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# The degree-three bounded cohomology of complex
Lie groups of classical type

Carlos De la Cruz Mengual  Faculty of Electrical and Computer Engineering
Technion, Haifa, Israel<EMAIL_ADDRESS>
###### Abstract.

We establish Monod’s isomorphism conjecture in degree-three bounded cohomology
for every complex simple Lie group of classical type. Our main ingredient is a
bounded-cohomological stability theorem with an optimal range in degree three
that we bootstrap from previous stability results by the author and Hartnick.
The bootstrapping procedure relies on the occurrence in our setting of a
variant of the recently observed phenomenon of secondary stability in the
sense of Galatius–Kupers–Randal-Williams.

##  1\. Introduction

###  1.1. Statement of main result

This article is the third one in a series concerned with the explicit
computation of the continuous bounded cohomology of Lie groups via
stabilization techniques (see prequels [11, 12]). The theory of continuous
bounded cohomology was developed in the early 2000s by Burger and Monod [6,
28] as a topological refinement of the celebrated notion for discrete groups
(conceived by Johnson, Trauber, and Gromov [19] independently), with powerful
applications in geometry, dynamics and rigidity theory [8, 14, 28].

Despite its power, the computation of continuous bounded cohomology remains a
notorious challenge. In the setting of Lie groups, the problem can be reduced
to the connected semisimple case, whose behavior is predicted by the prominent

######  Isomorphism Conjecture.

For any connected semisimple Lie group $G$ with finite center, the comparison
map $c^{\ast}:{\rm H}_{\rm b}^{\ast}(G)\to\operatorname{H}^{\ast}(G)$ is an
isomorphism in every degree.

Here we denote by $\operatorname{H}^{\ast}(G)$ the continuous group cohomology
of $G$ with coefficients on the trivial $G$-module $\mathbb{R}$, and by ${\rm
H}_{\rm b}^{\ast}(G)$ the corresponding continuous bounded cohomology. The
comparison map $c^{\ast}$ is defined by forgetting the boundedness of a class.
The isomorphism conjecture was formulated in this form by Monod in his ICM
address [27] in 2006. The surjectivity question, posed earlier by Dupont [9],
has been answered affirmatively in a variety of contexts, e.g. [21, 25]. We
emphasize that the continuous cohomology of Lie groups is well understood in
virtue of the van Est isomorphism (see e.g. [2, Cor. XI.5.6] or [33]) and
classical results on the cohomology of compact symmetric spaces (see e.g. [18,
26]).

Beyond the trivial degrees zero and one, the isomorphism conjecture has been
established fully in degree two by Burger and Monod [5]. However, in degrees
$\geq 3$, the conjecture has seen scarce progress in the last two decades. In
degree three, it has been verified for the groups
$\operatorname{SL}_{n}(\mathbb{R})$ [7, 30] and
$\operatorname{SL}_{n}(\mathbb{C})$ [1, 30, 17, 3]. In higher degrees, our
lack of understanding of the conjecture is remarkable, only known to hold in
degree four for $\operatorname{SL}_{2}(\mathbb{R})$ [20].

The goal of this paper is to expose further evidence in favor of the
isomorphism conjecture.

######  Theorem 1.

The isomorphism conjecture holds in degree three for every connected simple
complex Lie group $G$ of classical type. In particular, we have ${\rm H}_{\rm
b}^{3}(G)\cong\operatorname{H}^{3}(G)\cong\mathbb{R}$.

This statement encompasses the four classical complex families
$\mathbf{A}_{r},\,\mathbf{B}_{r},\,\mathbf{C}_{r},\,\mathbf{D}_{r}$, which
consist of the connected Lie groups locally isomorphic to

(1.1)
$\operatorname{SL}_{r+1}(\mathbb{C}),\quad\operatorname{SO}_{2r+1}(\mathbb{C}),\quad\operatorname{Sp}_{2r}(\mathbb{C}),\quad\operatorname{SO}_{2r}(\mathbb{C}).$

respectively. In each case, $r$ is the rank of the group, and all are simple
Lie groups whenever $r\geq 1$, except for $\operatorname{SO}_{2}(\mathbb{C})$
and $\operatorname{SO}_{4}(\mathbb{C})$ (abelian and with two simple factors,
respectively).

Non-vanishing results for the continuous bounded cohomology of Lie groups find
important applications in the study of representations of fundamental groups
of manifolds, and in particular, of their moduli spaces and deformations. The
case of surface group representations with Hermitian targets, studied
extensively in existing literature, serves as the most prominent example of
this claim (see, e.g., [4]). The relevant cohomological input in this setting
is the non-vanishing of ${\rm H}_{\rm b}^{2}$ for non-compact Lie groups of
Hermitian type.

A bounded-cohomological approach has been also applied successfully in the
study of 3-manifold group representations into
$\operatorname{PSL}_{n}(\mathbb{C})$—a group with non-vanishing ${\rm H}_{\rm
b}^{3}$—and has been instrumental in proofs of interesting rigidity phenomena
(see, e.g., [3, 13]). It is our hope that Theorem 1 be the foundation of
future investigations of representation varieties of 3-manifold groups into
more general complex Lie groups.

###  1.2. The role of bounded-cohomological stability

Each of the four sequences of groups indexed by $r$ and listed in (1.1) can be
organized as increasing chains of inclusions

(1.2)
$S_{0}\overset{\iota_{0}}{\hookrightarrow}S_{1}\overset{\iota_{1}}{\hookrightarrow}S_{2}\overset{\iota_{2}}{\hookrightarrow}\cdots\hookrightarrow
S_{r}\overset{\iota_{r}}{\hookrightarrow}S_{r+1}\hookrightarrow\cdots$

where $S_{r}$ is the group of rank $r$ and the inclusions $\iota_{r}$ are
block embeddings in appropriate bases. The following theorem concerning the
degree-three continuous bounded cohomology of the groups in these families
plays a key role in the proof of Theorem 1.

######  Theorem 2.

The embeddings $\iota_{r}$ in each of the classical families induce the
isomorphisms

$\begin{array}[]{ccccccccccc}(\mathbf{A}_{r})&&{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{2}(\mathbb{C}))&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{3}(\mathbb{C}))&\xleftarrow{\
\cong}&\cdots&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{r+1}(\mathbb{C}))&\xleftarrow{\ \cong}&\cdots\\\
(\mathbf{B}_{r})&&{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{3}(\mathbb{C}))&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{5}(\mathbb{C}))&\xleftarrow{\
\cong}&\cdots&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{2r+1}(\mathbb{C}))&\xleftarrow{\ \cong}&\cdots\\\
(\mathbf{C}_{r})&&{\rm H}_{\rm
b}^{3}(\operatorname{Sp}_{2}(\mathbb{C}))&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{Sp}_{4}(\mathbb{C}))&\xleftarrow{\
\cong}&\cdots&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{Sp}_{2r}(\mathbb{C}))&\xleftarrow{\ \cong}&\cdots\\\
(\mathbf{D}_{r})&&{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{6}(\mathbb{C}))&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{8}(\mathbb{C}))&\xleftarrow{\
\cong}&\cdots&\xleftarrow{\ \cong}&{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{2r}(\mathbb{C}))&\xleftarrow{\
\cong}&\cdots\\\\[5.0pt] \end{array}$

The property that the embeddings $\iota_{r}$ in a chain of inclusions as (1.2)
induce isomorphisms

(1.3) ${\rm H}_{\rm b}^{q}(S_{r_{0}})\xleftarrow{\ \cong}\ {\rm H}_{\rm
b}^{q}(S_{r_{0}+1})\xleftarrow{\ \cong}\ {\rm H}_{\rm
b}^{q}(S_{r_{0}+2})\xleftarrow{\ \cong}\ \cdots$

for every degree $q\in\mathbb{N}$, starting from an index $r_{0}=r_{0}(q)$, is
known as _bounded-cohomological stability_ (abbrv. _bc-stability_). An
assignment $q\mapsto r_{0}(q)$ as above is called _bc-stability range_.

Theorem 2 settles the _optimal_ (i.e. lowest possible) bc-stability range
$r_{0}(3)$ in degree $q=3$ for the four complex classical families. In the
case of the $\mathbf{A}_{r}$ family, Theorem 2 was proved by Bucher, Burger,
and Iozzi [3, Thm. 2], relying on work by Monod [30]. Establishing the
statement for the remaining three families is the crux of this article.

The isomorphism conjecture, if true, would imply bc-stability for all
classical families of Lie groups—for which continuous cohomology is known to
stabilize. Our derivation of Theorem 1 from Theorem 2 is a sort of converse of
this implication, based on the following principle. If a classical family
$(S_{r})_{r\in\mathbb{N}}$ is bc-stable with range $r_{0}(q)$, then proving
the degree-$q$ isomorphism conjecture for all the groups $(S_{r}\mid r\geq
r_{0}(q))$ within the range of bc-stability is reduced to verifying it only
for $S_{r_{0}(q)}$. The specific feature that favors the application of this
principle in our setting is that the range $r_{0}(3)$ established by Theorem 2
is low enough to reach groups whose ${\rm H}_{\rm b}^{3}$ is well understood.
Indeed, all the groups at the base of the respective isomorphism chains for
the families $\mathbf{B}_{r},\mathbf{C}_{r},\mathbf{D}_{r}$ are of type
$\mathbf{A}_{r}$ in disguise.

###  1.3. A secondary stability phenomenon

To explain the structure of our proof of Theorem 2 for the families
$\mathbf{B}_{r},\mathbf{C}_{r},\mathbf{D}_{r}$, we revisit the argument in
Bucher–Burger–Iozzi [3] for the $\mathbf{A}_{r}$ case.

In [30], Monod proved the bc-stability of the family $\mathbf{A}_{r}$, with an
addition that ensures a few extra injections—instead of isomorphisms—beyond
the regime of bc-stability. In degree three:

${\rm H}_{\rm b}^{3}(\operatorname{SL}_{2}(\mathbb{C}))\hookleftarrow{\rm
H}_{\rm b}^{3}(\operatorname{SL}_{3}(\mathbb{C}))\hookleftarrow{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\cong{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{5}(\mathbb{C}))\cong\cdots\cong{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{r+1}(\mathbb{C}))\cong\cdots\vspace{-1pt}$

Since ${\rm H}_{\rm b}^{3}(\operatorname{SL}_{2}(\mathbb{C}))\cong\mathbb{R}$
(see [1, 7]), one deduces that ${\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))$ must have _at most_ dimension one
for $n\geq 3$. That the dimension is _at least_ one is the content of [3],
which exhibits a non-zero class in ${\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))$ based on a construction by
Goncharov [17].

In analogy to (1.3), we call _weak_ bc-stability the property that the
embeddings $\iota_{r}$ in (1.2) induce injections starting from an index
$r_{1}=r_{1}(q)$ for every degree $q\in\mathbb{N}$. The (weak) bc-stability of
the complex $\mathbf{B}_{r},\mathbf{C}_{r},\mathbf{D}_{r}$—among other
classical families—was first proved by Hartnick and the author in [12, Thm. A,
Cor. B]. In recent work, Kastenholz and Sroka [22] improved the range of weak
bc-stability given in [12] for those families to $r_{1}(q)=2q+2$.

As opposed to Monod’s weak bc-stability result [30] for $\mathbf{A}_{r}$, the
ranges established in the references [12, 22] for the complex families
$\mathbf{B}_{r},\mathbf{C}_{r},\mathbf{D}_{r}$ do not yield a reduction to a
group whose ${\rm H}_{\rm b}^{3}$ is understood. Thus, our first step towards
Theorem 2 is the following range improvement in degree three.

######  Key Lemma.

The block embeddings $\iota_{r}$ induce injections

$\begin{array}[]{ccccccccccc}&&{\rm H}_{\rm
b}^{3}(\operatorname{O}_{3}(\mathbb{C}))&\hookleftarrow&{\rm H}_{\rm
b}^{3}(\operatorname{O}_{5}(\mathbb{C}))&\hookleftarrow&\cdots&\hookleftarrow&{\rm
H}_{\rm
b}^{3}(\operatorname{O}_{2r+1}(\mathbb{C}))&\hookleftarrow&\cdots\\\\[2.0pt]
&&{\rm H}_{\rm b}^{3}(\operatorname{Sp}_{2}(\mathbb{C}))&\hookleftarrow&{\rm
H}_{\rm
b}^{3}(\operatorname{Sp}_{4}(\mathbb{C}))&\hookleftarrow&\cdots&\hookleftarrow&{\rm
H}_{\rm
b}^{3}(\operatorname{Sp}_{2r}(\mathbb{C}))&\hookleftarrow&\cdots\\\\[2.0pt]
{\rm H}_{\rm b}^{3}(\operatorname{O}_{4}(\mathbb{C}))&\hookleftarrow&{\rm
H}_{\rm b}^{3}(\operatorname{O}_{6}(\mathbb{C}))&\hookleftarrow&{\rm H}_{\rm
b}^{3}(\operatorname{O}_{8}(\mathbb{C}))&\hookleftarrow&\cdots&\hookleftarrow&{\rm
H}_{\rm
b}^{3}(\operatorname{O}_{2r}(\mathbb{C}))&\hookleftarrow&\cdots\end{array}$

######  Remark .

The omission of the determinant-one condition in the orthogonal groups in the
Key Lemma is on purpose. Upgrading it to determinant-one groups is a technical
point that we will treat later. We ask from the reader to ignore that at this
stage, and assume for the sake of the introduction that the Key Lemma has been
proven for the special orthogonal groups.

A remarkable aspect of the Key Lemma is the next fact that lies at the heart
of its proof, given in Section 4. We will prove that if _any_ of the block
embeddings $\iota_{r}$ for the families
$\mathbf{B}_{r},\mathbf{C}_{r},\mathbf{D}_{r}$ induces an injection at the
level of ${\rm H}_{\rm b}^{3}$, then necessarily _all_ the $\iota_{r}$’s up to
the ranges produced by the Key Lemma must induce injections. In this sense,
any arbitrary range (e.g., [12, 22]) is as good as any other, for either of
them will allow us to bootstrap the optimal range _a posteriori_.

The proof of this intriguing fact relies on a principle reminiscent of
_secondary stability_ in the sense of Galatius–Kupers–Randal-Williams [15]. A
subject of ongoing research in the realm of ordinary homology, secondary
stability is the event in which “the failure of stability is itself stable”.
In our case, failure of weak bc-stability is measured by the kernels of the
maps induced by the embeddings $\iota_{r}$. We prove the stability of those
kernels based on a parametrization via cross-ratios of configuration spaces of
isotropic projective points, exposed in Section 3.

###  1.4. Non-triviality of ${\rm H}_{\rm b}^{3}$ and Gromov norms

In analogy with the argument for $\mathbf{A}_{r}$ given above, it remains to
show the non-triviality of ${\rm H}_{\rm b}^{3}$ for the relevant groups. We
will prove:

######  Theorem 3.

For any $r\in\mathbb{N}$, let $S_{r}$ be either
$\operatorname{SO}_{2r+1}(\mathbb{C}),\,\operatorname{Sp}_{2r}(\mathbb{C})$,
or $\operatorname{SO}_{2r}(\mathbb{C})$, and let $n=n(r)$ be the dimension of
the standard $S_{r}$-representation, so that
$S_{r}<\operatorname{SL}_{n}(\mathbb{C})$. Then, the restriction

(1.4) $\operatorname{res}_{r}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))\to{\rm H}_{\rm b}^{3}(S_{r})$

is an isomorphism, except if $S_{r}=\operatorname{SO}_{2r}(\mathbb{C})$ and
$r\in\\{1,2\\}$.

Since ${\rm H}_{\rm b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))$ is
1-dimensional for every $n\geq 2$, the injectivity of the map (1.4) implies
the desired non-triviality statement. For groups of types $\mathbf{B}_{r}$ and
$\mathbf{C}_{r}$, this follows from the next proposition, which is of
independent interest, for it also elucidates the norm structure of ${\rm
H}_{\rm b}^{3}$.

######  Proposition .

For $r\in\mathbb{N}$, if $S_{r}$ in Theorem 3 is
$\operatorname{SO}_{2r+1}(\mathbb{C})$ or
$\operatorname{Sp}_{2r}(\mathbb{C})$, then $\operatorname{res}_{r}$ is
isometric.

The proofs of subsection 1.4 and Theorems 2 and 3 for the families
$\mathbf{B}_{r}$ and $\mathbf{C}_{r}$, are given in Section 5. On the other
hand, subsection 1.4 does not hold in the $\mathbf{D}_{r}$ case (see
subsection 6.4). Thus, we give a separate argument for Theorems 2 and 3 for
that family in Section 6.

###  1.5. Outlook

In Section 8, we discuss the norm structure of ${\rm H}_{\rm b}^{3}$ for the
classical complex Lie groups, and provide a conjecture in the $\mathbf{D}_{r}$
case, which remains open.

With little extra bookkeeping of the various types of Levi factors of
parabolic subgroups, it should be possible to apply our ideas to prove the
degree-three isomorphism conjecture for the real split Lie groups
$\operatorname{SO}_{+}(r+1,r)$, $\operatorname{Sp}_{2r}(\mathbb{R})$, and
$\operatorname{SO}_{+}(r,r)$. In general, we expect the situation in degree
three for classical groups to be fully determined by what occurs in rank one.

Finally, an exciting and potentially fruitful research avenue is the further
usage of secondary stability techniques in bounded cohomology. This could
serve for optimizing existing stability ranges, and, eventually, lead to new
computations of bounded cohomology in higher degrees.

### Acknowledgments

I extend my gratitude to Marc Burger for suggesting the problem that led to
this work. I also thank Uri Bader for many helpful discussions. I am indebted
to Tobias Hartnick for the collaboration that led to the two prequels [11,
12], essential to this article, and grateful for his valuable feedback
throughout the development of this project. I thank Manuel Krannich for
pointing me to the notion of secondary stability and for his comments on a
previous version of this article. Finally, I thank Nicolas Monod for
suggesting the transit over discrete bounded cohomology in the proof of
section 6.

This work contains results of my doctoral thesis [10], and was completed
during a postdoctoral fellowship at the Weizmann Institute of Science, Israel.
I acknowledge support of the Swiss National Science Foundation, through the
Early Postdoc.Mobility Grant No. 188010.

##  2\. Notation and standing assumptions

###  2.1. Notation on formed spaces

In the rest of this article, we will adhere to the notation from [12, §2],
which we summarize below. The notation will be slightly simplified, as we only
consider symmetric or antisymmetric formed spaces over $\mathbb{C}$.

