned
references.

Denote by $\mathsf{edCH}$ the category of extremally disconnected compact
Hausdorff spaces (extremally disconnected meaning that the closure of any of
its open subsets is again open) and continuous maps. It can be made into a
site with coverings given by finite families of jointly surjective maps. For
$\mathsf{C}$ a category with finite limits, the category
$\mathsf{Cond}(\mathsf{C})$ of condensed objects in $\mathsf{C}$ is defined as
the category of sheaves on $\mathsf{edCH}$ with values in $\mathsf{C}$. It is
not hard to check that this identifies with the full subcategory of
$\operatorname{Fun}(\mathsf{edCH}^{\operatorname{op}},\mathsf{C})$ of finite
product preserving functors. Explicitly, a condensed object in $\mathsf{C}$ is
a functor

$X\colon\mathsf{edCH}^{\operatorname{op}}\to\mathsf{C}$

which satisfies

  1. (1)

$X(\emptyset)$ is the terminal object in $\mathsf{C}$;

  2. (2)

For any two objects $T_{1}$ and $T_{2}$ in $\mathsf{edCH}$ the natural map

$X(T_{1}\amalg T_{2})\to X(T_{1})\times X(T_{2})$

is an isomorphism.

The category $\mathsf{Cond}(\mathsf{Set})$ of condensed sets is an enlargement
of the category of compactly generated topological spaces in the following
sense:

######  Proposition 14.1 ([119, Proposition 1.7]).

The functor $\mathsf{Top}\to\mathsf{Cond}(\mathsf{Set}),\linebreak
X\mapsto\underline{X}$ given by the restricted Yoneda embedding admits a left
adjoint and is fully faithful when restricted to compactly generated
topological spaces.

The nicely behaved enlargement of topological abelian groups is now given by
the category $\mathsf{Cond}(\mathsf{Ab})$ of condensed abelian groups. It is a
complete and cocomplete abelian category additionally satisfying
Grothendieck’s axioms (AB5) and (AB6). We now summarise some additional
structure:

  1. (1)

The forgetful functor
$\mathsf{Cond}(\mathsf{Ab})\to\mathsf{Cond}(\mathsf{Set})$ admits a left
adjoint $X\mapsto{\mathbb{Z}}[X]$.

  2. (2)

The category $\mathsf{Cond}(\mathsf{Ab})$ admits a closed symmetric monoidal
structure with $M\otimes N$ given by the sheafification of the presheaf
$T\mapsto M(T)\otimes_{{\mathbb{Z}}}N(T)$.

  3. (3)

There is an equivalence
$D(\mathsf{Cond}(\mathsf{Ab}))\simeq\mathsf{Cond}(D({\mathbb{Z}}))$ between
the derived $\infty$-category of condensed abelian groups and condensed
objects in the derived $\infty$-category of ${\mathbb{Z}}$-modules. The
category $\mathsf{Cond}(D({\mathbb{Z}}))$ is a stable presentably symmetric
monoidal $\infty$-category with symmetric monoidal structure induced from the
one on $D({\mathbb{Z}})$. It can also be identified with the left derived
tensor product on condensed abelian groups.

We now want to outline approach (1) homology through solidification to define
$L^{2}$-Betti numbers of condensed sets. First we recall some background on
solid abelian groups from [119, Lecture 5 and 6].

######  Definition 14.2.

For a profinite set $T$ define

${\mathbb{Z}}[T]^{\blacksquare}=\underline{\operatorname{Hom}}_{\mathsf{Cond}(\mathsf{Ab})}(\underline{C(T,{\mathbb{Z}})},{\mathbb{Z}})$

where $C(T,{\mathbb{Z}})$ denotes the space of continuous maps endowed with
the compact open topology. It comes with a canonical map
${\mathbb{Z}}[T]\to{\mathbb{Z}}[T]^{\blacksquare}$. Equivalently, if
$T=\lim_{i}T_{i}$ is given as the cofiltered limit of finite sets, one can
identify ${\mathbb{Z}}[T]^{\blacksquare}\simeq\lim_{i}{\mathbb{Z}}[T_{i}]$
where the limit is formed in $\mathsf{Cond}(\mathsf{Ab})$.

A condensed abelian group $M$ is then called _solid_ if for all profinite sets
$T$ the induced map

$\operatorname{Hom}_{\mathsf{Cond}(\mathsf{Ab})}({\mathbb{Z}}[T]^{\blacksquare},M)\to\operatorname{Hom}_{\mathsf{Cond}(\mathsf{Ab})}({\mathbb{Z}}[T],M)$

is an equivalence. Denote by
$\mathsf{Solid}(\mathsf{Ab})\subset\mathsf{Cond}(\mathsf{Ab})$ the full
subcategory of solid abelian groups.

