# Flatmates and the bounded cohomology
of algebraic groups

Nicolas Monod  EPFL, Switzerland

###### Abstract.

For all algebraic groups over non-Archimedean local fields, the bounded
cohomology vanishes. This follows from the corresponding statement for
automorphism groups of Bruhat–Tits buildings, which hinges on the solution to
the flatmate conjecture raised in earlier work with Bucher. Vanishing and
invariance theorems for arithmetic groups are derived.

##  1\. Introduction

Our main goal is the following vanishing theorem.

######  Theorem A (Algebraic groups).

Let $\mathbf{G}$ be any algebraic group over a non-Archimedean local field
$k$.

Then the continuous bounded cohomology of $\mathbf{G}(k)$ with real
coefficients vanishes in every positive degree.

It is understood that the group $G=\mathbf{G}(k)$ of $k$-points of the scheme
$\mathbf{G}$ is endowed with the locally compact topology determined by the
local field $k$. Examples are (almost-)simple linear algebraic groups such as

$G=\mathbf{SL}_{d}(k)\text{ or }G=\mathbf{SO}_{d}(k),\kern
14.22636pt\text{both with }k=\mathbf{Q}_{p}\text{ or }k=\mathbf{F}_{q}((t)),$

for which A can be viewed as a strengthening of the classical vanishing
theorem of Garland, Casselman, Wigner and Harder [Gar73, Cas74, CW74, Har77],
noting that non-trivial coefficients were already treated in [Mon10].
Structure theory and general principles will reduce A to the case of such
simple algebraic groups; in view of Bruhat–Tits theory, that case is in turn
contained in the following statement.

######  Theorem B (Buildings).

Let $G$ be a locally compact group acting properly by automorphisms on a
locally finite Euclidean building.

If this action is strongly transitive, then the continuous bounded cohomology
of $G$ with real coefficients vanishes in every positive degree.

Previously, this statement was only known in the special case of trees [BM19].

### Motivations

Our first motive to establish A is that the bounded cohomology of _real or
complex_ algebraic groups, and more generally of Lie groups, remains very
mysterious to this day even though it has been subjected to intense scrutiny
ever since Gromov’s seminal work [Gro82]. The original impetus for that study
goes back to Milnor’s 1958 paper [Mil58] and the Milnor–Wood inequality: the
fact that some characteristic classes happen to be bounded translates into
non-trivial estimates for topological invariants of bundles and manifolds.
Gromov conceptualised this by introducing bounded cohomology and proved that
all characteristic classes of flat $G$-bundles, where $G$ is a
$\mathbf{R}$-algebraic group, are bounded. (Another proof was given by Bucher
in [Buc04, Buc07]; see also [HO12] for related results.)

Nonetheless, Dupont’s 1979 conjecture [Dup79] that all the real continuous
cohomology classes of connected simple Lie groups are bounded remains open for
many groups. In fact, if we exclude groups of Hermitian type, it is open for
almost all other simple Lie groups, see [HO12]. A stronger version of this
conjecture, open for _all_ semi-simple (non-compact) groups, is as follows:

######  Conjecture (Problem A in [Mon06]).

Let $G$ be a connected semi-simple Lie group with finite center. Then the
continuous bounded cohomology of $G$ with real coefficients is naturally
isomorphic to its ordinary (continuous) cohomology.

For semi-simple algebraic groups over non-Archimedean local fields, it is a
classical result that the ordinary continuous real cohomology vanishes [CW74,
Cor. 2], [BW80, §X]. In that sense, A answers the analogue of the above
conjecture in the non-Archimedean case.

This was previously only known for rank one groups, such as
$\mathbf{SL}_{2}(\mathbf{Q}_{p})$, because the statement of B was only
available in the particular case of trees [BM19]. (For higher rank groups,
vanishing in degree two was obtained [BM99] and the stability methods of
[Mon10] reduce degree three to the rank one case as proved in [GLMR23, Thm.
6.3.1]. Nothing was known in higher degrees.)

In the Lie case originaly considered for the above conjecture, the isomorphism
is only known in the following low degrees. In degree $2$ by [BM99]. In degree
$3$, for some groups by [Mon04, Pie18, BBI18] and very recently [DlCM23],
using [DlCMH23, DlCMH24], established degree $3$ for all classical complex
groups. In degree $4$ it is known for $\mathbf{SL}_{2}(\mathbf{R})$ only
[HO15].

A second motivation for A is that when ordinary cohomology is already known to
vanish, as for simple non-Archimedean groups, then the bounded vanishing is a
strict strengthening of vanishing. Indeed, it implies that even “almost-
cocycles” must be trivial, as illustrated by the case of quasi-morphisms. That
case, which concerns $n=2$ only, has a number of applications to rigidity.
Higher vanishing and bounded acyclicity have been much studied recently,
though mostly for “large” transformation groups without topology [MM85, Löh17,
Mon22, MN23, FFLM23, CFFLM23, FFMN24].