Let $\varepsilon\in\\{\pm 1\\}$. An $\varepsilon$_-symmetric formed space_ is
a pair $(V,\omega)$ of a finite-dimensional $\mathbb{C}$-vector space $V$ and
an $\varepsilon$-symmetric bilinear form $\omega:V\times V\to\mathbb{C}$. We
write $q(v)\coloneqq\omega(v,v)$ for the quadratic form. The rank of
$(V,\omega)$ is the maximal dimension of a totally isotropic subspace of $V$.
Any non-degenerate $\varepsilon$-symmetric space $(V,\omega)$ of dimension $n$
splits as the direct sum of formed spaces

(2.1) $V\cong\mathcal{H}_{\varepsilon}^{\oplus r}\oplus\mathcal{E}^{\oplus
d},\vspace{-2pt}$

where $r$ is the rank of $(V,\omega)$, $\mathcal{H}_{\varepsilon}$ is the
$\varepsilon$-hyperbolic plane, $\mathcal{E}$ is the Euclidean line, and
$d=n-2r\in\\{0,1\\}$. Furthermore, if $\varepsilon=-1$, then $d=0$. We denote
by $(V_{r}^{\varepsilon,d},\omega_{r}^{\varepsilon,d})$ the formed space on
the right-hand side of (2.1). An ordered basis
$\mathscr{B}_{r}^{\varepsilon,d}=\\{e_{r},\ldots,e_{1},dh,f_{1},\ldots,f_{r}\\}$
of $V_{r}^{\varepsilon,d}$ is an adapted basis of
$(V_{r}^{\varepsilon,d},\omega_{r}^{\varepsilon,d})$ if $\langle
h\rangle\cong\mathcal{E}$ and $\langle e_{i},f_{i}\rangle\cong\mathcal{H}$ for
every $i$. The matrix

$J_{r}^{\varepsilon,d}:=\begin{pmatrix}0&0&\varepsilon Q_{r}\\\\[-2.0pt]
0&1_{d}&0\\\\[-2.0pt] Q_{r}&0&0\end{pmatrix}\in{\rm M}_{n}(\mathbb{C}),$

represents the form $\omega_{r}^{\varepsilon,d}$ in the basis
$\mathscr{B}_{r}^{\varepsilon,d}$, where $Q_{r}\in{\rm M}_{r}(\mathbb{C})$ is
the matrix with 1’s on its antidiagonal and zero elsewhere (the middle row and
column are omitted if $d=0$).

Let $G_{r}^{\varepsilon,d}$ be the automorphism group of
$(V_{r}^{\varepsilon,d},\omega_{r}^{\varepsilon,d})$ and
$S_{r}^{\varepsilon,d}<G_{r}^{\varepsilon,d}$ its determinant-one subgroup.
Both $G_{r}^{\varepsilon,d}$ and $S_{r}^{\varepsilon,d}$ are complex Lie
subgroups of $\operatorname{GL}(V_{r}^{\varepsilon,d})$. Concretely,

$G_{r}^{\varepsilon,d}=\begin{cases}\operatorname{O}_{2r+1}(\mathbb{C}),&(\varepsilon,d)=(+1,1)\\\\[-2.0pt]
\operatorname{Sp}_{2r}(\mathbb{C}),&(\varepsilon,d)=(-1,0)\\\\[-2.0pt]
\operatorname{O}_{2r}(\mathbb{C}),&(\varepsilon,d)=(+1,0)\end{cases}$

Observe that

(2.2) $G_{r}^{(+1,1)}\cong\\{\pm I\\}\times
S_{r}^{(+1,1)}\quad\mathrm{and}\quad
G_{r}^{(-1,0)}=S_{r}^{(-1,0)}.\vspace{-2pt}$

For fixed $d$, the inclusions
$\mathscr{B}_{r}^{\varepsilon,d}\hookrightarrow\mathscr{B}_{r+1}^{\varepsilon,d}$
of adapted bases gives rise to the embeddings

(2.3)
$G_{0}^{\varepsilon,d}\overset{\iota_{0}}{\hookrightarrow}G_{1}^{\varepsilon,d}\overset{\iota_{1}}{\hookrightarrow}G_{2}^{\varepsilon,d}\hookrightarrow\cdots\hookrightarrow
G_{r}^{\varepsilon,d}\overset{\iota_{r}}{\hookrightarrow}G_{r+1}^{\varepsilon,d}\hookrightarrow\cdots\vspace{-2pt}$

of automorphism groups and of their corresponding determinant-one subgroups
$S_{r}^{\varepsilon,d}$. We abuse notation and denote by $\iota_{r}$ also the
embedding $S_{r}^{\varepsilon,d}\hookrightarrow S_{r+1}^{\varepsilon,d}$ (see
also (1.2)).

###  2.2. Standing assumptions

Throughout the article, we will keep a fixed choice of a pair

(2.4) $(\varepsilon,d)\in\\{(+1,1),(-1,0),(+1,0)\\},\vspace{-2pt}$

which is equivalent to the choice of the classical family
$\mathbf{B}_{r},\mathbf{C}_{r},\mathbf{D}_{r}$, respectively. We omit the
indices $\varepsilon,d$ whenever possible and write $V_{r}$, $G_{r}$, $S_{r}$,
etc. instead of $V_{r}^{\varepsilon,d}$, $G_{r}^{\varepsilon,d}$,
$S_{r}^{\varepsilon,d}$, etc. Finally, we set $n(r)\coloneqq\dim(V_{r})$, and

$\quad r_{0}\coloneqq\begin{cases}3,&\mbox{ if
}(\varepsilon,d)=(+1,0),\\\\[-4.0pt] 1,&\mbox{ otherwise,}\end{cases}\quad
r_{1}\coloneqq\begin{cases}2,&\mbox{ if }(\varepsilon,d)=(+1,0),\\\\[-4.0pt]
1,&\mbox{ otherwise.}\end{cases}$

###  2.3. Re-statement of main results

In the notation fixed in this section, our main results admit the following
compact re-statements (omitting the $\mathbf{A}_{r}$ case from Theorems 1 and
2):

######  Key Lemma _bis_.

For $r\geq r_{1}$, the induced map $\iota_{r}^{\ast}:{\rm H}_{\rm
b}^{3}(G_{r+1})\to{\rm H}_{\rm b}^{3}(G_{r})$ is injective.

######  Theorem 1 _bis_.

For $r\geq r_{0}$, the comparison map $c^{3}:{\rm H}_{\rm
b}^{3}(S_{r})\to\operatorname{H}^{3}(S_{r})$ is an isomorphism.

######  Theorem 2 _bis_.

For $r\geq r_{0}$, the induced map $\iota_{r}^{\ast}:{\rm H}_{\rm
b}^{3}(S_{r+1})\to{\rm H}_{\rm b}^{3}(S_{r})$ is an isomorphism.

######  Theorem 3 _bis_.

For $r\geq r_{0}$, the restriction $\operatorname{res}_{r}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n(r)}(\mathbb{C}))\to{\rm H}_{\rm b}^{3}(S_{r})$ is
an isomorphism.

##  3\. Configuration spaces of isotropic points

###  3.1. The variety of isotropic points

We will consider the following generalized flag variety.

######  Definition .

For any $r\geq r_{1}$, we call the irreducible complex projective variety

$\mathcal{P}_{r}=\mathcal{P}_{r}^{\varepsilon,d}\coloneqq\\{[v]\in\mathbb{P}(V_{r})\mid
q(v)=0\\}\vspace{-3pt}$

the _variety of isotropic points_ of $V_{r}$. By Witt’s lemma [35, Thm. 7.4],
it is a compact homogeneous $G_{r}$-space when endowed with the Hausdorff
topology of its set of $\mathbb{C}$-points, and the Lebesgue class is its
unique $G_{r}$-invariant measure class. The assumption $r\geq r_{1}$ excludes
two pathological cases: the set $\mathcal{P}_{0}$ is empty, and
$\mathcal{P}_{1}^{(+1,0)}$ is not irreducible as an algebraic set.

Let $Q_{r}$ be the stabilizer of $[e_{r}]\in\mathcal{P}_{r}$, a maximal
parabolic subgroup of $G_{r}$ with Levi decomposition $Q_{r}=U_{r}\rtimes
R_{r}$, where $U_{r}$ is unipotent and $R_{r}$ reductive. Represented in the
basis $\mathscr{B}_{r}$, the reductive factor
$R_{r}\cong\mathbb{C}^{\times}\times G_{r-1}$ consists of the block-diagonal
matrices

$\begin{pmatrix}\lambda&0&0\\\\[-2.0pt] 0&g&0\\\\[-2.0pt]
0&0&\lambda^{-1}\end{pmatrix}$

with $\lambda\in\mathbb{C}^{\times}$ and $g\in G_{r-1}$. If $Q_{r}^{-}$
denotes the stabilizer of $[f_{r}]\in\mathcal{P}_{r}$, then $Q_{r}\cap
Q_{r}^{-}=R_{r}$.

We are interested in parametrizing generic orbits of the diagonal
$G_{r}$-action on product spaces $\mathcal{P}_{r}^{k}$, for $k\in\\{3,4,5\\}$.
Genericity is understood in the measure-theoretical sense, and therefore, the
products $\mathcal{P}_{r}^{k}$ will be regarded as objects in the category of
_Lebesgue $G_{r}$-spaces_.

######  Definition .

We call _Lebesgue space_ a standard Borel space $X$ endowed with the measure
class of a Borel probability measure. If $X$ admits a Borel action of a lcsc
group $G$, and the measure class is $G$-invariant, we call $X$ a _Lebesgue
$G$-space_. Morphisms of Lebesgue ($G$-) spaces are equivalence classes (up to
null sets) of measure-class preserving Borel ($G$-)maps.

######  Example .

An algebraic $\mathbb{C}$-variety $X$ with the regular action of an algebraic
group $G$ is a Lebesgue $G$-space if we endow $X$ with its Lebesgue measure
class. The Borel $\sigma$-algebra of the set of orbits $G\\!\setminus\\!X$
with respect to the Hausdorff topology on $X$ is standard (see [38, Thm.
3.1.3]), and hence $G\\!\setminus\\!X$ is a Lebesgue space.

If $v_{0},\ldots,v_{k}$ are all non-zero vectors in $V_{r}$, we will write
$[v_{0},\ldots,v_{k}]$ as a shorthand for the tuple $([v_{0}],\ldots,[v_{k}])$
of points in $\mathbb{P}(V_{r})$. We will say the tuple is in general position
if so are the vectors $v_{0},\ldots,v_{k}$ (or any choices of lifts). We
define the subsets

(3.1) $\displaystyle\mathcal{P}_{r}^{(k+1)}\,$
$\displaystyle\coloneqq\left\\{[v_{0},\ldots,v_{k}]\in\mathcal{P}_{r}^{k+1}\mid\omega(v_{i},v_{j})\neq
0\ \ \forall\ i\neq j\right\\},$ (3.2)
$\displaystyle\mathcal{P}_{r}^{\\{k+1\\}}$
$\displaystyle\coloneqq\left\\{\vec{v}\in\mathcal{P}_{r}^{(k+1)}\mid\vec{v}\mbox{
is in general position}\right\\}$

Both are $G_{r}$-invariant, $\mathfrak{S}_{k}$-invariant111As customary in
homological algebra, we adopt the convention that $0\in\mathbb{N}$. The
symmetric group of the set $\\{0,\ldots,k\\}\subset\mathbb{N}$ will be denoted
by $\mathfrak{S}_{k}$., Zariski-open subsets of $\mathcal{P}_{r}^{k+1}$. In
particular, they are also Hausdorff-open and co-null. Moreover, if
$\partial_{i}\colon\mathcal{P}_{r}^{k+1}\to\mathcal{P}_{r}^{k}$ are the
($G_{r}$-equivariant) face maps that forget the $i$-th entry for
$i\in\\{0,\ldots,k\\}$, then
$\partial_{i}(\mathcal{P}_{r}^{(k+1)})=\mathcal{P}_{r}^{(k)}$ and
$\partial_{i}(\mathcal{P}_{r}^{\\{k+1\\}})=\mathcal{P}_{r}^{\\{k\\}}$.

Our first observation is that there is only one generic $G_{r}$-orbit of
triples of points in $\mathcal{P}_{r}$.

######  Lemma .

The group $G_{r}$ acts transitively on $\mathcal{P}_{r}^{\\{3\\}}$ for any
$r\geq r_{1}$. Moreover, the point stabilizer $M_{r}<G_{r}$ admits a
continuous surjection $M_{r}\twoheadrightarrow M^{\prime}_{r}$ with solvable
kernel, where

$M^{\prime}_{r}=\begin{cases}G_{r-1}^{(+1,0)}&\mbox{ if
}(\varepsilon,d)=(+1,1),\\\\[-3.0pt] G_{r-2}&\mbox{ if
}(\varepsilon,d)=(-1,0),\\\\[-3.0pt] G_{r-2}^{(+1,1)}&\mbox{ if
}(\varepsilon,d)=(+1,0).\end{cases}$

###### Proof.

For $r\geq r_{1}$, we set

(3.3) $\displaystyle\phi_{2}$ $\displaystyle\coloneqq
e_{r}+f_{r}+x_{0}\in\mathcal{P}_{r}\quad\mbox{where}$ $\displaystyle x_{0}$
$\displaystyle\coloneqq\begin{cases}0&\mbox{if }(\varepsilon,d)=(-1,0)\mbox{
and }r=1,\\\\[-4.0pt] {\scriptstyle\sqrt{-2}}\,h&\mbox{if
}(\varepsilon,d)=(+1,1)\mbox{ and }r=1,\\\\[-4.0pt]
e_{r-1}-f_{r-1}&\mbox{otherwise,}\end{cases}$

We show that for any
$\mathbf{p}=[v_{0},v_{1},v_{2}]\in\mathcal{P}_{r}^{\\{3\\}}$, there exists
$g\in G_{r}$ such that $g\,\mathbf{p}=[e_{r},\ f_{r},\ \phi_{2}]$. Indeed,
since $\langle v_{0},v_{1}\rangle<V_{r}$ is hyperbolic and
$(\mathbb{C}^{\times})^{2}=\mathbb{C}^{\times}$, there exists $g_{1}\in G_{r}$
such that

$g_{1}\,[v_{0},v_{1},v_{2}]=[e_{r},f_{r},e_{r}+f_{r}+x]$

for some $x\in V_{r-1}<V_{r}$. Here $x=0$ if and only if $n(r)=2$, which holds
only when $(\varepsilon,d)=(-1,0)$ and $r=1$. If $n(r)>2$, then
$q(x)=-(1+\varepsilon)=q(x_{0})$. Now, by Witt’s lemma, we find $g_{2}\in
G_{r-1}<G_{r}$ such that $g_{2}x=x_{0}$, proving the claim.

Up to index two, the stabilizer $M_{r}$ is isomorphic to the point stabilizer
of the $G_{r-1}$-action on

$\mathcal{Q}=\\{x\in V_{r-1}\mid q(x)=-(1+\varepsilon)\\}$

If $\varepsilon=-1$, then $e_{r-1}\in\mathcal{Q}$ and, hence, $M_{r}$ projects
onto $G_{r-2}$ with solvable kernel. If $\varepsilon=+1$ and $x\in\mathcal{Q}$
is arbitrary, the orthogonal decomposition $V_{r-1}=\langle
x\rangle\oplus\langle x\rangle^{\perp}$ implies that $M_{r}$ is isomorphic to
$\operatorname{Aut}(\langle x\rangle^{\perp},\omega|_{\langle
x\rangle^{\perp}})$ up to index two. This yields the claim, since $\langle
x\rangle^{\top}$ is non-degenerate and of codimension one in $V_{r-1}$. ∎

###  3.2. Cross-ratio coordinates on configurations of 4- and 5-tuples

We will describe the quotients $G_{r}\\!\setminus\\!\mathcal{P}_{r}^{4}$ and
$G_{r}\\!\setminus\\!\mathcal{P}_{r}^{5}$ as Lebesgue spaces in terms of the
_$\omega$ -cross-ratios_, introduced by Korányi and Reimann [23] in the realm
of complex hyperbolic geometry; see e.g. [16].

######  Definition .

For any $\mathbf{p}=(p_{0},p_{1},p_{2},p_{3})\in\mathcal{P}_{r}^{(4)}$ with
$p_{i}=[v_{i}]$, the ratios

$\displaystyle\operatorname{cr}_{0}(\mathbf{p})$
$\displaystyle\coloneqq\frac{\omega(v_{0},v_{2})\cdot\omega(v_{1},v_{3})}{\omega(v_{0},v_{3})\cdot\omega(v_{1},v_{2})},$
$\displaystyle\operatorname{cr}_{1}(\mathbf{p})$
$\displaystyle\coloneqq\operatorname{cr}_{0}(p_{1},p_{2},p_{0},p_{3})^{-1},$
$\displaystyle\operatorname{cr}_{2}(\mathbf{p})$
$\displaystyle\coloneqq\operatorname{cr}_{0}(p_{2},p_{0},p_{1},p_{3}),$

are independent of the choices of line representatives $v_{i}$, thus giving
rise to the $G_{r}$-invariant, holomorphic functions
$\operatorname{cr}_{j}:\mathcal{P}_{r}^{(4)}\to\mathbb{C}^{\times}$, called
$\omega$_-cross-ratios_. We will regard them as $G_{r}$-invariant morphisms
$\operatorname{cr}_{j}\colon\mathcal{P}_{r}^{4}\to\hat{\mathbb{C}}$ of
Lebesgue spaces. We also write $\operatorname{cr}_{j}^{-1}\coloneqq
1/\operatorname{cr}_{j}$.

The next lemma explains dependencies between $\omega$-cross-ratios, including
the effect of the permutation of their arguments. It shows that the only
$\omega$-cross-ratios associated to any generic choice of four points in
$\mathcal{P}_{r}$ are $\operatorname{cr}_{0}$, $\operatorname{cr}_{1}$,
$\operatorname{cr}_{2}$, and that any two of them determine the third one. Its
proof is a verification left as an exercise to the reader.

######  Lemma .

The following identities hold everywhere in $\mathcal{P}_{r}^{(4)}$:

  1. _(i)_

$\operatorname{cr}_{0}\,\circ\,(01)=\operatorname{cr}_{0}\,\circ\,(23)=\operatorname{cr}_{1}^{-1}\,\circ\,(02)=\operatorname{cr}_{1}^{-1}\,\circ\,(13)=\operatorname{cr}_{2}\,\circ\,(12)=\operatorname{cr}_{2}\,\circ\,(03)=\operatorname{cr}_{0}^{-1}$.

  2. _(ii)_

$\operatorname{cr}_{1}\,\circ\,(12)=\operatorname{cr}_{1}\,\circ\,(03)=\operatorname{cr}_{2}^{-1}\,\circ\,(01)=\operatorname{cr}_{2}^{-1}\,\circ\,(23)=\operatorname{cr}_{1}^{-1}$.

  3. _(iii)_

$\operatorname{cr}_{2}\,\circ\,(02)=\operatorname{cr}_{2}\,\circ\,(13)=\operatorname{cr}_{2}^{-1}$.

  4. _(iv)_

$\operatorname{cr}_{0}\cdot\operatorname{cr}_{1}^{-1}\cdot\operatorname{cr}_{2}=\varepsilon$.