Similarly, a derived condensed abelian group $C\in
D(\mathsf{Cond}(\mathsf{Ab}))\simeq\mathsf{Cond}(D({{\mathbb{Z}}}))$ is called
_solid_ if for all profinite sets $T$ the induced map

$\operatorname{Hom}_{D(\mathsf{Cond}(\mathsf{Ab}))}({\mathbb{Z}}[T]^{\blacksquare},C)\to\operatorname{Hom}_{D(\mathsf{Cond}(\mathsf{Ab}))}({\mathbb{Z}}[T],C)$

is an equivalence in $D({\mathbb{Z}})$. Denote by
$\mathsf{Solid}(D({\mathbb{Z}}))\subseteq\mathsf{Cond}(D({\mathbb{Z}}))$ the
full subcategory of solid derived ${\mathbb{Z}}$-modules.

The following theorem is the main result from [119, Lecture 5]

######  Theorem 14.3.

The full inclusion
$\mathsf{Solid}(D({\mathbb{Z}}))\subseteq\mathsf{Cond}(D({\mathbb{Z}}))$
admits a left adjoint $(-)^{L\blacksquare}$ called the solidification.
Furthermore, a condensed derived ${\mathbb{Z}}$-module is solid if and only if
all of its homology groups are solid abelian groups.

Now we come to the definition of $L^{2}$-Betti numbers of condensed sets. Let
$X$ be a condensed set with $G$-action, i.e., an element in
$\mathsf{Cond}(\mathsf{Set}^{BG})\simeq\mathsf{Cond}(\mathsf{Set})^{BG}$.
Consider the following composition

(14.1)
$\displaystyle\mathsf{Cond}(\mathsf{Set})^{BG}\xrightarrow{{\mathbb{Z}}[-]}\mathsf{Cond}(D({\mathbb{Z}}))^{BG}\xrightarrow{(-)^{L\blacksquare}}\mathsf{Cond}(D({\mathbb{Z}}))^{BG}$
$\displaystyle\simeq\ $
$\displaystyle\mathsf{Cond}(D({\mathbb{Z}})^{BG})\simeq\mathsf{Cond}(D({\mathbb{Z}}[G]))$

and denote the image of $X$ under (14.1) by ${\mathbb{Z}}[X]^{L\blacksquare}$.

######  Definition 14.4.

Let $R$ be a ${\mathbb{Z}}[G]$-algebra and $X$ a condensed set with
$G$-action. We define solid equivariant homology of $X$ with coefficients in
$R$ by

(14.2)
$H_{n}^{G,\blacksquare}(X;R)\coloneqq\Gamma(H_{n}({\mathbb{Z}}[X]^{L\blacksquare}\otimes_{{\mathbb{Z}}[G]}R)).$

Here, $\Gamma\colon\mathsf{Cond}(\mathsf{Mod}_{R})\to\mathsf{Mod}_{R}$ is the
global sections functor and $-\otimes_{{\mathbb{Z}}[G]}R\colon
D({\mathbb{Z}}[G])\linebreak\to D(R)$ denotes the (derived) base change.

Furthermore, we define the $L^{2}$-Betti numbers of $X$ by

(14.3)
$b^{(2)}_{n}(X;{\mathcal{N}}(G))\coloneqq\dim_{{\mathcal{N}}(G)}(H_{n}^{G,\blacksquare}(X;{\mathcal{N}}(G))).$

For free $G$-CW-complexes, we now compare equivariant homology and
$L^{2}$-Betti numbers from Definition 14.4 with the classical ones, keeping
Proposition 14.1 in mind. For this we need one further ingredient, giving a
description of the singular chain complex of a CW-complex in the condensed
world.

######  Proposition 14.5 ([119, Example 6.5]).

There is an equivalence of the two functors

$\displaystyle\mathsf{CW}\xrightarrow{\underline{(-)}}\mathsf{Cond}(\mathsf{Set})\xrightarrow{{\mathbb{Z}}[-]}\mathsf{Cond}(D({\mathbb{Z}}))\xrightarrow{(-)^{L\blacksquare}}\mathsf{Cond}(D({\mathbb{Z}}))\
\text{and}$
$\displaystyle\mathsf{CW}\xrightarrow{C_{\bullet}(-)}D({\mathbb{Z}})\xrightarrow{\underline{(-)}}\mathsf{Cond}(D({\mathbb{Z}})),$

where $\mathsf{CW}\subseteq\mathsf{Top}$ denotes the full subcategory of CW-
complexes, $C_{\bullet}(X)$ denotes the singular chain complex of $X$, and the
functor $\underline{(-)}\colon
D({\mathbb{Z}})\to\mathsf{Cond}(D({\mathbb{Z}}))$ sends an object $M$ to the
sheafification of the constant presheaf with value $M$.

We now explain why this generalises the $L^{2}$-Betti number of $G$-CW-
complexes defined in the body of this article.

######  Theorem 14.6.

Let $X$ be a $G$-space which is $G$-homotopy equivalent to a $G$-CW-complex
and $R$ a ${\mathbb{Z}}[G]$-algebra. Then Borel homology of $X$ with
coefficients in $R$ and homology of $X$ in the sense of Definition 14.4 are
equivalent. In particular, if $X$ is $G$-homotopy equivalent to a free $G$-CW-
complex, the $L^{2}$-Betti numbers of $X$ in the sense of (2.3) and Definition
14.3 agree.