### Arithmetic groups and non-trivial coefficients

Our third incentive is the cohomology of _discrete_ groups. A provides one of
the main missing pieces for the study of the bounded cohomology of
$S$-arithmetic groups. In the case of ordinary cohomology, this is the most
classical motivation and the main reason why continuous cohomology of
algebraic groups has been studied for general coefficients [BW80]. Indeed,
following Borel–Serre [BS76], to study the “abstract” cohomology of an
$S$-arithmetic group $\Gamma$, one realises it as a lattice $\Gamma<G$ in a
product of algebraic groups over various local fields (which is possible by
results of Borel [Bor63, §8], respectively Behr–Harder [Beh69, Har69] in
positive characteristic). For instance,

$\Gamma=\mathbf{SL}_{d}(\mathbf{Z}[1/p])\kern 14.22636pt\text{in
}G=\mathbf{SL}_{d}(\mathbf{R})\times\mathbf{SL}_{d}(\mathbf{Q}_{p}).$

In a suitable range of degrees, the cohomology of $\Gamma$ will then be
determined by the continuous cohomology of $G$ with coefficients in an
induction module. Therefore, vanishing results for $G$ with non-trivial
coefficients will imply _invariance theorems_ , namely the statement that
$\mathrm{H}^{n}(\Gamma)$ is isomorphic to $\mathrm{H}_{\mathrm{c}}^{n}(G)$ for
suitable $n$ [BS76, Ser71]. At that point, the vanishing for non-Archimedean
groups will further indicate that $\mathrm{H}^{n}(\Gamma)$ is given by the
continuous cohomology of the Archimedean factors, which is known from Lie
theory. In particular, if $G$ has no non-compact Archimedean factors, e.g. in
positive characteristic, then one concludes a vanishing result for $\Gamma$
(still for suitable $n$ only), as originally conjectured by Serre.

This classical picture has an analogue in bounded cohomology with notable
differences. For non-trivial coefficients, the ordinary vanishing holds below
the rank by work of Garland [Gar75, Gar73], Casselman [Cas74],
Casselman–Wigner [CW74], Borel–Wallach [BW80]. We established it for bounded
cohomology below twice the rank [Mon10], by different methods but still using
a form of Solomon–Tits theorem.

However, for trivial coefficients $\mathbf{R}$, the vanishing of A was
previously unknown because the Bruhat–Tits building methods from ordinary
cohomology fail in the bounded setting. To illustrate this, consider that
ordinary vanishing _above_ the rank for arbitrary coefficients clearly holds
since the rank is the dimension of this building, which is contractible. This
sort of soft and easy principles fail completely for bounded cohomology, which
is one of the reasons for its appeal, and for its difficulty. For instance,
the tree of $\mathbf{SL}_{2}(\mathbf{Q}_{p})$ is a contractible one-
dimensional simplicial complex [Ser77], but this does not preclude degree-two
bounded cohomology. Here is a folklore example:

######  Proposition C (Coefficients).

There exist irreducible continuous unitary representations $\pi$ of
$G=\mathbf{SL}_{2}(\mathbf{Q}_{p})$ with
$\mathrm{H}_{\mathrm{cb}}^{2}(G,\pi)\neq 0$.

Returning to arithmetic groups, we obtain a vanishing theorem for these
discrete groups in the spirit of Garland’s results on Serre’s conjecture by
combining A with our results from [Mon10].

######  Theorem D (Discrete groups).

Let $K$ be a global field and $\mathbf{G}$ a connected simple linear $K$-group
which is anisotropic over the Archimedean completions of $K$.

Let $S$ a finite set of valuation classes of $K$ and let
$\Gamma=\mathbf{G}(K(S))$ be the corresponding $S$-arithmetic group over the
ring $K(S)$ of $S$-integers.

Then $\mathrm{H}_{\mathrm{b}}^{n}(\Gamma)=0$ for all $0<n<2\sum_{v\in
S}\mathrm{rank}_{K_{v}}(\mathbf{G})$.

The assumption on Archimedean completions is trivially satisfied when $K$ has
positive characteristic. In characteristic zero, a concrete example is as
follows. Let $p$ be a prime $\equiv 1$ mod $4$ and let $d\geq 5$. Then

$\mathrm{H}_{\mathrm{b}}^{n}\big{(}\mathbf{SO}_{d}(\mathbf{Z}[1/p])\big{)}\
=0\kern 14.22636pt\forall 0<n<d-1$

and also $n=d-1$ for $d$ even, since the $\mathbf{Q}_{p}$-rank of
$\mathbf{SO}_{d}$ is $\lfloor d/2\rfloor$, using $p\equiv 1$ mod $4$ via
Gauß’s Theorem 108 [Gau01, §IV.108]. (The restriction $d\geq 5$ is to avoid
the non-simple case $d=4$ and the trivial range of $n$ for $d\leq 3$.)

More generally, when isotropic Archimedean places are allowed, we obtain an
invariance theorem:

######  Theorem Dbis (Discrete groups, bis).

Let $K$ be a global field, $\mathbf{G}$ a connected simple linear $K$-group
and $S$ a finite set of valuation classes of $K$ containing the set $S_{0}$ of
all Archimedean ones for which $\mathbf{G}$ is isotropic.

Let $\Gamma=\mathbf{G}(K(S))$ be the corresponding $S$-arithmetic group and
consider the semi-simple Lie group $L=\prod_{v\in S_{0}}\mathbf{G}(K_{v})$.

Then $\mathrm{H}_{\mathrm{b}}^{n}(\Gamma)\cong\mathrm{H}_{\mathrm{cb}}^{n}(L)$
for all $n<2\sum_{v\in S}\mathrm{rank}_{K_{v}}(\mathbf{G})$.

In general, only the case $n=2$ was previously known: this was the main result
of [BM99].