######  Remark .

Let us consider the rank-one examples (i.e. $r=1$) among the families treated
in Theorem 2. If $(\varepsilon,d)=(-1,0)$, then the $\omega$-cross-ratios
coincide with the classical cross-ratios on the complex projective line, since
$G_{1}=\operatorname{Sp}_{2}(\mathbb{C})=\operatorname{SL}_{2}(\mathbb{C})$
and $\mathcal{P}_{1}=\mathbb{P}(\mathbb{C}^{2})$. We will continue to refer to
this as the _classical setting_. subsection 3.1 recovers the well-known fact
that any triple in $\mathcal{P}_{1}^{\\{3\\}}$ is in the orbit of
$(\infty,0,1)\in\hat{\mathbb{C}}^{3}$. As usual, we identify
$\mathbb{P}(\mathbb{C}^{2})$ with $\hat{\mathbb{C}}$ by

(3.4) $[f_{1}]\equiv\infty,\quad\mathrm{and}\quad[e_{1}+z\cdot f_{1}]\equiv z\
\mbox{for any }z\in\mathbb{C}.$

Recall that for every $\mathbf{p}\in\mathcal{P}_{1}^{\\{4\\}}$, the cross-
ratio $\operatorname{cr}_{0}(\mathbf{p})$ is the only number
$a\in\mathbb{C}\smallsetminus\\{0,1\\}$ such that
$G_{1}\cdot\mathbf{p}=G_{1}\cdot(\infty,0,1,a)$. In the language of Lebesgue
spaces, this means that the map $\operatorname{cr}_{0}$ descends to a
isomorphism $G_{1}\\!\setminus\\!\mathcal{P}_{1}^{4}\cong\hat{\mathbb{C}}$. In
fact, due to the low dimension of $V_{1}$ in the classical setting, just one
of the $\omega$-cross-ratios determines the value of the remaining two.

The only other rank-one example appears for $(\varepsilon,d)=(+1,1)$, which
gives $G_{1}=\operatorname{O}_{3}(\mathbb{C})$. As in the classical setting,
we have an isomorphism
$G_{1}\\!\setminus\\!\mathcal{P}_{1}^{4}\cong\hat{\mathbb{C}}$.

In contrast to the rank-one situation, one single $\omega$-cross-ratio does
not suffice to parametrize generic 4-tuples in higher-rank. However, two of
them are enough.

######  Proposition .

For any $r\geq 2$, the $G_{r}$-invariant morphism
$\pi_{3}:\mathcal{P}_{r}^{4}\to\hat{\mathbb{C}}^{2}$ of Lebesgue spaces
defined as $\pi_{3}\coloneqq(\operatorname{cr}_{1},\,\operatorname{cr}_{2})$
descends to an isomorphism
$G_{r}\\!\setminus\\!\mathcal{P}_{r}^{4}\cong\hat{\mathbb{C}}^{2}$.

We explain now how the $\omega$-cross-ratios parametrize generic orbits of
5-tuples of points in $\mathcal{P}_{r}$.

######  Definition .

For $j\in\\{0,1,2\\}$, we define morphisms
$\alpha_{j},\,\beta_{j},\,\gamma_{j}\colon\mathcal{P}_{r}^{5}\to\hat{\mathbb{C}}$

$\displaystyle\alpha_{j}(\mathbf{p})$
$\displaystyle\coloneqq\operatorname{cr}_{j}\circ\,\partial_{4}(\mathbf{p})=\operatorname{cr}_{j}(p_{0},p_{1},p_{2},p_{3}),$
$\displaystyle\beta_{j}(\mathbf{p})$
$\displaystyle\coloneqq\operatorname{cr}_{j}\circ\,\partial_{3}(\mathbf{p})=\operatorname{cr}_{j}(p_{0},p_{1},p_{2},p_{4}),$
$\displaystyle\gamma_{j}(\mathbf{p})$
$\displaystyle\coloneqq\operatorname{cr}_{j}\circ\,\partial_{2}(\mathbf{p})=\operatorname{cr}_{j}(p_{0},p_{1},p_{3},p_{4}),$

for $\mathbf{p}=(p_{0},p_{1},p_{2},p_{3},p_{4})\in\mathcal{P}_{r}^{(5)}$,
where $\partial_{i}:\mathcal{P}_{r}^{5}\to\mathcal{P}_{r}^{4}$ denote the
usual face operators. As in subsection 3.2, we set $\alpha_{j}^{-1}\coloneqq
1/\alpha_{j}$ and define similarly $\beta_{j}^{-1}$, $\gamma_{j}^{-1}$.

We can express in terms of the $\omega$-cross-ratios from subsection 3.2 all
the remaining cross-ratios associated to a 5-tuple. This fact, recorded in the
next lemma, is readily verified.

######  Lemma .

The following identities hold everywhere in $\mathcal{P}_{r}^{(5)}$:

  1. _(i)_

$\operatorname{cr}_{0}\,\circ\,\partial_{1}=\alpha_{2}\beta_{2}^{-1}$

  2. _(ii)_

$\operatorname{cr}_{1}\,\circ\,\partial_{1}=\beta_{1}^{-1}\gamma_{1}$

  3. _(iii)_

$\operatorname{cr}_{2}\,\circ\,\partial_{1}=\varepsilon(\alpha_{2}^{-1}\beta_{1}^{-1}\beta_{2}\gamma_{1})$

  1. _(iv)_

$\operatorname{cr}_{0}\,\circ\,\partial_{0}=\alpha_{1}\beta_{1}^{-1}$

  2. _(v)_

$\operatorname{cr}_{1}\,\circ\,\partial_{0}=\varepsilon(\alpha_{1}\alpha_{2}^{-1}\beta_{1}^{-1}\gamma_{1})$

  3. _(vi)_

$\operatorname{cr}_{2}\,\circ\,\partial_{0}=\alpha_{2}^{-1}\gamma_{1}$

  1. _(vii)_

$\alpha_{0}\gamma_{0}=\beta_{0}$

We conclude from Lemmas 3.2 and 3.2 that at most five cross-ratios suffice to
describe generic orbits of 5-tuples. The next theorem states that five are
also necessary as long as $\dim(V_{r})$ is large. Recall that $r_{1}$ was
defined as a function of $(\varepsilon,d)$ at the end of Subsection 2.1.

######  Proposition .

For any $r\geq r_{1}+1$, the $G_{r}$-invariant morphism
$\pi_{4}\colon\mathcal{P}_{r}^{5}\to\hat{\mathbb{C}}^{5}$ of Lebesgue spaces
defined as
$\pi_{4}\coloneqq(\alpha_{1},\,\alpha_{2},\,\beta_{1},\,\beta_{2},\,\gamma_{1})$
descends to an isomorphism
$G_{r}\\!\setminus\\!\mathcal{P}_{r}^{5}\cong\hat{\mathbb{C}}^{5}$.

######  Remark .

In rank one, the isomorphism
$G_{1}\\!\setminus\\!\mathcal{P}_{1}^{5}\cong\hat{\mathbb{C}}^{2}$ holds for
$(\varepsilon,d)\in\\{(+1,1),(-1,0)\\}$. For instance, in the classical
setting (see subsection 3.2), any $\mathbf{p}\in\mathcal{P}_{1}^{\\{5\\}}$ is
in the $G_{1}$-orbit of the tuple $(\infty,0,1,a,b)\in\hat{\mathbb{C}}^{5}$
with $a=\alpha_{0}(\mathbf{p})$ and $b=\beta_{0}(\mathbf{p})$. In rank two for
$(\varepsilon,d)=(+1,0)$, the low dimension of $V_{2}$ yields one extra
relation, which forces the isomorphism
$G_{2}\\!\setminus\\!\mathcal{P}_{2}^{5}\cong\hat{\mathbb{C}}^{4}$.

Both Propositions 3.2 and 3.2 will be essential in our proof of Key Lemma.
Their proof, in turn, consist of elementary, yet fairly lengthy and technical
computations. For the sake of readibility, we have opted to present them in
Appendix A.

##  4\. Proof of Key Lemma

###  4.1. Cohomological characterization of $\ker\iota_{r}^{\ast}$

For any $r\geq 1$, we consider the complex

(4.1) $0\to
L^{\infty}(\mathcal{P}_{r})\xrightarrow{\operatorname{d}_{r}^{0}}L^{\infty}(\mathcal{P}_{r}^{2})\xrightarrow{\operatorname{d}_{r}^{1}}L^{\infty}(\mathcal{P}_{r}^{3})\rightarrow\cdots$

of dual Banach $G_{r}$-modules, with $G_{r}$-equivariant coboundary maps
defined by the alternating sum
$\operatorname{d}_{r}^{k}\coloneqq\sum_{i=0}^{k+1}(-1)^{i}\cdot\partial^{i}$.
Here $\partial^{i}$ is the operator induced by the face maps
$\partial_{i}:\mathcal{P}_{r}^{k+1}\to\mathcal{P}_{r}^{k}$. The complex (4.1)
is _acyclic_ in the sense that the homology of its augmented complex

(4.2)
$0\to\mathbb{R}\xrightarrow{\operatorname{d}_{r}^{-1}}L^{\infty}(\mathcal{P}_{r})\xrightarrow{\operatorname{d}_{r}^{0}}L^{\infty}(\mathcal{P}_{r}^{2})\xrightarrow{\operatorname{d}_{r}^{1}}L^{\infty}(\mathcal{P}_{r}^{3})\rightarrow\cdots$

vanishes in every degree. This follows from the observation that integration
over the first variable of $L^{\infty}(\mathcal{P}_{r}^{k+1})$ with respect to
a quasi-invariant probability measure on $\mathcal{P}_{r}$ is a contracting
homotopy of (4.2). We denote by ${\rm H}_{\rm
b}^{q}(G_{r}\curvearrowright\mathcal{P}_{r})$ the $q$-th homology of the
complex $L^{\infty}(\mathcal{P}_{r}^{\bullet+1})^{G_{r}}$ of
$G_{r}$-invariants of (4.1), and call it the _bounded cohomology of the
action_ $G_{r}\curvearrowright\mathcal{P}_{r}$.

In the rank-one cases, the action $G_{1}\curvearrowright\mathcal{P}_{1}$ is
amenable, and hence, there exists a canonical isomorphism ${\rm H}_{\rm
b}^{\ast}(G_{1})\cong{\rm H}_{\rm
b}^{\ast}(G_{1}\curvearrowright\mathcal{P}_{1});$ see [28, Thm. 7.5.3]. Beyond
rank one, the isomorphism needs not hold since the action is no longer
amenable. However, the bounded cohomology of that action—or any other
$G_{r}$-action on an acyclic complex—may still contain interesting information
about ${\rm H}_{\rm b}^{\ast}(G_{r})$. The precise relationship between ${\rm
H}_{\rm b}^{\ast}(G_{r})$ and ${\rm H}_{\rm
b}^{\ast}(G_{r}\curvearrowright\mathcal{P}_{r})$ is given by a spectral
sequence [11, Prop. 2.15], with which we prove the following

######  Lemma .

For any $r\in\mathbb{N}$, there exists a linear isomorphism
$\ker\iota_{r}^{\ast}\ \cong\ {\rm H}_{\rm
b}^{3}(G_{r+1}\curvearrowright\mathcal{P}_{r+1}).$

###### Proof.

There exists a spectral sequence
$\operatorname{E}_{\bullet}^{\bullet,\bullet}$ that abuts to zero, with first-
page terms

$\operatorname{E}_{1}^{p,q}={\rm H}_{\rm
b}^{q}(G_{r+1};L^{\infty}(\mathcal{P}_{r+1}^{p}))$

and differentials
$\operatorname{d}_{1}^{p,q}:\operatorname{E}_{1}^{p,q}\to\operatorname{E}_{1}^{p+1,q}$
induced by the operators $\operatorname{d}_{r+1}^{p-1}$ [11, Prop. 2.15]. Note
first that $\operatorname{E}_{2}^{p,0}\cong{\rm H}_{\rm
b}^{p-1}(G_{r+1}\curvearrowright\mathcal{P}_{r+1})$ for all $p\geq 1$.
Furthermore, as a consequence of the Eckmann–Shapiro lemma [28, Prop. 10.1.3],
subsection 3.1, and [28, Corollary 8.5.2], we have:

$\begin{array}[]{ll}\operatorname{E}_{1}^{0,q}&={\rm H}_{\rm b}^{q}(G_{r+1}),\\\\[4.0pt] \operatorname{E}_{1}^{1,q}&\cong{\rm H}_{\rm b}^{q}(Q_{r+1})\cong{\rm H}_{\rm b}^{q}(G_{r}),\\\\[4.0pt] \operatorname{E}_{1}^{2,q}&\cong{\rm H}_{\rm b}^{q}(R_{r+1})\cong{\rm H}_{\rm b}^{q}(G_{r}),\end{array}$ |  | $\operatorname{E}_{1}^{3,q}\cong{\rm H}_{\rm b}^{q}(M_{r+1})\cong\begin{cases}{\rm H}_{\rm b}^{q}(G_{r}^{(+1,0)})&\mbox{ if }(\varepsilon,d)=(+1,1),\\\ {\rm H}_{\rm b}^{q}(G_{r-1})&\mbox{ if }(\varepsilon,d)=(-1,0),\\\ {\rm H}_{\rm b}^{q}(G_{r-1}^{(+1,1)})&\mbox{ if }(\varepsilon,d)=(+1,0).\end{cases}$
---|---|---

Here, $Q_{r+1}$, $R_{r+1}$, $M_{r+1}$ are as defined in Subsection 3.1. For
$p\leq 3$, we have $\operatorname{E}_{2}^{p,1}=\operatorname{E}_{1}^{p,1}=0$
and $\operatorname{E}_{2}^{p,2}=\operatorname{E}_{1}^{p,2}=0.$ The latter
equality holds from the isomorphism conjecture in degree two [5, Lem. 6.1]. In
fact, for any $(\varepsilon,d)$ as in (2.4) and any $r$, the group
$G_{r}^{\varepsilon,d}$ is of non-Hermitian type, and therefore ${\rm H}_{\rm
b}^{2}(G_{r}^{\varepsilon,d})\cong\operatorname{H}^{2}(G_{r}^{\varepsilon,d})=0$.
Finally, the map $\operatorname{d}_{1}^{0,3}$ is conjugated by the
isomorphisms above to the map $\iota_{r}^{\ast}$ (e.g. by [11, Lem. 3.7]), and
hence, $\operatorname{E}_{2}^{0,3}\cong\ker\iota_{r}^{\ast}$. Summarizing, we
have showed that the second page $\operatorname{E}_{2}^{\bullet,\bullet}$ has
terms as displayed below.

3 |  $\ker\iota_{r}^{\ast}$ |  |  |  |
---|---|---|---|---|---
2 |  |  0 |  0 |  |
1 |  |  |  0 |  0 |
0 |  |  |  |  |  ${\rm H}_{\rm b}^{3}(G_{r+1}\curvearrowright\mathcal{P}_{r+1})$
|  0 |  1 |  2 |  3 |  4

Blank spaces indicate that the terms are not relevant to our computation.
Observe that all the displayed terms will remain unchanged until the fourth
page $\operatorname{E}_{4}^{\bullet,\bullet}$. Then, the arrow

$\operatorname{d}_{4}^{0,3}:\ker\iota_{r}^{\ast}\to{\rm H}_{\rm
b}^{3}(G_{r+1}\curvearrowright\mathcal{P}_{r+1})$

must be an isomorphism, for otherwise the limits
$\operatorname{E}^{3}_{\infty}\neq 0$ or $\operatorname{E}^{4}_{\infty}\neq 0$
would be non-zero. ∎

###  4.2. Secondary stability argument

Propositions 3.2 and 3.2 enter the proof of the Key Lemma _bis_ in the form of
the next statement, reminiscent of secondary stability in the sense of
Galatius–Kupers–Randal-Williams [15].

######  Lemma .

For every $r\geq r_{1}$, the linear isomorphism
$\ker\iota_{r}^{\ast}\cong\ker\iota^{\ast}_{r_{1}}$ holds.

###### Proof.

In virtue of subsection 4.1, we must prove the isomorphism

${\rm H}_{\rm b}^{3}(G_{r+1}\curvearrowright\mathcal{P}_{r+1})\cong{\rm
H}_{\rm b}^{3}(G_{r_{1}+1}\curvearrowright\mathcal{P}_{r_{1}+1}).$

By subsection 3.1, the space of invariants
$L^{\infty}(\mathcal{P}_{r+1}^{3})^{G_{r+1}}$ consists solely of constants,
and hence, there are no coboundaries in ${\rm H}_{\rm
b}^{3}(G_{r+1}\curvearrowright\mathcal{P}_{r+1})$. Let now $D_{r+1}^{3}$ be
the bounded operator defined by the commutative diagram

$\textstyle{L^{\infty}(\mathcal{P}_{r+1}^{4})^{G_{r+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{d}_{r+1}^{3}}$$\textstyle{L^{\infty}(\mathcal{P}_{r+1}^{5})^{G_{r+1}}}$$\textstyle{L^{\infty}(\hat{\mathbb{C}}^{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{r+1}^{3}}$$\scriptstyle{\pi_{3}^{\ast}}$$\scriptstyle{\cong}$$\textstyle{L^{\infty}(\hat{\mathbb{C}}^{5})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{4}^{\ast}}$$\scriptstyle{\cong}$

Here, $\pi_{3}^{\ast}$ and $\pi_{4}^{\ast}$ are the isomorphisms induced by
the morphisms of Lebesgue spaces from Propositions 3.2 and 3.2, respectively.
Observe that $\ker\operatorname{d}_{r+1}^{3}\cong\ker D_{r+1}^{3}$.

The proof is completed upon showing that $D_{r+1}^{3}$ is independent of
$r\geq r_{1}$. In fact, after applying subsection 3.2, the map $D_{r+1}^{3}$
can be expressed as

(4.3)
$D^{3}_{r+1}f(\mathbf{x})=f\\!\left(\varepsilon\frac{a_{1}c_{1}}{a_{2}b_{1}},\frac{c_{1}}{a_{2}}\right)-f\\!\left(\frac{c_{1}}{b_{1}},\varepsilon\frac{b_{2}c_{1}}{a_{2}b_{1}}\right)+f\\!\left(c_{1},\varepsilon\frac{a_{1}b_{2}c_{1}}{a_{2}b_{1}}\right)-f(b_{1},b_{2})+f(a_{1},a_{2})\vspace{2pt}$

for any $f\in L^{\infty}(\hat{\mathbb{C}}^{2})$ and a.e.
$\mathbf{x}=(a_{1},a_{2},b_{1},b_{2},c_{1})\in\hat{\mathbb{C}}^{5}$. ∎

###### Proof of Key Lemma _bis_.

The (weak) bc-stability of the complex classical families—proved in [12, Thm.
A]—implies that $\ker\iota_{r}^{\ast}=0$ for large enough $r$. This fact
combined with subsection 4.2 finishes the proof. ∎

In the classical setting (see subsection 3.2), the coboundary
$\operatorname{d}_{1}^{3}$ is conjugated to an operator
$D_{1}^{3}:L^{\infty}(\hat{\mathbb{C}})\to L^{\infty}(\hat{\mathbb{C}}^{2})$
in virtue of the isomorphisms discussed in Remarks 3.2 and 3.2. The functional
equation $D_{1}^{3}f=0$ for $f\in L^{\infty}(\mathbb{C})$ is known as the
_Spence–Abel functional equation_. Under a specific parametrization, it is
given by

(4.4)
$f\\!\left(\frac{a(1-b)}{b(1-a)}\right)-f\\!\left(\frac{1-b}{1-a}\right)+f\\!\left(\frac{b}{a}\right)-f(b)+f(a)=0\quad\mbox{
for a.e. }(a,b)\in\mathbb{C}^{2},\vspace{2pt}$

and its only solution up to a constant factor is the _Bloch–Wigner
dilogarithm_ $\mathcal{D}$ (see e.g. [37, §3]). The expression of
$D_{\\!\\!\infty}^{3}\coloneqq D_{r}^{3}$ given by (4.3) for any $r\geq
r_{1}+1$ produces a higher-rank analogue of the Spence–Abel equation. A
byproduct of the proof of the Key Lemma is the inexistence of non-trivial
solutions for this equation.