###### Proof.

Consider the diagram

(14.4)
${\mathsf{CW}^{BG}}$${D({\mathbb{Z}})^{BG}}$${D({\mathbb{Z}}[G])}$${D(R)}$${\mathsf{Cond}(D({\mathbb{Z}}))^{BG}}$${\mathsf{Cond}(D({\mathbb{Z}}))^{BG}}$${\mathsf{Cond}(D({\mathbb{Z}}[G]))}$${\mathsf{Cond}(D(R)).}$$\scriptstyle{C_{\bullet}(-)}$$\scriptstyle{{\mathbb{Z}}[-]}$$\scriptstyle{\simeq}$$\scriptstyle{\underline{(-)}}$$\scriptstyle{-\otimes_{{\mathbb{Z}}[G]}R}$$\scriptstyle{\underline{(-)}}$$\scriptstyle{\underline{(-)}}$$\scriptstyle{(-)^{L\blacksquare}}$$\scriptstyle{\simeq}$$\scriptstyle{-\otimes_{{\mathbb{Z}}[G]}R}$

The left square commutes by Proposition 14.5 and the rest of the diagram
commutes for obvious reasons. The composite of the upper horizontal arrows
recovers Borel homology of a $G$-CW-complex. It remains to note that taking
global sections $\Gamma\colon\mathsf{Cond}(D(R))\to D(R)$ is left inverse to
the functor $\underline{(-)}\colon D(R)\to\mathsf{Cond}(D(R))$.

For a free $G$-CW-complex $X$, Borel homology of $X$ with coefficients in $R$
can be computed as homology of the degreewise tensor product of the singular
chain complex of $X$ with $R$, as used in the definition of $L^{2}$-Betti
numbers in [85, Definition 6.50 on page 263].

To extend this result to all $G$-spaces $G$-homotopy equivalent to free
$G$-CW-complexes, observe that the functor
$\mathsf{Top}\to\mathsf{Cond}(D({\mathbb{Z}})),X\mapsto{\mathbb{Z}}[\underline{X}]^{L\blacksquare}$
sends homotopic maps to homotopic maps which formally follows from
${\mathbb{Z}}[[0,1]]^{L\blacksquare}\simeq{\mathbb{Z}}$. ∎

######  Remark 14.7 (General actions).

Note that the above definition does not recover $L^{2}$-Betti numbers for non-
free $G$-CW-complexes. To remedy this, one can work parametrised over the
orbit category $\mathsf{Or}(G)$ instead of $BG$.

We now briefly explain the two alternative definitions of $L^{2}$-Betti
numbers of condensed sets with $G$-action mentioned in the beginning of
Section 14.

######  Remark 14.8 (Condensed singular homology).

One alternative is to model the construction of singular homology inside
condensed sets defined as homology of the simplicial set

$\operatorname{Hom}_{\mathsf{Cond}(\mathsf{Set})}(\underline{\Delta^{\bullet}},X).$

This in fact agrees with singular homology for all topological spaces since
the spaces $\Delta^{n}$ are compactly generated so that the realization of the
condensed set $\underline{\Delta^{n}}$ is $\Delta^{n}$.

######  Remark 14.9 (A cohomological variant).

One can also define equivariant cohomology of condensed $G$-sets analogous to
Definition 14.4. For this, one assigns to a condensed set $X$ with $G$-action
the homology of

$\operatorname{Hom}_{\mathsf{Cond}(D({\mathbb{Z}}[G]))}({\mathbb{Z}}[X]^{L\blacksquare},\underline{R})\in
D(R).$

Then the same arguments as in Theorem 14.6 show that for free $G$-CW-complexes
this recovers Borel cohomology.

Note that as for any abelian group the induced constant condensed abelian
group is solid we have

$\displaystyle\operatorname{Hom}_{\mathsf{Cond}(D({\mathbb{Z}}[G]))}({\mathbb{Z}}[X]^{L\blacksquare},\underline{R})$
$\displaystyle\simeq\operatorname{Hom}_{\mathsf{Cond}(D({\mathbb{Z}}))}({\mathbb{Z}}[X]^{L\blacksquare},\underline{R})^{hG}$
$\displaystyle\simeq\operatorname{Hom}_{\mathsf{Cond}(D({\mathbb{Z}}))}({\mathbb{Z}}[X],\underline{R})^{hG}$
$\displaystyle\simeq\operatorname{Hom}_{\mathsf{Cond}(D({\mathbb{Z}}[G]))}({\mathbb{Z}}[X],\underline{R}).$

Because of that, this version of equivariant condensed cohomology looks most
natural to us. From the viewpoint of $L^{2}$-invariants, working with homology
instead of cohomology is often easier (for instance homology behaves better
with respect to colimits) and chose to present the dual version in more
detail.

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