As a concrete example, given an integer $m>1$, the inclusion of the
$S$-arithmetic group $\mathbf{SL}_{d}(\mathbf{Z}[1/m])$ into
$\mathbf{SL}_{d}(\mathbf{R})$ induces an isomorphism

$\mathrm{H}_{\mathrm{b}}^{n}\big{(}\mathbf{SL}_{d}(\mathbf{Z}[1/m])\big{)}\
\cong\
\mathrm{H}_{\mathrm{cb}}^{n}\big{(}\mathbf{SL}_{d}(\mathbf{R})\big{)}\kern
14.22636pt\forall n<2(d-1)(\omega(m)+1)$

where $\omega(m)$ denotes the number of distinct prime factors of $m$. Again,
this was previously only known in the special case of $d=2$ by the result for
trees [BM19, Cor. 4].

We note here that the ordinary (virtual) cohomological dimension of
$\mathbf{SL}_{d}(\mathbf{Z}[1/m])$ is $n=(d-1)(\omega(m)+d/2)$ by Borel–Serre
[BS76, §6], which lies in our range for $n$ in Dbis as soon as $m$ has more
than $d/2-2$ distinct prime factors.

######  Remark E (Equivalent formulation).

Theorems D and Dbis could instead be formulated for general irreducible
lattices in semi-simple groups as all results used in the proof hold in that
setting. This would however not really add any generality. Indeed, the range
for $n$ is void in rank one (since $\mathrm{H}_{\mathrm{b}}^{1}$ always
vanishes) and in rank $\geq 2$ Margulis’s arithmeticity theorem shows that all
lattices are commensurable to $S$-arithmetic groups. We find the above
statements more concrete as they focus on the structure of the groups $\Gamma$
under consideration.

### Flatmates and our approach

With Bucher, we proposed in [BM19] a strategy towards B and implemented it in
the special case of trees. The main difficulty in that strategy is to
understand the _flatmate complex_. This object, which will be detailed in
Section 3.B below, consists of all tuples of vertices that lie in a common
flat. More precisely, the question raised by Conjecture 10 in [BM19] is to
determine the “uniform homotopy type” of this complex. We can now answer this
problem as follows.

######  Theorem F (Flatmates).

The flatmate complex of any discrete irreducible Euclidean building is
uniformly acyclic.

In the special case of trees, where the flatmate complex is the “aligned
complex”, this statement was established in [BM19] by exhibiting a relatively
simple, geometrically meaningful, bounded homotopy. By contrast, in the
present case of buildings, the combinatorics of arbitrary configurations of
finitely many points seems far too complicated (for the present author).
Therefore, we shall prove F by introducing a few general simplicial tools
which will lead to a solution by general principles, commuting back and forth
between uniform and non-uniform homotopy arguments.

The new contributions of our approach are as follows. Contrary to ordinary
contractibility, Euclidean building are not uniformly acyclic. We show in
essence that they become so “modulo their flats” by considering the nerve of
the apartment system. In order to show that this nerve is uniformly acyclic,
we use on the one hand that the nerve is _non-uniformly_ homotopic to the
building, which is _non-uniformly_ acyclic. On the other hand, we introduce a
_support control_ principle which allows us to upgrade the acyclicity of the
nerve to its uniform counterpart using a _uniform nerve_ principle that we
provide. This requires a quantitative control on finite subcomplexes; using
building-theoretical arguments, we establish that this control holds in the
case of apartment systems (though not for Euclidean buildings themselves).

We expect these tools to be useful beyond the application to algebraic groups.

Regarding the uniform nerve principle, we recall that _topological_ nerve
theorems fail catastrophically in bounded cohomology. On the one hand, the
circle is boundedly acyclic but the nerve of finite good cover of the circle
is not. In the reverse direction, the nerve of a finite good cover of a
bouquet of two circles has no cohomology in dimension $>1$, while this bouquet
has infinite-dimensional bounded cohomology in dimensions $2$ and $3$.

######  Acknowledgements G (Gratitude).

I am very grateful to Pierre-Emmanuel Caprace and Francesco Fournier-Facio for
their comments.

##  2\. Simplicial methods

###  2.A. Notation

We use the standard notation where a simplicial complex $\Sigma$ is a set of
non-empty finite sets closed under passing to non-empty subsets. The set of
$q$-simplices is denoted $\Sigma_{q}$; strictly speaking we distinguish
$\Sigma_{0}$ from the vertex set
$\operatorname{Vert}(\Sigma)=\bigcup_{\sigma\in\Sigma}\sigma$. The full
simplex $\Delta X$ on a set $X$ is the collection of all non-empty finite
subsets $\sigma\subseteq X$. A subcomplex $\Sigma^{\prime}\subseteq\Sigma$ is
full if it contains every $\sigma\in\Sigma$ with
$\sigma\subseteq\operatorname{Vert}(\Sigma^{\prime})$. A simplicial map is a
map $f\colon\Sigma\to\Sigma^{\prime}$ that is induced by a vertex map
$\operatorname{Vert}\Sigma\to\operatorname{Vert}\Sigma^{\prime}$ also
abusively denoted by $f$.

Given a cover $\mathscr{F}$ of a set $X$, the nerve
$N\mathscr{F}\subseteq\Delta\mathscr{F}$ is the simplicial complex of all non-
empty finite subsets of $\mathscr{F}$ having non-empty intersection.