######  Corollary .

Let $\varepsilon\in\\{\pm 1\\}$. If $f\in L^{\infty}(\hat{\mathbb{C}}^{2})$
satisfies the functional equation $D_{\\!\\!\infty}^{3}f=0$ a.e., then $f$ is
a.e. the identically zero function. ∎

##  5\. Proof of Theorems 2 and 3 for the complex families $\mathbf{B}_{r}$
and $\mathbf{C}_{r}$

In Subsection 2.2, we fixed a pair
$(\varepsilon,d)\in\\{(+1,1),(-1,0),(+1,0)\\}$ as a standing assumption. In
this section, we will only consider the cases $(+1,1)$ and $(-1,0)$, which
correspond to the two complex classical families $\mathbf{B}_{r}$ and
$\mathbf{C}_{r}$, respectively. In both cases, we have $r_{0}=r_{1}=1$.

We first prove that $\operatorname{res}_{r}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n(r)}(\mathbb{C}))\to{\rm H}_{\rm b}^{3}(S_{r})$ is
isometric for any $r\in\mathbb{N}$, which is the statement of subsection 1.4.
We rely on the following result from [3]. Below, we denote by
$\Pi_{n}:\operatorname{SL}_{2}(\mathbb{C})\to\operatorname{SL}_{n}(\mathbb{C})$
the irreducible $n$-dimensional
$\operatorname{SL}_{2}(\mathbb{C})$-representation, and by $b_{n}$ the
_bounded Borel class_ , defined in [3], which generates ${\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))$ as a vector space.

######  Theorem 5.1 (see [3, Thm. 2]).

For any $n\geq 2$, the map $\Pi_{n}^{\ast}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))\to{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{2}(\mathbb{C}))$ induced by $\Pi_{n}$ is an
isometric isomorphism that maps the bounded Borel class $b_{n}$ to

$\Pi_{n}^{\ast}(b_{n})=\frac{n(n^{2}-1)}{6}\cdot b_{2}.$

In particular, the Gromov norm of $b_{n}$ is $\|b_{n}\|=n(n^{2}-1)/6\cdot
v_{3},$ where $v_{\mathbb{H}^{3}}\approx 1.0149...$ is the maximal volume of
an ideal tetrahedron in $\mathbb{H}^{3}$.

Consider now the irreducible representation
$\Pi_{n(r)}:\operatorname{SL}_{2}(\mathbb{C})\to\operatorname{SL}(V_{r})\cong\operatorname{SL}_{n(r)}(\mathbb{C})$
for $r\geq 1$. The parity of $n(r)$ depends on the choice of
$(\varepsilon,d)\in\\{(+1,1),(-1,0)\\}$. It is important to note that (up to
conjugation) $\Pi_{n(r)}$ preserves the form $\omega_{r}$ on $V_{r}$, which
implies that its image is contained in $S_{r}$. We will use the notation
$\bar{\Pi}_{r}:\operatorname{SL}_{2}(\mathbb{C})\to S_{r}$ for the co-
restriction of $\Pi_{n(r)}$.

###### Proof of subsection 1.4.

Note that
$\Pi_{n(r)}^{\ast}=\bar{\Pi}_{r}^{\ast}\,\circ\,\operatorname{res}_{r}$. Since
${\rm H}_{\rm b}^{3}(\operatorname{SL}_{n(r)}(\mathbb{C}))$ is generated by
the bounded Borel class $b_{n(r)}$, it suffices to show the equality
$\|\operatorname{res}_{r}(b_{n(r)})\|=\|b_{n(r)}\|$. Indeed, in virtue of
Theorem 5.1 and the fact that both $\operatorname{res}_{r}$ and
$\bar{\Pi}_{r}^{\ast}$ are norm non-increasing, we have

$\|b_{n(r)}\|=\|\Pi_{n(r)}^{\ast}(b_{n(r)})\|=\|\bar{\Pi}_{r}^{\ast}(\operatorname{res}_{r}(b_{n(r)}))\|\leq\|\operatorname{res}_{r}(b_{n(r)})\|\leq\|b_{n(r)}\|,$

completing the proof. ∎

For $(\varepsilon,d)$ as in this section, the restriction map ${\rm H}_{\rm
b}^{\ast}(G_{r})\to{\rm H}_{\rm b}^{\ast}(S_{r})$ is an isomorphism as a
result of (2.2) and [28, Cor. 8.5.2]. This implies immediately that Key Lemma
_bis_ holds _verbatim_ for $S_{r}$ instead of $G_{r}$ in these two cases:

######  Lemma .

For $r\geq r_{1}=1$, the induced map $\iota_{r}^{\ast}:{\rm H}_{\rm
b}^{3}(S_{r+1})\to{\rm H}_{\rm b}^{3}(S_{r})$ is injective. ∎

Recall also that, depending on $(\varepsilon,d)$, the group $S_{1}$ equals
$\operatorname{Sp}_{2}(\mathbb{C})$ or $\operatorname{SO}_{3}(\mathbb{C})$.
The former group equals $\operatorname{SL}_{2}(\mathbb{C})$ and the latter
admits an isogeny
$\operatorname{SL}_{2}(\mathbb{C})\twoheadrightarrow\operatorname{SO}_{3}(\mathbb{C})$,
as seen in, e.g., [36, 34]. Thus, $S_{1}$ is of type $\mathbf{A}_{1}$, and, in
particular, $\dim{\rm H}_{\rm b}^{3}(S_{1})=1$.

###### Proof of Theorem 3 _bis_ for $\mathbf{B}_{r},\mathbf{C}_{r}$.

If $r\geq r_{0}=1$, then $n(r)\geq 2$, so that $\|b_{n(r)}\|>0$. Together with
subsection 1.4, this establishes the injectivity of $\operatorname{res}_{r}$.
Surjectivity follows then from the fact that $\dim{\rm H}_{\rm
b}^{3}(S_{r})\leq 1$ for all $r\geq 1$. This, in turn, is a consequence of
section 5 and the equality $\dim{\rm H}_{\rm b}^{3}(S_{1})=1$. ∎

###### Proof of Theorem 2 _bis_ for $\mathbf{B}_{r},\mathbf{C}_{r}$.

Theorem 3 _bis_ implies that $\dim{\rm H}_{\rm b}^{3}(S_{r})=1$ for all $r\geq
1$. Thus, the induced maps $\iota_{r}^{\ast}:{\rm H}_{\rm
b}^{3}(S_{r+1})\to{\rm H}_{\rm b}^{3}(S_{r})$—injective by section 5—are
isomorphisms. ∎

##  6\. Proof of Theorems 2 and 3 for the complex family $\mathbf{D}_{r}$

In this section, we consider the case $(\varepsilon,d)=(+1,0)$, which
corresponds to the $\mathbf{D}_{r}$ family, and recall that $n(r)=2r$,
$r_{0}=3$, and $r_{1}=2$. Note that the images of the irreducible
representations
$\Pi_{2r}:\operatorname{SL}_{2}(\mathbb{C})\to\operatorname{SL}_{2r}(\mathbb{C})$
are not contained in $S_{r}=\operatorname{SO}_{2r}(\mathbb{C})$ for any $r$.
This limitation prevents the adaptation of the argument in Section 5 to the
$\mathbf{D}_{r}$ family.

We will establish Theorem 3 _bis_ by induction on the rank $r$ without relying
on norm considerations. The induction will be based on the following
statement.

######  Lemma .

The restriction map $\operatorname{res}_{2}:\,{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\to{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{4}(\mathbb{C}))$ is injective.

To prove section 6, we will realize $\operatorname{res}_{2}$ over the
Furstenberg boundaries of the involved groups. Subsection 6.1 collects
relevant background on the Goncharov–Bucher–Burger–Iozzi _volume cocycle_ of
$\operatorname{SL}_{n}(\mathbb{C})$, which represents the bounded Borel class
$b_{n}\in{\rm H}_{\rm b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))$. Then, in
Subsection 6.2, we will discuss the Furstenberg boundary of
$\operatorname{SO}_{4}(\mathbb{C})$.

###  6.1. The Goncharov–Bucher–Burger–Iozzi volume cocycle

The content of this subsection is extracted entirely from the references [17,
3]. We abbreviate by $\mathbb{P}$ the complex projective line
$\mathbb{P}(\mathbb{C}^{2})$, isomorphic to the boundary of the hyperbolic
3-space $\mathbb{H}^{3}$. We let
$\mathrm{Vol}_{\mathbb{P}}:\mathbb{P}^{4}\to\mathbb{R}$ be the function that
assigns to a 4-tuple $(p_{0},p_{1},p_{2},p_{3})\in\mathbb{P}^{4}$ the oriented
volume of the ideal tetrahedron in $\mathbb{H}^{3}$ with vertices $p_{i}$. Up
to rescaling, $\mathrm{Vol}_{\mathbb{P}}$ is the unique alternating,
$\operatorname{GL}_{2}(\mathbb{C})$-invariant, continuous bounded cocycle
$\mathbb{P}^{4}\to\mathbb{R}$; see [1, 7].

We may write $\mathrm{Vol}_{\mathbb{P}}$ in terms of the Bloch–Wigner
dilogarithm $\mathcal{D}:\mathbb{C}\to\mathbb{R}$ (see [37, §3]) as

(6.1) $\mathrm{Vol}_{\mathbb{P}}=\mathcal{D}\,\circ\,\operatorname{cr}_{0},$

where $\operatorname{cr}_{0}$ is the classical cross-ratio on $\mathbb{P}$
(see subsection 3.2 and subsection 3.2). Under the identification
$\mathbb{P}\cong\hat{\mathbb{C}}$ from (3.4), the equality (6.1) corresponds
to $\mathrm{Vol}_{\mathbb{P}}(\infty,0,1,z)=\mathcal{D}(z).$ Thus, the cocycle
equation for $\mathrm{Vol}_{\mathbb{P}}$ implies that $\mathcal{D}$ satisfies
Spence–Abel functional equation (4.4), and the alternation of
$\mathrm{Vol}_{\mathbb{P}}$ translates into the symmetries

(6.2)
$\mathcal{D}(z)=\mathcal{D}\left(1-\frac{1}{z}\right)=\mathcal{D}\left(\frac{1}{1-z}\right)=-\mathcal{D}\left(\frac{1}{z}\right)=-\mathcal{D}(1-z)=-\mathcal{D}\left(\frac{-z}{1-z}\right).\vspace{3pt}$

A complete flag in $\mathbb{C}^{n}$ is a sequence
$F^{0}\\!\subset\\!F^{1}\\!\subset\\!\cdots\\!\subset\\!F^{n-1}\\!\subset\\!F^{n}$
of subspaces $F^{j}\subset\mathbb{C}^{n}$ such that $\dim(F^{j})=j$. The group
$\operatorname{SL}_{n}(\mathbb{C})$ acts transitively on the variety
$\mathbb{F}_{n}$ of complete flags in $\mathbb{C}^{n}$, and the stabilizer of
the standard flag $\\{0\\}\\!\subset\\!\langle
e_{1}\rangle\\!\subset\\!\cdots\\!\subset\\!\langle
e_{1},\ldots,e_{n-1}\rangle\\!\subset\\!\mathbb{C}^{n}$ is the upper-
triangular subgroup $P_{n}<\operatorname{SL}_{n}(\mathbb{C})$, a minimal
parabolic subgroup. This identifies $\mathbb{F}_{n}$ with the Furstenberg
boundary of $\operatorname{SL}_{n}(\mathbb{C})$. Now, a complete _affine_ flag
$(F,\vec{v})$ is a pair that consists of a complete flag $F$, and a tuple of
spanning vectors $\vec{v}=(v^{1},\ldots,v^{n})$ in the sense that

$F^{j}=F^{j-1}\oplus\langle v^{j}\rangle\quad\mbox{for
}j=1,\ldots,n.\vspace{-3pt}$

We write $\mathbb{F}_{n,\mathrm{aff}}$ for the set of affine flags, which
comes equipped with a projection
$\mathbb{F}_{n,\mathrm{aff}}\twoheadrightarrow\mathbb{F}_{n}$.

######  Definition .

Let $\sigma_{3}$ denote the collection of equivalence classes
$[V;(v^{0},\ldots,v^{3})]$, where $V$ is a $\mathbb{C}$-vector space,
$v^{0},\ldots,v^{3}\in V$ are vectors that span $V$. The equivalence
$[V,(v^{j})]\sim[W,(w^{j})]$ holds if there exists a linear isomorphism

$\phi:V\xrightarrow{\cong\ }W\mbox{ with }\ \phi(v^{j})=w^{j}\ \ \mbox{for all
}j\in\\{0,1,2,3\\}.$

Note that $\sigma_{3}$ is a set since every class $[V;(v^{j})]$ admits a
representative $[\mathbb{C}^{m};(\bar{v}^{j})]$ for $0\leq m\leq 4$.

Now, for any $J=(j_{0},\ldots,j_{3})\in\\{0,\ldots,n-1\\}^{4}$, we set
$T_{J}:\mathbb{F}_{\mathrm{aff}}(\mathbb{C}^{n})^{4}\to\sigma_{3}$ as

(6.3)
$T_{J}\big{(}(F_{0},\vec{v}_{0}),\ldots,(F_{3},\vec{v}_{3})\big{)}\coloneqq\left[\frac{\langle
F_{0}^{j_{0}+1},\ldots,F_{3}^{j_{3}+1}\rangle}{\langle
F_{0}^{j_{0}},\ldots,F_{3}^{j_{3}}\rangle};\left(v_{0}^{j_{0}+1},\ldots,v_{3}^{j_{3}+1}\right)\right],$

where each $(F_{i},\vec{v}_{i})\in\mathbb{F}_{n,\mathrm{aff}}$ is such that
$F_{i}=(F_{i}^{0}\subset\cdots\subset F_{i}^{n})$ and
$\vec{v}_{i}=(v_{i}^{1},\ldots,v_{i}^{n})$, and the vectors in the right-hand
side of (6.3) are regarded as classes modulo $\langle
F_{0}^{j_{0}},\ldots,F_{3}^{j_{3}}\rangle$.

######  Definition .

Let $n\geq 2$, and let $\mathrm{Vol}:\sigma_{3}\to\mathbb{R}$ be the map
defined as

$\mathrm{Vol}\left[\mathbb{C}^{m};(v^{0},\ldots,v^{3})\right]\coloneqq\begin{cases}\mathrm{Vol}_{\mathbb{P}}[v^{0},\ldots,v^{3}]&\mbox{if
}m=2\mbox{ and all }v^{i}\neq 0,\\\\[-3.0pt] 0&\mbox{otherwise}.\end{cases}$

We let $\tilde{B}_{n}^{J}\colon\mathbb{F}_{n,\mathrm{aff}}^{4}\to\mathbb{R}$
be the composition $\mathrm{Vol}\,\circ\,T_{J}$ for any
$J\in\\{0,\ldots,n-1\\}^{4}$. This function is independent of the choices of
spanning vectors, and thus, descends to a bounded Borel function
$B_{n}^{J}:\mathbb{F}_{n}^{4}\to\mathbb{R}$. Moreover, the sum

$B_{n}\coloneqq\sum_{J}B_{n}^{J}\vspace{-4pt}$

descends to an alternating, $\operatorname{GL}_{n}(\mathbb{C})$-invariant,
bounded strict Borel cocycle $B_{n}:\mathbb{F}_{n}^{4}\to\mathbb{R}$ (see [3,
Cor. 13]). Observe that $B_{2}=\mathrm{Vol}_{\mathbb{P}}$, so by analogy, we
will call $B_{n}$ the (Goncharov–Bucher–Burger–Iozzi) _volume cocycle_. The
_bounded Borel class_ $b_{n}\in{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))\cong{\rm H}_{\rm
b}^{3}(\operatorname{GL}_{n}(\mathbb{C}))$, mentioned in Theorem 5.1 above, is
defined as $b_{n}\coloneqq[B_{n}]$.

Whilst $B_{n}$ is not everywhere continuous, it is so in the following generic
locus.

######  Definition .

We say that a 4-tuple $(F_{0},F_{1},F_{2},F_{3})\in\mathbb{F}_{n}^{4}$ is in
_general position_ if

$\dim\langle F_{0}^{j_{0}},\ldots,F_{3}^{j_{3}}\rangle=j_{0}+\cdots+j_{3}$

as long as the right-hand side is at most $n$. We will denote by
$\mathbb{F}_{n}^{\\{4\\}}$ the Zariski-open (hence Hausdorff-open and co-null)
subset of $\mathbb{F}_{n}^{4}$ of 4-tuples in general position.

######  Lemma .

For any $n\geq 2$ and $J=(j_{0},\ldots,j_{3})\in\\{0,\ldots,n-1\\}^{4}$, the
restriction of $B_{n}^{J}$ to the open subset $\mathbb{F}_{n}^{\\{4\\}}$ is
continuous, and vanishes identically unless the equality

(6.4) $\|J\|_{1}=j_{0}+\ldots+j_{3}=n-2$

holds. In particular, the volume cocycle
$B_{n}:\mathbb{F}_{n}^{4}\to\mathbb{R}$ is continuous on
$\mathbb{F}_{n}^{\\{4\\}}$.

###### Proof.

See discussion preceding [3, Lem. 16]. The restriction
$B_{n}^{J}|_{\mathbb{F}_{n}^{\\{4\\}}}$ reduces to a hyperbolic volume
function $\mathrm{Vol}_{\mathbb{P}}$ when (6.4) holds, and vanishes otherwise.
This implies the continuity statement. Note that there are precisely
$n(n^{2}-1)/6$ tuples $J$ that fulfill (6.4). ∎

###  6.2. The Furstenberg boundary of $\operatorname{SO}_{4}(\mathbb{C})$

The content of this subsection is classical, and can be found e.g. in [36,
§I.7]. We continue to abbreviate
$\mathbb{P}\coloneqq\mathbb{P}(\mathbb{C}^{2})$. Observe first that
$\operatorname{SO}_{4}(\mathbb{C})$ acts on the variety of totally isotropic
_projective lines_ in $\mathbb{P}(\mathbb{C}^{4})$ (i.e. 2-dimensional linear
subspaces of $\mathbb{C}^{4}$) with two orbits, $O^{+}$ and $O^{-}$.
Respectively, these consist of elements

$\displaystyle l^{+}_{a}$
$\displaystyle\coloneqq\big{\\{}[1,-t,a,at]^{\top}\in\mathbb{P}(\mathbb{C}^{4})\
\mid t\in\hat{\mathbb{C}}\big{\\}}\in O^{+},$ $\displaystyle l^{-}_{a}$
$\displaystyle\coloneqq\big{\\{}[1,a,-t,at]^{\top}\in\mathbb{P}(\mathbb{C}^{4})\
\mid t\in\hat{\mathbb{C}}\big{\\}}\in O^{-},$

with $a\in\hat{\mathbb{C}}\cong\mathbb{P}$, given in coordinates with respect
to the basis $\mathscr{B}_{2}^{(+1,0)}=\\{e_{1},e_{2},f_{2},e_{1}\\}$. These
lines satisfy the properties

$\begin{cases}l^{+}_{a}\cap l_{b}^{+}\\!\\!\\!\\!&=l_{a}^{-}\cap
l_{b}^{-}=\emptyset\quad\mbox{for }a\neq b,\\\ l^{+}_{a}\cap
l_{b}^{-}\\!\\!\\!\\!&=[1,b,-a,ab]^{\top}.\end{cases}$

The $\operatorname{SO}_{4}(\mathbb{C})$-action on the product
$\mathcal{F}\coloneqq O^{+}\times O^{-}$ is transitive, and the stabilizer of
the pair $(l_{0}^{+},l_{0}^{-})\in\mathcal{F}$ equals the subgroup of upper-
triangular orthogonal matrices. Thus, we identify the Furstenberg boundary of
$\operatorname{SO}_{4}(\mathbb{C})$ with $\mathcal{F}$.

There exists an exceptional isogeny
$\theta:\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})\twoheadrightarrow\operatorname{SO}_{4}(\mathbb{C})$
that descends to an isomorphism of Lebesgue
$\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})$-spaces,

(6.5)
$\partial\theta:\mathbb{P}\times\mathbb{P}\to\mathcal{F},\quad\partial\theta(a,b)\coloneqq(l_{a}^{+},l_{b}^{-}),\
\ a,b\in\hat{\mathbb{C}}.$

The following embedding of the Furstenberg boundary $\mathcal{F}$ into the
full flag variety of $\mathbb{C}^{4}$ plays a key role in our proof of section
6.

######  Definition .

Let $\rho:\mathcal{F}\to\mathbb{F}_{4}$ be the
$\operatorname{SO}_{4}(\mathbb{C})$-equivariant embedding defined as

$\displaystyle\rho(l^{+},l^{-})$ $\displaystyle\coloneqq\big{(}\\{0\\}\subset\
l^{+}\cap l^{-}\ \subset\ l^{+}\ \subset l^{+}+l^{-}\ \subset\
\mathbb{C}^{4}\big{)}.$

An analogous map can be obtained if we choose $l^{-}$ instead of $l^{+}$ as
the 2-dimensional subspace.

######  Remark .

The image of $\rho$ is a null set. In particular, $\rho$ does not preserve
measure classes.

###  6.3. Generic values of the volume cocycle of
$\operatorname{SO}_{4}(\mathbb{C})$

We introduce a notion of genericity for tuples of points in the Furstenberg
boundary $\mathcal{F}$.

######  Definition .

For any $k\in\mathbb{N}$, we set

$\displaystyle\mathcal{F}^{\\{k+1\\}}$
$\displaystyle\coloneqq\partial\theta^{\,\times(k+1)}\big{(}\mathbb{P}^{\\{k+1\\}}\times\mathbb{P}^{\\{k+1\\}}\big{)}\subset\mathcal{F}^{k+1},$

where the brackets in the upper index of $\mathbb{P}$ refer to the generic
subsets (3.2) in the classical setting. The product
$\mathbb{P}^{\\{k+1\\}}\times\mathbb{P}^{\\{k+1\\}}$ is regarded as a subset
of $(\mathbb{P}\times\mathbb{P})^{k+1}$ in the obvious way.

######  Lemma .

The following statements hold:

  1. _(i)_

The set $\mathcal{F}^{\\{k+1\\}}$ is an
$\operatorname{SO}_{4}(\mathbb{C})$-invariant, co-null Borel subset of
$\mathcal{F}^{k+1}$, and the equality
$\partial_{i}(\mathcal{F}^{\\{k+1\\}})=\mathcal{F}^{\\{k\\}}$ holds for every
$i\in\\{0,\ldots,k+1\\}$.

  2. _(ii)_

The action
$\operatorname{SO}_{4}(\mathbb{C})\curvearrowright\mathcal{F}^{\\{3\\}}$ is
transitive.

  3. _(iii)_

The image $\rho^{\,\times 4}(\mathcal{F}^{\\{4\\}})$ is contained in the set
$\mathbb{F}_{4}^{\\{4\\}}$ of 4-tuples in general position.

###### Proof.

Item (i) follows from the fact that $\partial\theta$ is an isomorphism of
Lebesgue
$\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})$-spaces,
and $\mathbb{P}^{\\{k+1\\}}\times\mathbb{P}^{\\{k+1\\}}$ is invariant, Borel
and co-null in $(\mathbb{P}\times\mathbb{P})^{k+1}$. Item (ii) is a
consequence of the transitivity of the action
$\operatorname{SL}_{2}(\mathbb{C})\curvearrowright\mathbb{P}^{\\{3\\}}$. For
(iii), we let $\mathbf{a}=(a_{0},\ldots,a_{3})$ and
$\mathbf{b}=(b_{0},\ldots,b_{3})\in\mathbb{P}^{\\{4\\}}$. The genericity of
these tuples means that their entries are pairwise distinct when regarded
under the identification $\mathbb{P}\cong\hat{\mathbb{C}}$. Set
$l_{i}^{+}\coloneqq l_{a_{i}}^{+}$ and $l_{i}^{-}\coloneqq l_{b_{i}}^{-}$, so
that

$\big{(}(l_{0}^{+},l_{0}^{-}),\ldots,(l_{3}^{+},l_{3}^{-})\big{)}=(\partial\theta_{0})^{4}(\mathbf{a},\mathbf{b})\in\mathcal{F}^{4}.$

We claim that the image under $\rho^{\,\times 4}$ of the tuple
$((l_{0}^{+},l_{0}^{-}),\ldots,(l_{3}^{+},l_{3}^{-}))$ is in general position.
Indeed, if we set $p_{i}\coloneqq l_{i}^{+}\cap l_{i}^{-}\in\mathcal{P}_{2}$,
then, for instance, for distinct $i,j\in\\{0,1,2,3\\}$, we have

$\displaystyle\dim(p_{i}+p_{j})$ $\displaystyle=1+1-\dim(l_{i}^{+}\cap
l_{i}^{-}\cap l_{j}^{+}\cap l_{j}^{-})=2,$
$\displaystyle\dim(p_{i}+l_{j}^{+})$ $\displaystyle=1+2-\dim(l_{i}^{+}\cap
l_{i}^{-}\cap l_{j}^{+})=3,$ $\displaystyle\dim(p_{i}+(l_{j}^{+}+l_{j}^{-}))$
$\displaystyle=1+3-\dim(l_{i}^{+}\cap l_{i}^{-}\cap(l_{j}^{+}+l_{j}^{-}))=4,$

where we used that $l_{i}^{+}\cap l_{j}^{+}=l_{i}^{-}\cap l_{j}^{-}=\\{0\\}$
and

$l_{i}^{+}\cap l_{i}^{-}\cap(l_{j}^{+}+l_{j}^{-})\subset(l_{i}^{+}\cap
l_{i}^{-}\cap l_{j}^{+})+(l_{i}^{+}\cap l_{i}^{-}\cap l_{j}^{-})=\\{0\\}.$

The remaining verifications are completely analogous to the three given above.
∎

In the following lemma, we examine the values of the volume cocycle $B_{n}$
over the image of the generic set $\mathcal{F}^{\\{4\\}}$ via the inclusion
$\rho^{\,\times 4}:\mathcal{F}^{4}\to\mathbb{F}_{4}^{4}$. For simplicity, in
its proof we will write 4-tuples of indices $(j_{0},\ldots,j_{3})$ as
$j_{0}\cdots j_{3}$, omitting parentheses and commas.

######  Lemma .

For any $(a,b)\in\hat{\mathbb{C}}^{2}$, let
$F_{(a,b)}\coloneqq\rho\big{(}\partial\theta(a,b)\big{)}$ and $F_{a}\coloneqq
F_{(a,a)}$. Then for all $(a,b)\in(\mathbb{C}\smallsetminus\\{0,1\\})^{2}$,
the equality

$B_{4}(F_{\infty},F_{0},F_{1},F_{(a,b)})=2\left(\mathcal{D}(a)+\mathcal{D}(b)\right)$

holds. In particular, the map $B_{4}\,\circ\,\rho^{\,\times
4}|_{\mathcal{F}^{\\{4\\}}}:\mathcal{F}^{\\{4\\}}\to\mathbb{R}$ is non-zero.

###### Proof.

We claim the following values for the non-vanishing summands $B_{4}^{J}$ of
$B_{4}$ (see subsection 6.1):

(6.6)
$B_{4}^{J}(F_{\infty},F_{0},F_{1},F_{(a,b)})=\left\\{\begin{array}[]{ll}\phantom{-}\mathcal{D}(b)&\mbox{
if }J\in\\{2000,0200,0020,0002\\},\\\ -\mathcal{D}(b/a)&\mbox{ if
}J\in\\{1100,0011\\},\\\
\phantom{-}\mathcal{D}\left(\frac{1-b}{1-a}\right)&\mbox{ if
}J\in\\{1010,0101\\},\\\\[4.0pt]
-\mathcal{D}\left(\frac{a(1-b)}{b(1-a)}\right)&\mbox{ if
}J\in\\{1001,0110\\}.\\\ \end{array}\right.$

First, if we let $\phi_{ab}\coloneqq e_{1}+b\,e_{2}-a\,f_{2}+ab\,f_{1}$ for
any $a,b\in\hat{\mathbb{C}}$, then

$l_{a}^{+}=\langle e_{1}-a\,f_{2},\ e_{2}+a\,f_{1}\rangle,\quad
l_{b}^{-}=\langle e_{1}+b\,e_{2},\
b\,f_{1}-f_{2}\rangle,\quad\mathrm{and}\quad l_{a}^{+}\cap
l_{b}^{-}=\langle\phi_{ab}\rangle,$

and therefore, for $a,b\in\mathbb{C}\smallsetminus\\{0,1\\}$,

$\displaystyle F_{\infty}$ $\displaystyle=\big{(}\\{0\\}\subset\langle
f_{1}\rangle\subset\langle f_{1},f_{2}\rangle\subset\langle
f_{1},f_{2},e_{2}\rangle\subset\mathbb{C}^{4}\big{)},$ $\displaystyle F_{0}$
$\displaystyle=\big{(}\\{0\\}\subset\langle e_{1}\rangle\subset\langle
e_{1},e_{2}\rangle\subset\langle
e_{1},e_{2},f_{2}\rangle\subset\mathbb{C}^{4}\big{)},$ $\displaystyle
F_{(a,b)}$
$\displaystyle=\big{(}\\{0\\}\subset\langle\phi_{ab}\rangle\subset\langle\phi_{ab},e_{1}-a\,f_{2}\rangle\subset\langle\phi_{ab},e_{1}-a\,f_{2},e_{1}+b\,e_{2}\rangle\subset\mathbb{C}^{4}\big{)}.$

Now, we have:

$\displaystyle B_{4}^{2000}(F_{\infty},F_{0},F_{1},F_{(a,b)})$
$\displaystyle=\mathrm{Vol}\ \bigg{[}\frac{\langle
f_{1},f_{2},e_{2}\rangle+\langle
e_{1}\rangle+\langle\phi_{11}\rangle+\langle\phi_{ab}\rangle}{\langle
f_{1},f_{2}\rangle+\\{0\\}+\\{0\\}+\\{0\\}};\big{(}e_{2},e_{1},\phi_{11},\phi_{ab}\big{)}\bigg{]}$
$\displaystyle=\mathrm{Vol}\ \big{[}V^{(+1,0)}_{2}/\langle
f_{1},f_{2}\rangle;\big{(}e_{2},e_{1},\phi_{11},\phi_{ab}\big{)}\big{]}$
$\displaystyle\overset{(\ast)}{=}\mathrm{Vol}\left[\mathbb{C}^{2};\begin{pmatrix}0&1&1&1\\\\[-3.0pt]
1&0&1&b\end{pmatrix}\right]$
$\displaystyle=\mathrm{Vol}_{\mathbb{P}}(\infty,0,1,b)=\mathcal{D}(b),\quad\mathrm{and}\quad$
$\displaystyle B_{4}^{0011}$
$\displaystyle(F_{\infty},F_{0},F_{1},F_{(a,b)})=$
$\displaystyle=\mathrm{Vol}\ \bigg{[}\frac{\langle f_{1}\rangle+\langle
e_{1}\rangle+\langle\phi_{11},e_{1}\\!-\\!f_{2}\rangle+\langle\phi_{ab},e_{1}\\!-\\!af_{2}\rangle}{\\{0\\}+\\{0\\}+\langle\phi_{11}\rangle+\langle\phi_{ab}\rangle};\big{(}f_{1},e_{1},e_{1}-f_{2},e_{1}-af_{2}\big{)}\bigg{]}$
$\displaystyle=\mathrm{Vol}\
\big{[}V_{2}/\langle\phi_{11},\phi_{ab}\rangle;\big{(}f_{1},e_{1},e_{1}-f_{2},e_{1}-af_{2}\big{)}\big{]}$
$\displaystyle\\!\overset{(\ast\ast)}{=}\\!\mathrm{Vol}\left[\mathbb{C}^{2};\begin{pmatrix}0&\frac{a-b}{a-1}&1&b\\\\[-2.0pt]
-b^{-1}&0&1&a\end{pmatrix}\right]$
$\displaystyle=\mathrm{Vol}_{\mathbb{P}}(\infty,0,1,a/b)=\mathcal{D}(a/b)=-\mathcal{D}(b/a).$

The last equality in the case $J=0011$ follows from one of the symmetries in
(6.2). The equalities $(\ast)$ and $(\ast\ast)$ follow from the notion of
equivalence in $\sigma_{3}$, implemented respectively by the epimorphisms
$T_{1},T_{2}:V_{2}\twoheadrightarrow\mathbb{C}^{2}$ defined as:

$\displaystyle T_{1}$
$\displaystyle:e_{1}\mapsto\big{(}1,0\big{)}^{\top},\qquad\\!e_{2}\mapsto\big{(}0,1\big{)}^{\top},\qquad
f_{2}\mapsto\big{(}0,0\big{)}^{\top},\qquad\
f_{1}\mapsto\big{(}0,0\big{)}^{\top};$ $\displaystyle T_{2}$
$\displaystyle:e_{1}\mapsto\left({\textstyle\frac{\scriptstyle
a-b}{\scriptstyle
a-1}},0\right)^{\top},\quad\\!\\!\\!e_{2}\mapsto\left(-1,{\textstyle\frac{\scriptstyle
1-b}{\scriptstyle
b}}\right)^{\top},\,f_{2}\mapsto\left({\textstyle\frac{\scriptstyle
1-b}{\scriptstyle a-1}},-1\right)^{\top},\
f_{1}\mapsto\left(0,-b^{-1}\right)^{\top}.$

Note that $\ker T_{1}=\langle f_{1},f_{2}\rangle$ and $\ker
T_{2}=\langle\phi_{11},\phi_{ab}\rangle$. To conclude, we add the terms in
(6.6) and use the fact that $\mathcal{D}$ satisfies the Spence–Abel equation
(4.4):

$\displaystyle B_{4}(F_{\infty},F_{0},F_{1},F_{(a,b)})$
$\displaystyle=4\,\mathcal{D}(b)+2\left(-\mathcal{D}\\!\left({\textstyle\frac{a(1-b)}{b(1-a)}}\right)+\mathcal{D}\\!\left({\textstyle\frac{1-b}{1-a}}\right)-\mathcal{D}\\!\left({\textstyle\frac{b}{a}}\right)\right)$
$\displaystyle=4\,\mathcal{D}(b)+2\big{(}\mathcal{D}(a)-\mathcal{D}(b)\big{)}=2\big{(}\mathcal{D}(a)+\mathcal{D}(b)\big{)}.\qed$

###  6.4. Proof of section 6

If $\rho:\mathcal{F}\to\mathbb{F}_{4}$ induced a morphism of
$L^{\infty}$-complexes, then it would, by virtue of functoriality [28, Prop.
8.4.2], implement the map $\operatorname{res}_{2}$ from section 6 at the
cocycle level, and the injectivity of $\operatorname{res}_{2}$ would be a
consequence of subsection 6.3. However, since $\rho$ does not preserve measure
classes, it cannot possibly induce a map of $L^{\infty}$-spaces.

To circumvent this measure-theoretical difficulty, we will consider the
commutative diagram

(6.7) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
28.24171pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
43.93477pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8625pt\hbox{$\scriptstyle{\operatorname{res}_{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 77.48615pt\raise 0.0pt\hbox{\hbox{\kern
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where the upper indices $(-)^{\delta}$ on groups indicate that they are being
regarded as discrete. In words, the map $\operatorname{res}_{2}$ “factors”
over its discrete analogue $\operatorname{res}_{2}^{\delta}$. This observation
proves advantageous for us, as the latter map is more tractable than the
former at the level of cocycles.

As actions of discrete groups,
$\operatorname{SL}_{4}(\mathbb{C})^{\delta}\curvearrowright\mathbb{F}_{4}$ and
$\operatorname{SO}_{4}(\mathbb{C})^{\delta}\curvearrowright\mathcal{F}$ are
amenable. In fact, their point-stabilizers are isomorphic to upper-triangular
subgroups, and hence abstractly solvable. Thus, we have canonical isomorphisms

${\rm H}_{\rm
b}^{\ast}(\operatorname{SL}_{4}(\mathbb{C})^{\delta})\cong\operatorname{H}^{\ast}(\ell^{\infty}(\mathbb{F}_{n}^{\bullet+1})^{\operatorname{SL}_{4}(\mathbb{C})})\quad\mathrm{and}\quad{\rm
H}_{\rm
b}^{\ast}(\operatorname{SO}_{4}(\mathbb{C})^{\delta})\cong\operatorname{H}^{\ast}(\ell^{\infty}(\mathcal{F}^{\bullet+1})^{\operatorname{SO}_{4}(\mathbb{C})}).$

Moreover, the now well-defined morphism of complexes

(6.8)
$\rho^{\ast}:\ell^{\infty}(\mathbb{F}_{4}^{\bullet+1})\to\ell^{\infty}(\mathcal{F}^{\bullet+1})\vspace{-3pt}$

implements the restriction $\operatorname{res}_{2}^{\delta}:{\rm H}_{\rm
b}^{\ast}(\operatorname{SL}_{4}(\mathbb{C})^{\delta})\to{\rm H}_{\rm
b}^{\ast}(\operatorname{SO}_{4}(\mathbb{C})^{\delta})$ (see [14, Thms. 4.23,
4.15]).

Our last ingredient in the proof of section 6 is an “equivariant lifting”
theorem, due to Monod [29], that will enable the realization at the level of
cocycles of the maps $\operatorname{id}^{\ast}$ in the diagram (6.7). For any
measure space $X$, we denote the Banach space of bounded measurable functions
$X\to\mathbb{R}$ (without identification modulo null sets) by
$\mathscr{L}^{\infty}(X)$. If $\varphi\in\mathscr{L}^{\infty}(X)$, we write
$[\varphi]$ for its equivalence class in $L^{\infty}(X)$.