Given a poset $S$, that is, a set endowed with a partial order, the
corresponding order complex is the simplicial complex $BS\subseteq\Delta S$
consisting of all non-empty finite chains. Thus an element of $BS_{q}$ is of
the form $\\{s_{0}<\cdots<s_{q}\\}$. Both isotone and antitone (i.e. order
preserving/reversing) poset maps induce simplicial maps since the definition
of $BS$ is self-dual.

Our notation reflects the fact that $BS$ represents a classifying space of $S$
viewed as a category (compare [Qui78]). We warn the reader that $BS$ is
sometimes called a “nerve” (and its realisation a classifying space) [Seg68];
adding to the confusion, the nerve that we defined above is itself a
classifying space of a category of inclusions associated to the cover.

Finally, since a simplicial complex $\Sigma$ is a poset under inclusion, we
can form its order complex $B\Sigma$, which is the barycentric subdivision of
$\Sigma$.

We write $C_{q}(\Sigma)$ for the group of real-valued $q$-chains of usual
simplicial homology. This is a normed vector space for the following norm. A
basis of $C_{q}(\Sigma)$ is obtained by choosing an oriented simplex for every
$q$-simplex. Consider the $\ell^{1}$-norm associated to this basis, i.e. the
sum of the absolute values of the coefficients in this basis; this norm does
not depend on the choice of orientation since orientations only affect signs.
The boundary maps $\partial\colon C_{q+1}(\Sigma)\to C_{q}(\Sigma)$ are
augmented by the sum of coefficients $\epsilon\colon
C_{0}(\Sigma)\to\mathbf{R}$.

The simplicial complex $\Sigma$ is uniformly acyclic if its (augmented) chain
complex admits a contracting homotopy $h_{\bullet}$ such that each
$h_{q}\colon C_{q}(\Sigma)\to C_{q+1}(\Sigma)$ is bounded (in the sense of
linear maps between normed vector spaces). This notion was used in various
forms since [MM85] and was recently systematically developed in [KS23] in the
semi-simplicial setting. Examples include all simplicial cones, in particular
any full simplex, where the bound on $h_{q}$ can be chosen to be $1$.

For bounded cohomology (simplicial and beyond), we refer to the founding paper
of Gromov [Gro82] and to [Fri17, Iva20]. Uniform acyclicity implies the
vanishing of simplicial bounded cohomology (“bounded acyclicity”), though the
two are not equivalent.

###  2.B. Support control

The first tool that we introduce is a “support-to-norm” principle leveraging
the fact that the norm on homology cycles is not just any norm, but a
$\ell^{1}$-norm. This allows to control operator norms on a basis, which is an
elementary form of projectivity; in fact, a theorem of Köthe [Köt66] shows
that projective Banach spaces are precisely $\ell^{1}$-spaces.

######  Theorem 2.1.

Let $\Sigma$ be a simplicial complex. Suppose that there is a function
$\varphi\colon\mathbf{N}\to\mathbf{N}$ such that every set of $n$ vertices is
contained in some acyclic subcomplex of $\Sigma$ with at most $\varphi(n)$
vertices.

Then $\Sigma$ is uniformly acyclic.

In preparation for the proof, we introduce an auxiliary notion.

######  Lemma 2.2.

Given $q,r\in\mathbf{N}$, there is a constant $U_{q}(r)$ with the following
property. For any simplicial complex $\Phi$ on at most $r$ vertices and for
any $(q+1)$-chain $\beta\in C_{q+1}(\Phi)$, there is $\beta^{\prime}\in
C_{q+1}(\Phi)$ with $\partial\beta=\partial\beta^{\prime}$ and
$\|\beta^{\prime}\|\leq U_{q}(r)\|\partial\beta\|$.

Moreover, there is a smallest such constant.

######  Definition 2.3.

This smallest constant will be called the universal constant $U_{q}(r)$.

(We could choose a coarser constant depending on $r$ only since finitely many
$q$ are relevant for every given $r$, but our notation allows the proof of 2.1
to extend to a more general setting, recorded in 2.7 below.)

###### Proof of 2.2.

Given any finite simplicial complex $\Phi$, the homology boundary map
$\partial\colon C_{q+1}(\Phi)\to C_{q}(\Phi)$ is a linear map between finite-
dimensional vector spaces and therefore it is an open map (for any norm, in
particular the given $\ell^{1}$-norm). This means that there is a constant $C$
(depending on $\Phi$ and $q$) such that any boundary $\partial\beta$, where
$\beta\in C_{q+1}(\Phi)$, can be written
$\partial\beta=\partial\beta^{\prime}$ for a chain $\beta^{\prime}$ with
$\|\beta^{\prime}\|\leq C\|\partial\beta\|$.

We define $U_{q}(r)$ as the infimum of those $C$ that have this property
simultaneously for all simplicial complexes $\Phi$ on at most $r$ vertices.
This is well-defined since there are only finitely many isomorphism types of
such complexes.

We note that the finite dimensionality of $C_{q+1}(\Phi)$ implies also that
$U_{q}(r)$ itself works as a constant $C$ above, despite the use of the
infimum. ∎

###### Proof of 2.1.

For brevity, we shall say that a chain $\omega\in C_{q}(\Sigma)$ has support
at most $m$ if $\omega$ is a linear combination of at most $m$ oriented
$q$-simplices. We construct by induction on $q\geq-1$ a sequence $h_{q}$ of
linear maps

$\textstyle{0}$$\textstyle{\mathbf{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{-1}}$$\textstyle{C_{0}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\scriptstyle{h_{0}}$$\textstyle{C_{1}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{h_{1}}$$\textstyle{C_{2}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{h_{2}}$$\textstyle{C_{3}(\Sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{h_{3}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$

and a sequence of functions $\psi_{q}\colon\mathbf{N}\to\mathbf{N}$. The
inductive claims are:

  1. (i)

Boundedness: the linear map $h_{q}$ is bounded.