######  Theorem 6.1 (see [29]).

Let $G$ be a locally compact group with an amenable $C^{1}$-action on a
differentiable manifold $X$. Then for every $k\in\mathbb{N}$, there exists a
$G$-equivariant lifting

$s^{k}:L^{\infty}(X^{k+1})\to\mathscr{L}^{\infty}(X^{k+1})\vspace{-3pt}$

of the canonical projection $\mathscr{L}^{\infty}(X^{k+1})\twoheadrightarrow
L^{\infty}(X^{k+1})$ such that:

  1. _(i)_

$s^{\bullet}$ is a morphism of complexes, and

  2. _(ii)_

if $x\in X^{k+1}$ is a continuity point of a function
$\varphi\in\mathscr{L}^{\infty}(X^{k+1})$, then $(s[\varphi])(x)=\varphi(x)$.

###### About the proof.

The existence of the $s^{k}$ is [29, Thm. A]. For (i), see [29, Rem. 1]. For
(ii), see the comment at the end of the proof of [29, Thm. B]. ∎

Let us fix a morphism of complexes $s$ that results from the composition of an
equivariant lifting as in Theorem 6.1 and the surjection
$\mathscr{L}^{\infty}\twoheadrightarrow\ell^{\infty}$:

(6.9) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
17.54724pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
29.54724pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\ell^{\infty}(\mathbb{F}_{4}^{\bullet+1})}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}\vspace{-6pt}$

Again by functoriality, the morphism $s$ implements
$\operatorname{id}^{\ast}:{\rm H}_{\rm
b}^{\ast}(\operatorname{SL}_{4}(\mathbb{C}))\to{\rm H}_{\rm
b}^{\ast}(\operatorname{SL}_{4}(\mathbb{C})^{\delta})$ at the level of
cocycles. Observe that $s[B_{4}]=B_{4}$ on $\mathbb{F}_{4}^{\\{4\\}}$ by
subsection 6.1.

###### Proof of section 6.

Since ${\rm H}_{\rm b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\cong\mathbb{R}$,
it suffices to prove that the composition
$\operatorname{res}^{\delta}\circ\operatorname{id}^{\ast}$ in the diagram
(6.7) is not the zero map. This will follow from exhibiting a non-trivial
3-cocycle in the image of $\rho^{\ast}\circ s$, for $\rho^{\ast}$ and $s$ as
introduced in (6.8) and (6.9), respectively.

Let $[B_{4}]$ be the class up to null sets of the volume cocycle
$B_{4}\in\mathscr{L}^{\infty}(\mathbb{F}_{4}^{4})$, and set
$\mathcal{B}_{2}\coloneqq(\rho^{\ast}\circ
s)[B_{4}]\in\ell^{\infty}(\mathcal{F}^{4})$. Observe that $\mathcal{B}_{2}$ is
an $\operatorname{SO}_{4}(\mathbb{C})$-invariant bounded 3-cocycle. Moreover,

$\mathcal{B}_{2}(x)=s[B_{4}]\big{(}\rho^{\,\times
4}(x)\big{)}=B_{4}\big{(}\rho^{\,\times 4}(x)\big{)}\quad\mbox{for all
}x\in\mathcal{F}^{\\{4\\}},$

in virtue of subsection 6.1 and subsection 6.3 (iii). Thus, we deduce from
subsection 6.3 that $\mathcal{B}_{2}$ does not vanish identically on
$\mathcal{F}^{\\{4\\}}$, and this implies that $\mathcal{B}_{2}$ is not a
coboundary. Indeed, by subsection 6.3 (ii), any
$\operatorname{SO}_{4}(\mathbb{C})$-invariant function
$\varphi\in\ell^{\infty}(\mathcal{F}^{3})$ must take a constant value $c$ on
$\mathcal{F}^{\\{3\\}}$, and therefore, for every $x\in\mathcal{F}^{\\{4\\}}$,
any coboundary $\operatorname{d}\\!\varphi$ must satisfy the equality

$\operatorname{d}\\!\varphi(x)={\textstyle\sum_{i=0}^{3}}(-1)^{i}\,\varphi(\partial_{i}(x))=c-c+c-c=0.\qed$

We finish this subsection by computing the norm of the restricted class
$\operatorname{res}_{2}(b_{4})\in{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{4}(\mathbb{C}))$. In particular, this proves that
the restrictions $\operatorname{res}_{r}$ need not be isometric in the
$\mathbf{D}_{r}$ case.

######  Proposition .

The restricted class $\operatorname{res}_{2}(b_{4})\in{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{4}(\mathbb{C}))$ has Gromov norm equal to
$4\,v_{\mathbb{H}^{3}}$, and the map $\operatorname{res}_{2}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\hookrightarrow{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{4}(\mathbb{C}))$ has operator norm equal to $2/5$.

###### Proof.

Let $\mathcal{B}_{2}\in\ell^{\infty}(\mathcal{F}^{4})$ be the cocycle defined
in the proof of section 6. Since $\mathcal{B}_{2}$ is the restriction of a
Borel cocycle, its class $[\mathcal{B}_{2}]\in L^{\infty}(\mathcal{F}^{4})$ is
well-defined. Now, as in (6.9), let

$s^{\prime}:L^{\infty}(\mathcal{F}^{\bullet+1})\to\ell^{\infty}(\mathcal{F}^{\bullet+1})\vspace{-3pt}$

be the $\operatorname{SO}_{4}(\mathbb{C})$-equivariant morphism of complexes
defined by the composition of a lifting as in Theorem 6.1 and the surjection
$\mathscr{L}^{\infty}\twoheadrightarrow\ell^{\infty}$. Moreover, the equality
$s^{\prime}[\mathcal{B}_{2}]=\mathcal{B}_{2}$ holds since $\mathcal{B}_{2}$ is
continuous on the co-null set $\mathcal{F}^{\\{4\\}}\subset\mathcal{F}^{4}$.
By [28, Thm. 7.5.3] and the commutativity of the diagram (6.7), we deduce that
the norm of $\operatorname{res}_{2}(b_{4})$ must equal the norm of the
cohomology class defined by $[\mathcal{B}_{2}]$.

Now, by the essential transitivity of the action
$\operatorname{SO}_{4}(\mathbb{C})\curvearrowright\mathcal{F}^{3}$ (see
subsection 6.3 (ii)), there are no non-trivial invariant 3-coboundaries in
$L^{\infty}(\mathcal{F}^{\bullet+1})$. Hence,
$\|\operatorname{res}_{2}(b_{4})\|=\|[\mathcal{B}_{2}]\|_{\infty}=4\cdot
v_{3}$ by subsection 6.3. For the operator norm of $\operatorname{res}_{2}$,
note simply that $\|b_{4}\|=10\cdot v_{3}$. ∎

###  6.5. End of proof of Theorems 2 and 3

For any $r\geq r_{1}=2$, consider the commutative cube

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where the horizontal arrows correspond to the restriction maps

$\operatorname{res}_{s}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n(s)}(\mathbb{C}))\to{\rm H}_{\rm
b}^{3}(S_{s})\quad\mathrm{and}\quad{\rm H}_{\rm
b}^{3}(\operatorname{GL}_{n(s)}(\mathbb{C}))\to{\rm H}_{\rm
b}^{3}(G_{s}),\vspace{-2pt}$

the arrows pointing frontwards to the restriction maps

$\sigma_{n(s)}:{\rm H}_{\rm
b}^{3}(\operatorname{GL}_{n(s)}(\mathbb{C}))\to{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n(s)}(\mathbb{C}))\quad\mathrm{and}\quad\tau_{s}:{\rm
H}_{\rm b}^{3}(G_{s})\to{\rm H}_{\rm b}^{3}(S_{s})\vspace{-2pt}$

for $s\in\\{r,r+1\\}$, and the vertical arrows to the maps induced by the
block inclusions

$\displaystyle\iota_{r}:S_{r}\hookrightarrow S_{r+1},\quad$
$\displaystyle\quad
j_{r}:\operatorname{SL}_{n(r)}(\mathbb{C})\hookrightarrow\operatorname{SL}_{n(r+1)}(\mathbb{C}),$
$\displaystyle\iota_{r}:\,G_{r}\hookrightarrow G_{r+1},\quad$
$\displaystyle\quad
j_{r}:\operatorname{GL}_{n(r)}(\mathbb{C})\hookrightarrow\operatorname{GL}_{n(r+1)}(\mathbb{C}),$

Observe that $\sigma_{n(s)}$ is an isomorphism by [28, Cor. 8.5.5] and that
$\tau_{s}$ is an injection by [28, Prop. 8.6.2] for both $s\in\\{r,r+1\\}$,
since $S_{s}<G_{s}$ is a finite-index normal subgroup. Moreover, the maps
$j_{r}^{\ast}$ induced in degree three by $j_{r}$, both for
$\operatorname{GL}$ and $\operatorname{SL}$, are isomorphisms due to Theorem 2
for the $\mathbf{A}_{r}$ case. By Key Lemma _bis_, the map
$\iota_{r}^{\ast}:{\rm H}_{\rm b}^{3}(G_{r+1})\to{\rm H}_{\rm b}^{3}(G_{r})$
is an injection.

The next lemma will serve as the induction step in the proofs of Theorems 2
_bis_ and 3 _bis_.

######  Lemma .

If, for any $r\geq 1$, the map $\operatorname{res}_{r}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n(r)}(\mathbb{C}))\to{\rm H}_{\rm b}^{3}(S_{r})$ is
an isomorphism, then all the arrows in the diagram (6.10) are isomorphisms.

###### Proof.

Note, by examining the bottom face of the cube, that if
$\operatorname{res}_{r}$ is an isomorphism, then $\tau_{r}$ is surjective.
This, in turn, implies that all the arrows in the bottom face are
isomorphisms. Relying on these isomorphisms, a similar analysis of, first, the
back face; then, the left face; and finally, the front face of (6.10),
complete the proof of the lemma. ∎

###### Proof of Theorem 3 _bis_ for $\mathbf{D}_{r}$.

We prove by induction that $\operatorname{res}_{r}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n(r)}(\mathbb{C}))\to{\rm H}_{\rm b}^{3}(S_{r})$ is
an isomorphism for every $r\geq r_{0}=3$. By subsection 6.5, if
$\operatorname{res}_{r}$ is an isomorphism for any $r\geq 3$, then so is
$\operatorname{res}_{r+1}$. Thus, the claim follows once we establish that

$\operatorname{res}_{r_{0}}=\operatorname{res}_{3}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{6}(\mathbb{C}))\to{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{6}(\mathbb{C}))$

is an isomorphism. The group $\operatorname{SO}_{6}(\mathbb{C})$ admits an
exceptional isogeny
$\operatorname{SL}_{4}(\mathbb{C})\twoheadrightarrow\operatorname{SO}_{6}(\mathbb{C})$,
as in e.g. [36, 34], which induces an isomorphism in bounded cohomology [28,
Cor. 8.5.2]. After applying Theorem 2 for $\mathbf{A}_{r}$, we deduce the
chain of isomorphisms

$\displaystyle{\rm H}_{\rm b}^{3}(\operatorname{SO}_{6}(\mathbb{C}))$
$\displaystyle\cong{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\cong{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{6}(\mathbb{C}))\cong\mathbb{R}.$

Thus, it suffices to prove that $\operatorname{res}_{3}$ is not identically
zero. For this purpose, consider the front face of the commutative cube
(6.10), with $r=2$. Since the bottom arrow $\operatorname{res}_{2}$ is
injective by section 6, we deduce our claim. ∎

###### Proof of Theorem 2 _bis_ for $\mathbf{D}_{r}$.

By subsection 6.5, if $\operatorname{res}_{r}$ is an isomorphism for any
$r\geq r_{0}=3$, then so is the map $\iota_{r}^{\ast}:{\rm H}_{\rm
b}^{3}(S_{r+1})\to{\rm H}_{\rm b}^{3}(S_{r})$. Theorem 3 _bis_ finishes the
proof. ∎

######  Remark .

We note that $\operatorname{res}_{2}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\hookrightarrow{\rm H}_{\rm
b}^{3}(\operatorname{SO}_{4}(\mathbb{C}))$ is not an isomorphism, since ${\rm
H}_{\rm b}^{3}(\operatorname{SL}_{4}(\mathbb{C}))\cong\mathbb{R}$, and
$\operatorname{SO}_{4}(\mathbb{C})$ is isogenous to
$\operatorname{SL}_{2}(\mathbb{C})\times\operatorname{SL}_{2}(\mathbb{C})$,
which has 2-dimensional ${\rm H}_{\rm b}^{3}$.

##  7\. Proof of Theorem 1

Let $(\varepsilon,d)\in\\{(+1,1),(-1,0),(+1,0)\\}$. We will prove Theorem 1
_bis_, that is, that the comparison map $c^{3}:{\rm H}_{\rm
b}^{3}(S_{r})\to\operatorname{H}^{3}(S_{r})$ is an isomorphism for all $r\geq
r_{0}$, by induction on $r$.

The base case $r=r_{0}$ holds due to the exceptional Lie algebra isomorphisms
in low rank, which place $S_{r_{0}}$ inside the $\mathbf{A}_{r}$ family. For
the induction step, consider the commutative diagram

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.46742pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm H}_{\rm
b}^{3}(S_{r+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.54167pt\hbox{$\scriptstyle{\iota_{r}^{\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-15.76886pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-19.89996pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.95279pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15138pt\hbox{$\scriptstyle{c^{3}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 80.46742pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
80.46742pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{H}^{3}(S_{r+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
100.7793pt\raise-15.76886pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.54167pt\hbox{$\scriptstyle{\iota_{r}^{\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
89.5293pt\raise-15.76886pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
100.7793pt\raise-19.89996pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
H}_{\rm
b}^{3}(S_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.87502pt\raise-36.6891pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15138pt\hbox{$\scriptstyle{c^{3}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.76465pt\raise-25.38773pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\cong}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
84.04521pt\raise-31.53772pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
84.04521pt\raise-31.53772pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{H}^{3}(S_{r})}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

The induction hypothesis is the fact that the bottom arrow an isomorphism.
That the left arrow is an isomorphism is Theorem 2. That the right one is an
isomorphism is the stability of classical families in continuous cohomology,
for which we argue as follows. For any non-compact semisimple Lie group $G$,
the continuous cohomology $\operatorname{H}^{\ast}(G)$ is isomorphic to the
cohomology of its compact symmetric space (see e.g. [33, §5]). In the cases
covered by Theorem 1, such compact symmetric spaces are diffeomorphic to
compact classical groups. The stability of the (space) cohomology of those
groups is recorded in [26, Thm. III.6.5]. ∎

##  8\. About the Gromov norm on ${\rm H}_{\rm b}^{3}$ of complex classical
groups

For the purpose of this final section, let us adapt the notation introduced in
Section 2 to cover also the complex family $\mathbf{A}_{r}$, as follows. For
$\mathscr{F}\in\\{\mathbf{A,\,B,\,C,\,D}\\}$, let $S_{\mathscr{F}_{r}}$
denote, respectively, the group

$\operatorname{SL}_{r+1}(\mathbb{C}),\quad\operatorname{SO}_{2r+1}(\mathbb{C}),\quad\operatorname{Sp}_{2r}(\mathbb{C}),\quad\operatorname{SO}_{2r}(\mathbb{C}),$

in the complex family $\mathscr{F}_{r}$. As explained in Subsection 2.2, the
choice of $\mathscr{F}\in\\{\mathbf{B,\,C,\,D}\\}$ corresponds, respectively,
to choosing $(\varepsilon,d)\in\\{(+1,1),(-1,0),(+1,0)\\}$, and we will assume
the notation introduced in Section 2 for those cases.

Let $b_{\mathscr{F}_{r}}\in{\rm H}_{\rm b}^{3}(S_{\mathscr{F}_{r}})$ be the
continuous bounded cohomology class defined as

$b_{\mathscr{F}_{r}}\coloneqq\begin{cases}b_{r+1}&\mbox{ if
}\mathcal{F}=\mathbf{A},\\\\[-2.0pt]
\frac{1}{2}\,\operatorname{res}_{r}(b_{2r+1})&\mbox{ if
}\mathcal{F}=\mathbf{B},\\\ \operatorname{res}_{r}(b_{2r})&\mbox{ if
}\mathcal{F}=\mathbf{C},\\\ \frac{1}{2}\,\operatorname{res}_{r}(b_{2r})&\mbox{
if }\mathcal{F}=\mathbf{D},\\\ \end{cases}$

where $b_{n}\in{\rm H}_{\rm b}^{3}(\operatorname{SL}_{n}(\mathbb{C}))$ is the
bounded Borel class, and $\operatorname{res}_{r}:{\rm H}_{\rm
b}^{3}(\operatorname{SL}_{n(r)}(\mathbb{C}))\to{\rm H}_{\rm
b}^{3}(S_{\mathscr{F}_{r}})$ is the restriction map.

The next statement is an immediate corollary of Bucher–Burger–Iozzi [3, Thm.
2] (see Theorem 5.1), which gives the computation of the Gromov norm of the
bounded Borel class, and of subsection 1.4.

######  Corollary .

For $\mathscr{F}\in\\{\mathbf{A,\,B,\,C}\\}$ and $r\geq 1$, the Gromov norm of
$b_{\mathscr{F}_{r}}\in{\rm H}_{\rm b}^{3}(S_{\mathscr{F}_{r}})$ is

$\|b_{\mathscr{F}_{r}}\|=\mathcal{I}(\mathscr{F}_{r})\cdot
v_{\mathbb{H}^{3}},$

where $v_{\mathbb{H}^{3}}$ is the maximal volume of an ideal tetrahedron in
$\mathbb{H}^{3}$, and the number $\mathcal{I}(\mathscr{F}_{r})\in\mathbb{Z}$
is given in the table

$\begin{array}[]{|c||c|c|c|c|}\hline\cr\mathscr{F}_{r}&\mathbf{A}_{r}&\mathbf{B}_{r}&\mathbf{C}_{r}\\\
\hline\cr\mathcal{I}(\mathscr{F}_{r})&\frac{r(r+1)(r+2)}{6}&\frac{r(r+1)(2r+1)}{3}&\frac{r(4r^{2}-1)}{3}\\\
\hline\cr\end{array}$

It was indicated to us by Marc Burger that the number
$\mathcal{I}(\mathscr{F}_{r})$ equals the Dynkin index of the principal
$\operatorname{SL}_{2}(\mathbb{C})$-homomorphism
$\operatorname{SL}_{2}(\mathbb{C})\to S_{\mathcal{F}_{r}}$. We refer the
reader to [31] for the definition of Dynkin index, and [24] for the theory of
principal $\operatorname{SL}_{2}(\mathbb{C})$-homomorphisms. The Dynkin index
of the principal $\operatorname{SL}_{2}(\mathbb{C})$-homomorphisms has been
computed for all complex simple Lie groups [32], including exceptional ones.
For $\mathcal{F}=\mathbf{D}$, it is given by the formula

$\mathcal{I}(\mathbf{D}_{r})=\frac{r(r-1)(2r-1)}{3}.$

Relying on the equality
$\|b_{\mathbf{D}_{2}}\|=2\,v_{\mathbb{H}^{3}}=\mathcal{I}(\mathbf{D}_{2})\,v_{\mathbb{H}^{3}}$
established in subsection 6.4, we conjecture that section 8 holds also for
$\mathscr{F}=\mathbf{D}$.

##  Appendix A Proofs of cross-ratio parametrization statements

This appendix is devoted to the proof of the parametrizations via
$\omega$-cross-ratios of the configuration spaces of four and five points in
the variety $\mathcal{P}_{r}$ of isotropic points. These were stated in the
main text as subsection 3.2 and subsection 3.2, respectively. We continue to
adhere to the notation and standing assumptions in Subsections 2.1 and 2.2.

###  A.1. Proof of subsection 3.2

Let us assume that $r\geq 2$. We consider the Borel functions
$\Gamma,\,\Delta,\,\Delta^{1/2},\,\varphi_{-},\,\varphi_{+}:\mathbb{C}^{2}\to\mathbb{C}$,
defined as

(A.1) $\displaystyle\Gamma(z_{1},z_{2})$ $\displaystyle\coloneqq
1-z_{1}-z_{2},$ (A.2) $\displaystyle\Delta(z_{1},z_{2})$
$\displaystyle\coloneqq\Gamma(z_{1},z_{2})^{2}-2(1+\varepsilon)\cdot
z_{1}z_{2},$ (A.3) $\displaystyle\Delta^{1/2}(z_{1},z_{2})$
$\displaystyle\coloneqq\begin{cases}\Gamma(z_{1},z_{2})&\mbox{ if
}\varepsilon=-1,\\\ \sqrt{\Delta(z_{1},z_{2})}&\mbox{ if
}\varepsilon=+1,\end{cases}$ (A.4) $\displaystyle\varphi_{\eta}(z_{1},z_{2})$
$\displaystyle\coloneqq\left(\frac{\Delta^{1/2}\ +\
\eta\,\Gamma}{2}\right)(z_{1},z_{2}),\quad\eta\in\\{\pm 1\\},$

where $\sqrt{-}:\mathbb{C}\to\mathbb{C}$ is a Borel choice of complex square
root. Abusively, we will use the same symbols to denote the Borel functions
$\mathcal{P}_{r}^{4}\to\mathbb{C}$ defined a.e. by precomposing with
$\pi_{3}=(\operatorname{cr}_{1},\operatorname{cr}_{2})$.

######  Definition .

We let $\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$ be the subset of
$\mathcal{P}_{r}^{\\{4\\}}$ consisting of tuples $(p_{0},p_{1},p_{2},p_{3})$
such that the subspace $\langle p_{0},p_{1},p_{2},p_{3}\rangle<V_{r}$ is non-
degenerate (see (3.2) for the definition of $\mathcal{P}_{r}^{\\{4\\}}$) .

######  Lemma .

A tuple $[v_{0},v_{1},v_{2},v_{3}]\in\mathcal{P}_{r}^{(4)}$ belongs to
$\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$ if and only if the associated Gram
matrix $(\omega(v_{i},v_{j}))_{i,j}$ is non-singular. Moreover,

$\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}=\\{(p_{0},p_{1},p_{2},p_{3})\in\mathcal{P}_{r}^{(4)}\mid\Delta(p_{0},p_{1},p_{2},p_{3})\neq
0\\},$

and in particular, $\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$ is a
$G_{r}$-invariant, co-null Borel subset of $\mathcal{P}_{r}^{4}$.

###### Proof.

The matrix $(\omega(v_{i},v_{j}))_{i,j}$ is singular if and only if there
exists a non-trivial linear combination $v=\sum_{i}\alpha_{i}v_{i}$ such that
$\omega(v_{k},v)=0$ for every $k\in\\{0,\ldots,3\\}$. This means precisely
that either $v=0$ and the vectors $v_{0},\ldots,v_{3}$ are linearly dependent,
or $\langle v_{0},\ldots,v_{3}\rangle$ is degenerate. The second part of the
lemma follows from the fact that the Gram determinant of
$\\{v_{0},v_{1},v_{2},v_{3}\\}$ equals the quotient
$\Delta[v_{0},v_{1},v_{2},v_{3}]\,/(\omega(v_{0},v_{1})\,\omega(v_{2},v_{3}))^{2}.$
∎

The subspace $L<V_{r}$ generated by any four lines
$(p_{0},p_{1},p_{2},p_{3})\in\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$ is
isomorphic to the sum $\mathcal{H}\oplus\mathcal{H}$. In subsection A.1 below,
we will construct an adapted basis of $L$ in terms of any such points
$p_{0},p_{1},p_{2},p_{3}$. For that purpose, we introduce perpendicular
projections with respect to $\omega$ onto hyperbolic planes, and list relevant
properties.

######  Definition .

Let $\mathcal{H}_{0}<V_{r}$ be a hyperbolic plane and $\\{v_{0},v_{1}\\}$ a
basis of isotropic vectors of $\mathcal{H}_{0}$. The _perpendicular
projection_ $\mbox{proj}=\mbox{proj}_{\mathcal{H}_{0}}\colon
V_{r}\twoheadrightarrow\mathcal{H}_{0}$ onto $\mathcal{H}_{0}$ is given by

$\mbox{proj}(v)\coloneqq{\displaystyle\frac{\omega(v_{1},v)}{\omega(v_{1},v_{0})}\,v_{0}+\frac{\omega(v_{0},v)}{\omega(v_{0},v_{1})}\,v_{1},}$

with kernel $\mathcal{H}_{0}^{\perp}$. We also let $\widehat{\,\cdot\,}\colon
V_{r}\to\mathcal{H}_{0}^{\perp}$ be the _complement map_ , defined as
$\widehat{v}\coloneqq v-\mbox{proj}(v)$.

######  Lemma .

Let $[v_{0},v_{1},v_{2},v_{3}]\in\mathcal{P}_{r}^{(4)}$, let
$\widehat{\,\cdot\,}$ denote the complement map to $\langle
v_{0},v_{1}\rangle$, let $v,w\in V_{r}$ be any two vectors, and let $u\in
V_{r}$ be isotropic. Then the following identities hold:

  1. _(i)_

$\omega(\widehat{v},\widehat{w})=\omega(\widehat{v},w)=\omega(v,\widehat{w})$.

  2. _(ii)_

$q(\widehat{u})=-(1+\varepsilon)\,\big{(}\omega(v_{1},u)\,\omega(u,v_{0})/\omega(v_{0},v_{1})\big{)}$.

  3. _(iii)_

$\omega(\widehat{v_{2}},\widehat{v_{3}})=\omega(v_{2},v_{3})\cdot\Gamma[v_{0},v_{1},v_{2},v_{3}]$.

  4. _(iv)_

$\omega(\widehat{v_{2}},\widehat{v_{3}})^{2}-q(\widehat{v_{2}})\,q(\widehat{v_{3}})=\omega(v_{2},v_{3})^{2}\cdot\Delta[v_{0},v_{1},v_{2},v_{3}]$.

###### Proof.

Since $\omega(\widehat{v},\mbox{proj}(w))=0$, we have
$\omega(\widehat{v},\widehat{w})=\omega(\widehat{v},w)-\omega(\widehat{v},\mbox{proj}(w))=\omega(\widehat{v},w)$,
the first equality in (i). The second one is analogous. Items (ii) and (iii)
follow immediately from (i) and the definition of the complement map. For
$\varepsilon=-1$, the identity (iv) holds since $\Delta=\Gamma^{2}$ and
$q(\widehat{v_{2}})\,q(\widehat{v_{3}})=0$. If $\varepsilon=+1$, then (iv)
follows from (iii) and the next computation:

$\displaystyle q(\widehat{v_{2}})\,q(\widehat{v_{3}})$
$\displaystyle=4\cdot\frac{\omega(v_{1},v_{2})\,\omega(v_{2},v_{0})}{\omega(v_{0},v_{1})}\cdot\frac{\omega(v_{1},v_{3})\,\omega(v_{3},v_{0})}{\omega(v_{0},v_{1})}$
$\displaystyle=4\cdot\omega(v_{2},v_{3})^{2}\cdot\frac{\omega(v_{1},v_{3})\,\omega(v_{2},v_{0})}{\omega(v_{1},v_{0})\,\omega(v_{2},v_{3})}\cdot\frac{\omega(v_{2},v_{1})\,\omega(v_{0},v_{3})}{\omega(v_{2},v_{3})\,\omega(v_{0},v_{1})}$
$\displaystyle=4\cdot\omega(v_{2},v_{3})^{2}\cdot(\operatorname{cr}_{1}^{-1}\cdot\operatorname{cr}_{2})[v_{0},v_{1},v_{2},v_{3}].\qed$

The following technical lemma plays a key role in the proof of subsection 3.2.

######  Lemma .

Let
$\mathbf{p}=[v_{0},v_{1},v_{2},v_{3}]\in\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$,
let $\mathcal{H}_{0}\coloneqq\langle v_{0},v_{1}\rangle$ be the hyperbolic
plane generated by the first two vectors, and let $\widehat{\,\cdot\,}$ denote
the complement map to $\mathcal{H}_{0}$. We also let the quantities
$\Pi=\Pi(v_{0},v_{1},v_{2})$, $\lambda=\lambda(v_{0},v_{1},v_{2})$,
$\mu=\mu(v_{0},v_{1},v_{2},v_{3})$ be defined as

(A.5) $\displaystyle\Pi$
$\displaystyle\coloneqq\omega(v_{0},v_{1})\,\omega(v_{1},v_{2})\,\omega(v_{2},v_{0}),$
(A.6) $\displaystyle\lambda$
$\displaystyle\coloneqq\frac{\sqrt{\Pi}}{\omega(v_{1},v_{0})}=\frac{\omega(v_{1},v_{2})\,\omega(v_{0},v_{2})}{\sqrt{\Pi}},$
(A.7) $\displaystyle\mu$
$\displaystyle\coloneqq\lambda^{-1}\cdot\omega(v_{3},v_{2}).$

Then $\mathcal{H}_{0}^{\prime}\coloneqq\mathcal{H}_{0}^{\perp}\cap\langle
v_{0},v_{1},v_{2},v_{3}\rangle=\langle\widehat{v_{2}},\widehat{v_{3}}\rangle$
is a hyperbolic plane, and the unique solutions
$e^{\prime},f^{\prime}\in\mathcal{H}_{0}^{\prime}$ of the next linear systems
form an adapted basis of $\mathcal{H}_{0}^{\prime}$:

(A.8) $\left\\{\begin{array}[]{rl}\omega(e^{\prime},v_{2})&=\lambda\\\
\omega(e^{\prime},v_{3})&=\mu\,\varphi_{-}(\mathbf{p})\end{array}\right\\}\quad\mathrm{and}\quad\left\\{\begin{array}[]{rl}\omega(f^{\prime},v_{2})&=-\lambda,\\\
\omega(f^{\prime},v_{3})&=\varepsilon\mu\,\varphi_{+}(\mathbf{p}),\end{array}\right\\}$

###### Proof.

After writing $e^{\prime}$ and $f^{\prime}$ as linear combinations of the
basis $\\{\widehat{v_{2}},\widehat{v_{3}}\\}$ of $\mathcal{H}^{\prime}$, one
obtains that the coefficient matrix of both systems in (A.8) has determinant

$q(\widehat{v_{2}})\,q(\widehat{v_{3}})-\omega(\widehat{v_{2}},\widehat{v_{3}})^{2}=-\omega(v_{2},v_{3})^{2}\cdot\Delta(\mathbf{p}).$

This quantity is non-zero in virtue of subsection A.1. Their explicit
solutions are:

(A.11) $\displaystyle\mbox{For }\varepsilon=-1:$ $\displaystyle\ \
\left\\{\\!\\!\begin{array}[]{rl}e^{\prime}&=(\lambda/\omega(\widehat{v_{3}},\widehat{v_{2}}))\cdot\widehat{v_{3}},\\\
f^{\prime}&=(-\mu/\omega(v_{2},v_{3}))\cdot\widehat{v_{2}}-(\lambda/\omega(\widehat{v_{3}},\widehat{v_{2}}))\cdot\widehat{v_{3}}.\end{array}\right.$
(A.14) $\displaystyle\mbox{For }\varepsilon=+1:$ $\displaystyle\ \
\left\\{\\!\\!\begin{array}[]{rl}e^{\prime}&=\phantom{-}\lambda/q(\widehat{v_{2}})\cdot(\widehat{v_{2}}-r),\\\
f^{\prime}&=-\lambda/q(\widehat{v_{2}})\cdot(\widehat{v_{2}}+r),\end{array}\\!\\!\right.$

where

$r\coloneqq\frac{\omega(\widehat{v_{2}},\widehat{v_{3}})\cdot\widehat{v_{2}}-q(\widehat{v_{2}})\cdot\widehat{v_{3}}}{\omega(v_{2},v_{3})\cdot\Delta^{1/2}(\mathbf{p})}\in\mathcal{H}^{\prime}.\vspace{4pt}$

The next identities are consequences of subsection A.1 and facilitate the
computations that verify that $e^{\prime}$ and $f^{\prime}$ as above are
actually solutions of (A.8) and an adapted basis of
$\mathcal{H}^{\prime}_{0}$:

$\displaystyle(\varepsilon=-1)\quad$
$\displaystyle\omega(\widehat{v_{2}},v_{2})=q(\widehat{v_{2}})=0,\quad\omega(\widehat{v_{3}},v_{3})=q(\widehat{v_{3}})=0,$
$\displaystyle\omega(\widehat{v_{2}},v_{3})=\omega(v_{2},\widehat{v_{3}})=\omega(\widehat{v_{2}},\widehat{v_{3}})=\omega(v_{2},v_{3})\cdot\Gamma(\mathbf{p})$
$\displaystyle\qquad\quad\ \
=\omega(v_{2},v_{3})\cdot\Delta^{1/2}(\mathbf{p})\neq 0.$
$\displaystyle(\varepsilon=+1)\quad$
$\displaystyle\omega(r,v_{2})=\omega(r,\widehat{v_{2}})=0,\quad
q(r)=-q(\widehat{v_{2}})=2\lambda^{2}\neq 0,$
$\displaystyle\omega(r,v_{3})=\omega(r,\widehat{v_{3}})=\omega(v_{2},v_{3})\cdot\Delta^{1/2}(\mathbf{p})\neq
0.\qed$

###### Proof of subsection 3.2.

Let $r\geq 2$. The theorem is proven once we establish the existence of a co-
null subset $\Omega_{3}\subset\mathbb{C}^{2}$ and a Borel map
$\Phi_{3}:\Omega_{3}\to\mathcal{P}_{r}^{4}$ with the next three properties:

  1. (i)

$\pi_{3}(\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}})\subset\Omega_{3}$.

  2. (ii)

$\Phi_{3}(\Omega_{3})\subset\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$ and
$\pi_{3}\,\circ\,\Phi_{3}=\operatorname{id}$.

  3. (iii)

For all $\mathbf{p}\in\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$, there exists
$g\in G_{r}$ such that $\Phi_{3}\,\circ\,\pi_{3}(\mathbf{p})=g\,\mathbf{p}$.

We proceed now to their construction. Let us define

(A.15)
$\Omega_{3}\coloneqq\left\\{(a_{1},a_{2})\in(\mathbb{C}^{\times})^{2}\mid\
\Delta(a_{1},a_{2})\neq 0\right\\}.$

Note that $\Omega_{3}$ is a non-empty Zariski-open subset of $\mathbb{C}^{2}$,
hence co-null, and condition (i) follows immediately from subsection A.1. We
define the isotropic vector $\phi_{2}$ and the Borel map
$\phi_{3}\colon\Omega_{3}\to\langle e_{r},e_{r-1},f_{r-1},f_{r}\rangle$ by

$\displaystyle\phi_{2}$ $\displaystyle\coloneqq e_{r}+e_{r-1}-f_{r-1}+f_{r},$
$\displaystyle\phi_{3}(a_{1},a_{2})$ $\displaystyle\coloneqq a_{1}\
e_{r}+\varphi_{-}(a_{1},a_{2})\ e_{r-1}+\varepsilon\,\varphi_{+}(a_{1},a_{2})\
f_{r-1}+\varepsilon\,a_{2}\ f_{r}.$

The vector $\phi_{3}(a_{1},a_{2})$ is isotropic for any
$(a_{1},a_{2})\in\Omega_{3}$. We let now
$\Phi_{3}:\Omega_{3}\to\mathcal{P}_{r}^{4}$ be the map

(A.16) $\Phi_{3}(a_{1},a_{2})\coloneqq\left[e_{r},\,f_{r},\,\phi_{2},\,\
\phi_{3}(a_{1},a_{2})\right].$

The first three points in the triple are precisely the representative generic
3-tuple used in the proof of subsection 3.1. After abbreviating
$\phi_{3}\equiv\phi_{3}(a_{1},a_{2})$, the identities given below hold true,
implying that none of those quantities is zero and, hence, that $\Phi_{3}$
ranges onto $\mathcal{P}_{r}^{(4)}$.

(A.17)
$\begin{array}[]{lllllll}\omega(e_{r},f_{r})=1,&&\omega(e_{r},\phi_{2})=1,&&\omega(e_{r},\phi_{3})=\varepsilon
a_{2},\\\
\omega(f_{r},\phi_{2})=\varepsilon,&&\omega(f_{r},\phi_{3})=\varepsilon
a_{1},&&\omega(\phi_{2},\phi_{3})=\varepsilon.\end{array}$

From (A.17), we also derive that
$\operatorname{cr}_{i}\big{(}\Phi_{3}(a_{1},a_{2})\big{)}=a_{i}$ for $i=1,2$.
Together with subsection A.1, this implies that $\Phi_{3}(a_{1},a_{2})$ is in
$\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$ and that
$\pi_{4}\,\circ\,\Phi_{3}(a_{1},a_{2})=(a_{1},a_{2})$, establishing (ii).