  2. (ii)

Homotopy: the identity map can be written $h_{q-1}\partial+\partial h_{q}$ for
$q>0$, $h_{-1}\epsilon+\partial h_{0}$ for $q=0$ and $\epsilon h_{-1}$ for
$q=-1$.

  3. (iii)

Support: if $q\geq 0$ and $\omega\in C_{q}(\Sigma)$ has support at most
$m\in\mathbf{N}$, then $h_{q}(\omega)$ has support at most $\psi_{q}(m)$.

To start the induction at $q=-1$, we select a vertex $v_{0}$ and define
$h_{-1}(t)=t\\{v_{0}\\}$ for $t\in\mathbf{R}$. We set $\psi_{-1}\equiv 1$. The
first two inductive conditions are satisfied. The last one was not formulated
for $q=-1$, but in view of the inductive step we note that its _conclusion_
still holds since $h_{-1}(t)$ has support at most $1$.

We now address the inductive step for any $q\geq 0$, abusively writing
$\partial$ for $\epsilon$ in the special case $q=0$.

We consider the basis of $C_{q}(\Sigma)$ given by some choice of an oriented
simplex $\dot{\sigma}$ for every $q$-simplex $\sigma$. We shall first define
$h_{q}$ on each $\dot{\sigma}$ (but assuming that $h_{q-1}$ is already given
on its entire domain of definition).

The second inductive assumption implies that
$\alpha=\dot{\sigma}-h_{q-1}\partial\dot{\sigma}$ is a cycle because

$\partial(h_{q-1}\partial\dot{\sigma})=(\partial
h_{q-1})\partial\dot{\sigma}=(\mathrm{Id}-h_{q-2}\partial)\partial\dot{\sigma}=\partial\dot{\sigma}$

for $q>0$, whereas for $q=0$ we have $\partial
h_{-1}\partial\dot{\sigma}=\partial\dot{\sigma}$ from $\partial=\epsilon$.

Since $\partial\dot{\sigma}$ has support at most $q+1$, the third inductive
assumption shows that $\alpha$ has support at most $1+\psi_{q-1}(q+1)$. Thus
at most $n$ vertices are involved, where

$n=(q+1)\big{(}1+\psi_{q-1}(q+1)\big{)}.$

The assumption on $\Sigma$ implies that $\alpha=\partial\beta$ for some
$\beta\in C_{q+1}(\Sigma)$ such that $\beta$ (and hence also $\alpha$) is
supported on a subcomplex $\Phi\subseteq\Sigma$ with at most $\varphi(n)$
vertices. By 2.2, upon possibly replacing $\beta$ by another chain in
$C_{q+1}(\Phi)$, we can assume that $\beta$ has norm at most
$U_{q}(\varphi(n))\|\alpha\|$.

We now define $h_{q}(\dot{\sigma})=\beta$ and extend it by linearity to define
$h_{q}$ on all of $C_{q}(\Sigma)$; this is possible since the various
$\dot{\sigma}$ form a basis. As for $\psi_{q}$, we define it for
$m\in\mathbf{N}$ by

$\psi_{q}(m)=m\cdot\binom{\varphi(n)}{q+2}\,,\kern
8.53581pt\text{recalling}\kern
8.53581ptn=(q+1)\big{(}1+\psi_{q-1}(q+1)\big{)}.$

Let us proceed to verify the inductive claims for $q$.

For the boundedness condition (i), let $\omega\in C_{q}(\Sigma)$. Since
$\omega$ is a finite sum of the form
$\sum_{\sigma\in\Sigma_{q}}\omega(\sigma)\dot{\sigma}$, we have

$\|h_{q}(\omega)\|\leq\sum_{\sigma\in\Sigma_{q}}|\omega(\sigma)|\cdot\|h_{q}(\dot{\sigma})\|\leq\Big{(}\sup_{\sigma\in\Sigma_{q}}\|h_{q}(\dot{\sigma})\|\Big{)}\Big{(}\sum_{\sigma\in\Sigma_{q}}|\omega(\sigma)|\Big{)}.$

Since $\sum_{\sigma\in\Sigma_{q}}|\omega(\sigma)|$ is the $\ell^{1}$-norm of
$\omega$ defined on $C_{q}(\Sigma)$, it suffices to show that the supremum in
the above expression is finite. This follows because the bound obtained above
for $\beta$, namely

$\|h_{q}(\dot{\sigma})\|\leq U_{q}(\varphi(n))\,\|\alpha\|\leq
U_{q}(\varphi(n))\,\left(\|\dot{\sigma}\|+\|h_{q-1}\|\cdot\|\partial\|\cdot\|\dot{\sigma}\|\right),$

is independent of $\dot{\sigma}$, in view of the definition of $n$ and
recalling that $\|\dot{\sigma}\|=1$.