For (iii), we fix an arbitrary
$\mathbf{p}=[v_{0},v_{1},v_{2},v_{3}]\in\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$.
As in subsection A.1, we let $\lambda,\Pi,\mu$ be the quantities defined in
function of $(v_{0},v_{1},v_{2},v_{3})$ from (A.5)–(A.7), and
$e^{\prime},\,f^{\prime}$ be the solutions of the systems (A.8). Then we set

(A.18)
$e^{\prime}_{r}\coloneqq\frac{\lambda\,v_{1}}{\omega(v_{1},v_{2})},\quad
e^{\prime}_{r-1}\coloneqq e^{\prime},\quad f^{\prime}_{r-1}\coloneqq
f^{\prime},\quad
f^{\prime}_{r}\coloneqq\frac{\omega(v_{1},v_{2})\,v_{0}}{\sqrt{\Pi}}.$

Observe that $\\{e^{\prime}_{r},\,f^{\prime}_{r}\\}$ is an adapted basis of
the hyperbolic plane $\mathcal{H}_{r}\coloneqq\langle
v_{0},v_{1}\rangle<V_{r}$, and that $\\{e^{\prime}_{r-1},f^{\prime}_{r-1}\\}$
is an adapted basis of
$\mathcal{H}_{r-1}\coloneqq\mathcal{H}_{r}^{\perp}\cap\langle
v_{0},v_{1},v_{2},v_{3}\rangle$. By Witt’s lemma, we extend
$\\{e^{\prime}_{r},e^{\prime}_{r-1},f^{\prime}_{r-1},f^{\prime}_{r}\\}$ to a
full adapted basis $\mathscr{B}_{r}^{\prime}$ of $V_{r}$. The automorphism
$T\in\operatorname{GL}(V_{r})$ that maps the standard adapted basis
$\mathscr{B}_{r}$ to $\mathscr{B}^{\prime}_{r}$ lies in $G_{r}$, and
therefore, so does

(A.19) $g=g(v_{0},v_{1},v_{2},v_{3})\coloneqq T^{\top}J_{r}.$

Note that for every $i\in\\{0,\ldots,3\\}$, we have

$\displaystyle g\cdot v_{i}$
$\displaystyle=\omega(e^{\prime}_{r},v_{i})\,e_{r}+\omega(e^{\prime}_{r-1},v_{i})\,e_{r-1}+\omega(f^{\prime}_{r-1},v_{i})\,f_{r-1}+\omega(f^{\prime}_{r},v_{i})\,f_{r}.$

Writing in coordinates with respect to the standard adapted basis
$\mathscr{B}_{r}$, we have the equalities below, where the second one follows
after multiplying the fourth vector by $\mu^{-1}$:

$g\cdot\mathbf{p}=\left[\begin{array}[]{cccc}1&0&\lambda&\lambda\,\omega(v_{1},v_{3})/\omega(v_{1},v_{2})\\\
0&0&\lambda&\mu\,\varphi_{-}(\mathbf{p})\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\\
0&0&-\lambda&\varepsilon\,\mu\,\varphi_{+}(\mathbf{p})\\\
0&1&\lambda&\omega(v_{1},v_{2})\,\omega(v_{0},v_{3})/\sqrt{\Pi}\end{array}\right]=\Phi_{3}\,\circ\,\pi_{3}(\mathbf{p}).\vspace{-20pt}$

∎

###  A.2. Proof of subsection 3.2

Recall the cross-ratios
$\alpha_{j},\beta_{j},\gamma_{j}:\mathcal{P}_{r}^{5}\to\mathbb{C}$ from
subsection 3.2. With the notation established in (A.1)–(A.4), we set
$\psi_{-},\psi_{+}:\mathbb{C}^{6}\to\mathbb{C}$ to be a.e.

(A.20)
$\psi_{\eta}(\mathbf{a},\mathbf{b},\mathbf{c})\coloneqq\eta\,\varphi_{\eta}(\mathbf{a})\cdot\Gamma(\mathbf{b})\
\ +\ \ a_{2}b_{1}c_{1}^{-1}\cdot\Gamma(\mathbf{c}),\quad\eta\in\\{\pm 1\\},$

for
$\mathbf{a}=(a_{1},a_{2}),\,\mathbf{b}=(b_{1},b_{2}),\,\mathbf{c}=(c_{1},c_{2})$.
Let also
$\Gamma_{\alpha},\,\Gamma_{\beta},\,\Gamma_{\gamma}:\,\mathcal{P}_{r}^{5}\to\mathbb{C}$
be the functions defined a.e. as

$\Gamma_{\alpha}\coloneqq\Gamma\,\circ\,(\alpha_{1},\alpha_{2}),\quad\Gamma_{\beta}\coloneqq\Gamma\,\circ\,(\beta_{1},\beta_{2}),\quad\Gamma_{\gamma}\coloneqq\Gamma\,\circ\,(\gamma_{1},\gamma_{2}).$

In this subsection, we will consider
$\Delta,\,\Delta^{1/2},\,\varphi_{\eta},\,\psi_{\eta}$ a.e. defined functions
$\mathcal{P}_{r}^{5}\to\mathbb{C}$ of 5-tuples after precomposing the
expressions from (A.2)–(A.4) with the function $(\alpha_{1},\alpha_{2})$, and
the expression from (A.20) with
$(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2},\gamma_{1},\gamma_{2})$.

The proof of subsection 3.2 builds up on the computations of the previous
subsection. The next lemma establishes the result of pairing the vectors
$e^{\prime}$, $f^{\prime}$ obtained in subsection A.1 from a tuple
$[v_{0},v_{1},v_{2},v_{3}]\in\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$ with a
fifth vector $v_{4}$. Its proof is a direct computation that we shall omit,
relying on the explicit expressions of $e^{\prime}$ and $f^{\prime}$ given in
(A.11) and (A.14).

######  Lemma .

Let $[v_{0},v_{1},v_{2},v_{3}]\in\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$, let
$\widehat{\cdot}$ be the complement map onto the hyperbolic plane $\langle
v_{0},v_{1}\rangle$ spanned by the first two vectors, and let
$\lambda,\mu,e^{\prime},f^{\prime}$ be as in subsection A.1. If an isotropic
vector $v_{4}\in V_{r}\smallsetminus\\{0\\}$ is such that
$\mathbf{p}\coloneqq[v_{0},v_{1},v_{2},v_{3},v_{4}]$ lies in
$\mathcal{P}_{r}^{(5)}$, then

$\omega(e^{\prime},v_{4})=\varepsilon\frac{\omega(v_{2},v_{4})}{\lambda}\cdot\frac{\psi_{-}(\mathbf{p})}{\Delta^{1/2}(\mathbf{p})}\quad\mathrm{and}\quad\omega(f^{\prime},v_{4})=\frac{\omega(v_{2},v_{4})}{\lambda}\cdot\frac{\psi_{+}(\mathbf{p})}{\Delta^{1/2}(\mathbf{p})}.$

We give our definition of genericity for 5-tuples at this point, and then
prove subsection 3.2.

######  Definition .

We let $\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}}$ be the set of 5-tuples
$(p_{0},p_{1},p_{2},p_{3},p_{4})\in\mathcal{P}_{r}^{\\{5\\}}$ such that every
4-subtuple is in $\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$. In virtue of
subsection 3.2, such a tuple is in $\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}}$ if
and only if

$(p_{0},p_{1},p_{i},p_{j})\in\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}\quad\mbox{for
}i,j\in\\{2,3,4\\}\mbox{ with }i<j.\vspace{-1pt}$

One verifies that $\mathcal{P}_{5}^{\\{\\!\\{5\\}\\!\\}}$ is a
$G_{r}$-invariant, co-null subset of $\mathcal{P}_{r}^{5}$.

###### Proof of subsection 3.2.

For every $r\geq r_{1}+1$, we must define a co-null subset
$\Omega_{4}\subset\mathbb{C}^{5}$ and a Borel map
$\Phi_{4}:\Omega_{4}\to\mathcal{P}_{r}^{5}$ such that:

  1. (i)

$\pi_{4}(\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}})\subset\Omega_{4}$.

  2. (ii)

$\Phi_{4}(\Omega_{4})\subset\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}}$ and
$\pi_{4}\,\circ\,\Phi_{4}=\operatorname{id}$.

  3. (iii)

For all $\mathbf{p}\in\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}}$, there exists
$g\in G_{r}$ such that $\Phi_{4}\,\circ\,\pi_{4}(\mathbf{p})=g\,\mathbf{p}$.

We leave out the case $(\varepsilon,d)=(+1,1)$, $r=2$ for individual
consideration, and assume that

$r\geq\begin{cases}3&\mbox{ if }\varepsilon=+1,\\\\[-4.0pt] 2&\mbox{ if
}\varepsilon=-1,\vspace{-3pt}\end{cases}$

Similarly to the definition of $\Omega_{3}$ in (A.15), we let

(A.21) $\displaystyle\Omega_{4}$
$\displaystyle\coloneqq\left\\{(a_{1},a_{2},b_{1},b_{2},c_{1})\in(\mathbb{C}^{\times})^{5}\mid\Delta(a_{1},a_{2}),\,\Delta(b_{1},b_{2}),\,\Delta(c_{1},c_{2})\neq
0\right\\},$

where $c_{2}\coloneqq\varepsilon(a_{1}b_{2}c_{1})/(a_{2}b_{1})$. Being an
intersection of finitely many non-empty Zariski-open subsets of
$\mathbb{C}^{5}$, the set $\Omega_{4}$ is itself non-empty and Zariski-open,
in particular co-null. subsection A.2 and subsection A.1 imply that
$\Delta(\alpha_{1}(\mathbf{p}),\alpha_{2}(\mathbf{p})),\,\Delta(\beta_{1}(\mathbf{p}),\beta_{2}(\mathbf{p})),\,\Delta(\gamma_{1}(\mathbf{p}),\gamma_{2}(\mathbf{p}))\neq
0$ for every $\mathbf{p}\in\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}}$,
establishing (i).

We define the functions $\tilde{\phi}_{4}\colon\Omega_{4}\to\langle
e_{r},e_{r-1},f_{r-1},f_{r}\rangle$ and $\phi_{4}:\Omega_{4}\to V_{r}$ as

$\displaystyle\tilde{\phi}_{4}(\mathbf{a},\mathbf{b},c_{1})$
$\displaystyle\coloneqq b_{1}\ e_{r}\ +\
\frac{\psi_{-}(\mathbf{a},\mathbf{b},\mathbf{c})}{\Delta^{1/2}(\mathbf{a})}\
e_{r-1}\ +\
\varepsilon\,\frac{\psi_{+}(\mathbf{a},\mathbf{b},\mathbf{c})}{\Delta^{1/2}(\mathbf{a})}\
f_{r-1}\ +\ \varepsilon\,b_{2}\ f_{r},$ (A.22)
$\displaystyle\phi_{4}(\mathbf{a},\mathbf{b},c_{1})$
$\displaystyle\coloneqq\tilde{\phi}_{4}(\mathbf{a},\mathbf{b},c_{1})\
+\left\\{e_{r-2}-\left(\frac{q\big{(}\tilde{\phi}_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}}{2}\right)\cdot
f_{r-2}\right\\},$

with $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ as in (A.20) and
$c_{2}\coloneqq\varepsilon(a_{1}b_{2}c_{1})/(a_{2}b_{1})$. Observe that if
$\varepsilon=+1$, the vector $\tilde{\phi}_{4}(\mathbf{a},\mathbf{b},c_{1})$
is not necessarily isotropic. In virtue of the assumption that $r\geq 3$ in
that setting, we are able to incorporate in
$\phi_{4}(\mathbf{a},\mathbf{b},c_{1})$ a correction that makes it isotropic
for any $(\mathbf{a},\mathbf{b},c_{1})$.

We define $\Phi_{4}\colon\Omega_{4}\to\mathcal{P}_{r}^{5}$,

$\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})\coloneqq\left[e_{r},\,f_{r},\,\phi_{2},\
\phi_{3}(\mathbf{a}),\ \phi_{4}(\mathbf{a},\mathbf{b},c_{1})\right].$

Note that the first four lines in $\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})$
correspond to the tuple $\Phi_{3}(\mathbf{a})$ introduced in (A.16).
Abbreviating $\phi_{3}\equiv\phi_{3}(\mathbf{a})$ and
$\phi_{4}\equiv\phi_{4}(\mathbf{a},\mathbf{b},c_{1})$, we obtain, in addition
to (A.17), the identities

(A.23) $\omega(e_{r},\phi_{4})=\varepsilon
b_{2},\quad\omega(f_{r},\phi_{4})=\varepsilon
b_{1},\quad\omega(\phi_{2},\phi_{4})=\varepsilon,\quad\omega(\phi_{3},\phi_{4})=a_{2}b_{1}c_{1}^{-1}.$

In particular, this proves that $\Phi_{4}$ ranges in $\mathcal{P}_{r}^{(5)}$.
From (A.17) and (A.23), we derive

$\displaystyle\alpha_{1}\big{(}\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}$
$\displaystyle=a_{1},\qquad\beta_{1}\big{(}\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}=b_{1},\qquad\gamma_{1}\big{(}\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}=c_{1},$
$\displaystyle\alpha_{2}\big{(}\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}$
$\displaystyle=a_{2},\qquad\beta_{2}\big{(}\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}=b_{2}.$

This implies that $\pi_{4}\,\circ\,\Phi_{4}=\operatorname{id}$ and that every
4-subtuple of $\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})$ lies in
$\mathcal{P}_{r}^{\\{\\!\\{4\\}\\!\\}}$. In order to conclude that
$\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})\in\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}}$
holds, it suffices to observe that if $\dim(V_{r})=2r+d\geq 5$, then
necessarily $r>2$, so that the correction term in (A.22) is non-zero, and
hence, the lines in $\Phi_{4}(\mathbf{a},\mathbf{b},c_{1})$ are linearly
independent. This proves point (ii).

For (iii), fix an arbitrary
$\mathbf{p}=[v_{0},v_{1},v_{2},v_{3},v_{4}]\in\mathcal{P}_{r}^{\\{\\!\\{5\\}\\!\\}}$
and let
$\\{e^{\prime}_{r},e^{\prime}_{r-1},f^{\prime}_{r-1},f_{r}^{\prime}\\}$ be the
adapted basis of the subspace $\langle v_{0},v_{1},v_{2},v_{3}\rangle$,

$e^{\prime}_{r}\coloneqq\frac{\lambda\,v_{1}}{\omega(v_{1},v_{2})},\quad
e^{\prime}_{r-1}\coloneqq e^{\prime},\quad f^{\prime}_{r-1}\coloneqq
f^{\prime},\quad
f^{\prime}_{r}\coloneqq\frac{\omega(v_{1},v_{2})\,v_{0}}{\sqrt{\Pi}},$

defined as in (A.18), where $\Pi,\lambda,\mu,e^{\prime},f^{\prime}$ are
functions of $(v_{0},v_{1},v_{2},v_{3})$ as in subsection A.1. Let also
$g=g(v_{0},v_{1},v_{2},v_{3})\in G_{r}$ be the element from (A.19), so that

$g\cdot[v_{0},v_{1},v_{2},v_{3}]=\Phi_{3}(\pi_{3}(\mathbf{p})).$

Let $x_{0}\in\langle v_{0},v_{1},v_{2},v_{3}\rangle^{\perp}$ be the vector
such that

$g\cdot
v_{4}=x_{0}+\omega(e^{\prime}_{r},v_{4})\,e_{r}+\omega(e^{\prime}_{r-1},v_{4})\,e_{r-1}+\omega(f^{\prime}_{r-1},v_{4})\,f_{r-1}+\omega(f^{\prime}_{r},v_{4})\,f_{r}.\\\
$

Since the span of the lines in $g[v_{0},v_{1},v_{2},v_{3}]$ equals $\langle
e_{r},e_{r-1},f_{r-1},f_{r}\rangle$, we have that $x_{0}\neq 0$ whenever
$r>2$. In coordinates with respect to $\mathscr{B}_{r}$, we have

$\displaystyle g\cdot[v_{4}]$
$\displaystyle=\left[\begin{array}[]{c}\omega(e^{\prime}_{r},v_{4})\\\\[5.0pt]
\omega(e^{\prime}_{r-1},v_{4})\\\\[4.0pt] x_{0}\\\\[4.0pt]
\omega(f^{\prime}_{r-1},v_{4})\\\\[4.0pt]
\omega(f^{\prime}_{r},v_{4})\end{array}\right]=\left[\begin{array}[]{c}\lambda\,\omega(v_{1},v_{4})/\omega(v_{1},v_{2})\\\\[5.0pt]
\varepsilon\frac{\omega(v_{2},v_{4})}{\lambda}\cdot\frac{\psi_{-}(\mathbf{p})}{\Delta^{1/2}(\mathbf{p})}\\\\[4.0pt]
x_{0}\\\\[4.0pt]
\frac{\omega(v_{2},v_{4})}{\lambda}\cdot\frac{\psi_{+}(\mathbf{p})}{\Delta^{1/2}(\mathbf{p})}\\\\[4.0pt]
\omega(v_{1},v_{2})\,\omega(v_{0},v_{4})/\sqrt{\Pi}\end{array}\right]=\left[\begin{array}[]{c}\beta_{1}(\mathbf{p})\\\\[5.0pt]
\frac{\psi_{-}(\mathbf{p})}{\Delta^{1/2}(\mathbf{p})}\\\\[5.0pt]
x_{0}^{\prime}\\\\[5.0pt]
\varepsilon\,\frac{\psi_{+}(\mathbf{p})}{\Delta^{1/2}(\mathbf{p})}\\\\[5.0pt]
\varepsilon\,\beta_{2}(\mathbf{p})\end{array}\right]=\left[\tilde{\phi}_{4}\,(\pi_{4}(\mathbf{p}))+x_{0}^{\prime}\right]$

where $x^{\prime}_{0}\coloneqq\varepsilon\lambda/\omega(v_{2},v_{4})\cdot
x_{0}\neq 0$. Indeed, the last equality holds by subsection 3.2, which implies
that
$\gamma_{2}(\mathbf{p})=\varepsilon(\alpha_{1}\alpha_{2}^{-1}\beta_{1}^{-1}\beta_{2}\gamma_{1})(\mathbf{p})$.

Note that there exists an element $g^{\prime}\in G_{r-2}<G_{r}$ such that
$g^{\prime}x_{0}=e_{r-2}+(q(x_{0}^{\prime})/2)\,f_{r-2}$, and that

$q(x^{\prime}_{0})=-q\big{(}\tilde{\phi}_{4}\,(\pi_{4}(\mathbf{p})\big{)}.$

Therefore,
$(g^{\prime}g)\cdot\mathbf{p}=\Phi_{4}\,\circ\,\pi_{4}(\mathbf{p})$,
completing the proof of (iii).

The proof for the case $(\varepsilon,d)=(+1,1)$, $r=2$ is analogous, but
requires a few modifications. We must replace everywhere in the argument above
the set $\Omega_{4}$ as in (A.21) by its co-null subset
$\Omega_{4}^{\prime}\subset\Omega_{4}$ defined below, and the map $\phi_{4}$
from (A.22) by $\phi^{\prime}_{4}:\Omega_{4}^{\prime}\to V_{r}$:

$\displaystyle\Omega_{4}^{\prime}\coloneqq\left\\{(\mathbf{a},\mathbf{b},c_{1})\in\Omega_{4}\left|\
q\big{(}\tilde{\phi}_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}\neq
0\right.\right\\},$
$\displaystyle\phi_{4}^{\prime}(\mathbf{a},\mathbf{b},c_{1})\coloneqq\tilde{\phi}_{4}(\mathbf{a},\mathbf{b},c_{1})\
+\sqrt{-q\big{(}\tilde{\phi}_{4}(\mathbf{a},\mathbf{b},c_{1})\big{)}}\cdot
h.\qed$

######  Remark .

The isomorphism $G_{1}\\!\setminus\\!\mathcal{P}_{1}^{5}\cong\mathbb{C}^{2}$
of Lebesgue spaces holds for the parameters
$(\varepsilon,d)\in\\{(+1,1),(-1,0)\\}$. Indeed, in the classical setting (see
subsection 3.2), any 5-tuple
$\mathbf{p}\in\mathcal{P}_{1}^{\\{\\!\\{5\\}\\!\\}}$ is in the $G_{1}$-orbit
of the tuple $(\infty,0,1,a,b)\in\widehat{\mathbb{C}}^{5}$ with
$a=\alpha_{0}(\mathbf{p})$ and $b=\beta_{0}(\mathbf{p})$.

In the other case that is not covered by subsection 3.2, namely