The homotopy condition (ii) is linear and therefore holds by construction
because, on our basis,

$\dot{\sigma}=h_{q-1}\partial\dot{\sigma}+\alpha=h_{q-1}\partial\dot{\sigma}+\partial\beta=(h_{q-1}\partial+\partial
h_{q})\dot{\sigma}.$

Finally, for the support condition (iii), let $m\in\mathbf{N}$ and consider
$\omega\in C_{q}(\Sigma)$ with support at most $m$. Thus $\omega$ is a linear
combination of at most $m$ oriented simplices $\dot{\sigma}$. For each
$\dot{\sigma}$, our construction of $h_{q}(\dot{\sigma})$ is a $(q+1)$-chain
on a complex with at most $\varphi(n)$ vertices, and thus it has support at
most $\binom{\varphi(n)}{q+2}$. It follows as claimed that $h_{q}(\omega)$ has
support at most $\psi_{q}(m)$. ∎

We wrote the above proof in such a way that it shows a formally stronger
statement, recorded in 2.7 below. Only the simpler statement of 2.1 will be
used in this article; the reader can ignore the rest of this subsection and
any mention of semi-simplicial sets.

First we note that the vertex-count $\varphi$, which will be the relevant
quantity in our applications to buildings, only served to bound the support,
in terms of $(q+1)$-simplices, of a $(q+1)$-chain $\beta$ bounding a $q$-cycle
$\alpha$ (whence the binomial coefficients). Therefore we can formalise this
in the more general semi-simplicial setting where simplices are not determined
by vertices:

######  Definition 2.4.

A semi-simplicial set $(\Sigma_{q})_{q\geq 0}$ has support control (below some
$\overline{q}\leq+\infty$) if for every $q<\overline{q}$ there is a function
$\varphi_{q}\colon\mathbf{N}\to\mathbf{N}$ such that every $q$-cycle with
support at most $m$ is the boundary of a $(q+1)$-chain with support at most
$\varphi_{q}(m)$. (The restriction $\overline{q}$ on the range can be useful
in non-acyclic settings such as the spherical Solomon–Tits theorem.) We call
$\varphi_{\bullet}$ the control function.

Accordingly, we replace the universal constants $U_{q}(r)$ by a semi-
simplicial analogue: define the semi-simplicial universal constant
$U^{\mathrm{ss}}_{q}(p)$ by considering all semi-simplicial sets
$\Phi_{\bullet}$ such that $\Phi_{q+1}$ has at most $p$ elements and then
considering the same constant $C$ as in the proof of 2.2 but for the linear
map of finite-dimensional spaces $\partial\colon C_{q+1}(\Phi_{\bullet})\to
C_{q}(\Phi_{\bullet})$.

In the special case of simplicial complexes, the relation to the earlier
constants is thus $U_{q}(r)\leq U^{\mathrm{ss}}_{q}(\binom{r}{q+2})$ and
$\varphi_{q}(m)\leq\binom{\varphi(m(q+1))}{q+2}$.

######  Example 2.5.

On can check that support control below $\overline{q}=1$ is equivalent to:
connected with finite diameter.

######  Remark 2.6.

The definition of support control is given in terms of _real_ cochains. We
will establish it, however, in the stonger form of quantitative
contractibility as in 2.1, which implies support control with any
coefficients. Accordingly we can speak of integral support control for
$\mathbf{Z}$ coefficients, etc.

######  Theorem 2.7 (Technical variant).

Let $(\Sigma_{q})_{q\geq 0}$ be a semi-simplicial set and let
$1\leq\overline{q}\leq+\infty$. Suppose that $\Sigma_{\bullet}$ has support
control below $\overline{q}$.

Then $\Sigma_{\bullet}$ admits a bounded contracting homotopy
$(h_{q})_{q\geq-1}$ up to $q<\overline{q}$.

In particular, the real simplicial bounded cohomology
$\mathrm{H}_{\mathrm{b}}^{q}(\Sigma_{\bullet})$ vanishes in degrees
$0<q<\overline{q}$ and
$\mathrm{H}_{\mathrm{b}}^{\overline{q}}(\Sigma_{\bullet})$ injects into
$\mathrm{H}^{\overline{q}}(\Sigma_{\bullet})$ when $\overline{q}<+\infty$.

###### Proof.

The inductive proof given for 2.1 holds almost unchanged with the adaptations
introduced above. Thus, since $\alpha$ has support at most
$1+\psi_{q-1}(q+1)$, it follows that $\beta$ has support at most
$\varphi_{q}(1+\psi_{q-1}(q+1))$ and therefore the inductive definition of
$\psi_{q}$ becomes

$\psi_{q}(m)=m\cdot\varphi_{q}(1+\psi_{q-1}(q+1)).$

The rest of the proof is unchanged.

If $\overline{q}\neq+\infty$, we stop the inductive argument at
$q=\overline{q}-1$. The statements for bounded cohomology follow by duality,
see Theorems 2.3 and 2.8 in [MM85]. ∎

######  Remark 2.8.

This proof shows that in hindsight we can take the control function to be
linear for every given $q$ since $\psi_{\bullet}$ is in particular also a
control function.

We believe that there are many circumstances where support control is a
helpful method to establish (and strengthen) uniform acyclicity. For instance,
it is well-suited to combinatorial arguments such as glueing:

######  Lemma 2.9.

Let $(\Sigma_{q})_{q\geq 0}$ be a semi-simplicial set and let
$1\leq\overline{q}\leq+\infty$. Suppose that $\Sigma_{\bullet}$ is the union
of two semi-simplicial subsets $\Sigma^{+}_{\bullet}$, $\Sigma^{-}_{\bullet}$.

If $\Sigma^{+}_{\bullet}$ and $\Sigma^{-}_{\bullet}$ have support control
below $\overline{q}$ and $\Sigma^{+}_{\bullet}\cap\Sigma^{-}_{\bullet}$ has
support control below $\overline{q}-1$, then $\Sigma_{\bullet}$ has support
control below $\overline{q}$.

Moreover, the control function for $\Sigma_{\bullet}$ can be taken to depend
only on the control functions for $\Sigma^{\pm}_{\bullet}$ and
$\Sigma^{+}_{\bullet}\cap\Sigma^{-}_{\bullet}$.

If the support is replaced by the norm, then the analogous statement is given
in [KS23, 7.13].

###### Proof.

A $q$-cycle $\alpha$ on $\Sigma_{\bullet}$ can be written
$\alpha^{+}-\alpha^{-}$ for $\alpha^{\pm}\in C_{q}(\Sigma^{\pm}_{\bullet})$
without introducing new simplices in the supports. Then
$\partial\alpha^{+}=\partial\alpha^{-}$ is a $(q-1)$-cycle on
$\Sigma^{+}_{\bullet}\cap\Sigma^{-}_{\bullet}$. If
$\partial\alpha^{\pm}=\partial\omega$ for $\omega\in
C_{q}(\Sigma^{+}_{\bullet}\cap\Sigma^{-}_{\bullet})$, then
$\alpha^{\pm}-\omega$ is a $q$-cycle on $\Sigma^{\pm}_{\bullet}$ and hence can
be written $\partial\beta^{\pm}$ for $\beta^{\pm}\in
C_{q+1}(\Sigma^{\pm}_{\bullet})$. By assumption, the supports of $\omega$ and
$\beta^{\pm}$ can be bounded in terms of the support of $\alpha$. Finally,
$\alpha=\partial(\beta^{+}-\beta^{-})$. ∎

###  2.C. Uniform nerve principles

Leray established the classical correspondance between the homotopy type of a
space and that of the nerve of a good cover; a simplicial version is due to
Borsuk. Our next tool is a uniform version of the simplicial nerve lemma. We
write $\operatorname{Sub}(\Sigma)$ for the poset of subcomplexes of a
simplicial complex $\Sigma$. In order to facilitate the control of the
constants, we make a strong assumption on the intersections (which will be
granted in our applications). We begin with a statement for finite covers (of
generally infinite complexes).

######  Theorem 2.10.

Let $\Sigma$ be a simplicial complex and let
$\mathscr{F}\subseteq\operatorname{Sub}(\Sigma)$ be a finite cover of $\Sigma$
by subcomplexes.

Suppose that every non-empty intersection of elements of $\mathscr{F}$ is a
full simplex.

Then $\Sigma$ is uniformly homotopy equivalent to the nerve complex
$N\mathscr{F}$. Moreover, all bounds can be chosen independently of $\Sigma$
and $\mathscr{F}$, i.e. they depend only on the homology degree.

The overall structure of the argument is similar to a strategy used for
ordinary nerve principles such as in [Bjö81, Lem. 1.1]. One ingredient is the
_carrier lemma_ , for which a uniform version was established in [FFMN24] (in
the greater generality of semi-simplicial sets). Recall that given simplicial
complexes $\Omega,\Omega^{\prime}$ a carrier is an isotone map
$C\colon\Omega\to\operatorname{Sub}(\Omega^{\prime})$; it is uniformly acyclic
if for each $q$, every complex $C(\sigma)$ is uniformly acyclic, uniformly
over $\sigma\in\Omega_{q}$. A simplicial map
$\varphi\colon\Omega\to\Omega^{\prime}$ is carried by $C$ if
$\forall\sigma:\varphi(\sigma)\in C(\sigma)$.

######  Lemma 2.11 ([FFMN24]).

If $\varphi,\psi\colon\Omega\to\Omega^{\prime}$ are carried by the same
uniformly acyclic carrier, then they are boundedly homotopic with constants
depending only on the carrier and the homology degree.

In particular, if $\varphi,\psi\colon S\to S^{\prime}$ are two isotone (or two
antitone) poset maps with $\forall s:\varphi(s)\leq\psi(s)$, then the
corresponding simplicial maps $BS\to BS^{\prime}$ are boundedly homotopic with
constants depending only on the homology degree.

###### Reference for the proof.

The first statement is (the simplicial case of) Lemma 4.11 in [FFMN24]. The
second one is a version of Theorem 4.12 therein and follows from the first by
exhibiting a cone as carrier. ∎

###### Proof of 2.10.

Considering $\Sigma$ and $N\mathscr{F}$ as posets, we define an antitone map

$f\colon\Sigma\longrightarrow N\mathscr{F},\kern
8.53581ptf(\sigma)=\\{F\in\mathscr{F}:\sigma\in F\\}.$

Next, for every $\alpha\in N\mathscr{F}$, we choose some vertex $x_{\alpha}$
of $\cap\alpha$. Define

$g\colon N\mathscr{F}\longrightarrow\Sigma,\kern
8.53581ptg(\beta)=\big{\\{}x_{\alpha}:\alpha\in N\mathscr{F}\text{ with
}\beta\subseteq\alpha\big{\\}}.$

Note that $g(\beta)$ is indeed a simplex of $\Sigma$ since all those
$x_{\alpha}$ are in $\cap\beta$, which is a full simplex. The map $g$ is an
antitone map of posets.

We can now consider $f$ and $g$ as simplicial maps between the corresponding
order complexes $B\Sigma$ and $BN\mathscr{F}$, which are none other than the
barycentric subdivisions of $\Sigma$ and $N\mathscr{F}$. We claim that these
simplicial maps are bounded homotopy inverses to each other with all bounds
depending only on the homology degree. This claim will complete the proof of
the theorem because any simplicial complex is uniformly homotopy equivalent to
its barycentric subdivision with bounds depending only on the homology degree.
Indeed, the classical homotopy equivalences are given by explicit sums
depending only on the degree, see e.g. [Mun84, §17]. (This fact has also been
established in [KS23, §7.7] for the generality of semi-simplicial sets.)

Turning to the claim, we consider the composition $gf$. This is an isotone map
on the poset $\Sigma$ (the vertex set of $B\Sigma$). We have

$gf(\sigma)=\big{\\{}x_{\alpha}:\alpha\in N\mathscr{F}\text{ such that
}\forall F\in\mathscr{F},\sigma\in F\Rightarrow F\in\alpha\big{\\}}$

and $gf(\sigma)$ is a simplex of the subcomplex $\cap f(\sigma)$ of $\Sigma$.
Note that the map which to each $\sigma$ associates $\cap f(\sigma)$ is a
carrier from $\Sigma$ to itself. We define a further carrier map

$C\colon B\Sigma\longrightarrow\operatorname{Sub}(B\Sigma),\kern
8.53581ptC(\\{\sigma_{0}\subsetneqq\cdots\subsetneqq\sigma_{p}\\})=B\big{(}\cap
f(\sigma_{p})\big{)}.$

This carrier $C$ carries the simplicial map $gf$; indeed:

$gf(\\{\sigma_{0}\subsetneqq\cdots\subsetneqq\sigma_{p}\\})=\\{gf(\sigma_{0})\subseteq\cdots\subseteq
gf(\sigma_{p})\\}\in B\big{(}\cap f(\sigma_{p})\big{)}$

since each $gf(\sigma_{j})$ is in $\cap f(\sigma_{j})$ which is a subset of
$\cap f(\sigma_{p})$. On the other hand, $C$ also carries the identity because
$\sigma_{p}\in\cap f(\sigma_{p})$ by definition of $f$. In order to conclude
from the uniform carrier lemma that $gf$ and the identity are uniformly
homotopy equivalent with the desired uniformity of constants, it remains only
to justify that the carrier $C$ is uniformly acyclic with constants depending
only on the degree. But this last point follows from the fact that the carrier
is the barycentric subdivision of $\cap f(\sigma_{p})$, which is a full
simplex.

We now consider the other composition, $fg$, which is simpler. We have

$fg(\beta)=\big{\\{}F\in\mathscr{F}:\forall\alpha\in
N,\beta\subseteq\alpha\Rightarrow x_{\alpha}\in F\big{\\}}.$

Unravelling all definitions, we see that $\beta\subseteq fg(\beta)$. In other
words, $fg$ dominates the identity (as poset maps); therefore, the uniform
carrier lemma applies. ∎

The covers that will appear in the proof of our main result are not finite, in
fact not even locally finite (they have locally the power of continuum), but
turn out to have uniformly acyclic nerves. Therefore, we shall need the
following variant of 2.10. We caution the reader that a difficulty resides in
the fact that successive nested finite subcovers will a priori give distinct
homotopy equivalence maps; we will argue that they must be boundedly homotopic
to each other.

######  Theorem 2.12.

Let $\Sigma$ be a simplicial complex set and let
$\mathscr{F}\subseteq\operatorname{Sub}(\Sigma)$ be a cover of $\Sigma$ by
subcomplexes.

Suppose that every non-empty finite intersection of elements of $\mathscr{F}$
is a full simplex.

Then $\Sigma$ is uniformly acyclic if and only if the nerve $N\mathscr{F}$ is
so.

###### Proof.

Suppose that $N\mathscr{F}$ is uniformly acyclic. Fix $q\in\mathbf{N}$ and
consider any cycle $\omega\in C_{q}(\Sigma)$. Then there is a finite subset
$\mathscr{F}^{\prime}\subseteq\mathscr{F}$ which covers all simplices involved
in $\omega$. We consider the subcomplex $\Sigma^{\prime}$ of $\Sigma$ given by
the union of $\mathscr{F}^{\prime}$ and view $\omega$ as a cycle on
$\Sigma^{\prime}$. The homotopy equivalence of 2.10, applied to
$\Sigma^{\prime}$, means that there are bounded linear chain maps
$C_{q}(\Sigma^{\prime})\to C_{q}(N\mathscr{F}^{\prime})$ and in the opposite
direction which induce mutually inverse isomorphisms in homology. Consider the
cycle $\eta\in C_{q}(N\mathscr{F}^{\prime})$ corresponding to $\omega$ as a
cycle on $N\mathscr{F}$. The uniform acyclicity assumption implies that we can
write $\eta=\partial\vartheta$ for $\vartheta\in C_{q+1}(N\mathscr{F})$ with
$\|\vartheta\|\leq c\|\eta\|$, where $c$ depends only on $q$ and on
$N\mathscr{F}$.

Again $\vartheta$ is supported on $N\mathscr{F}^{\prime\prime}\subseteq
N\mathscr{F}$ for some finite $\mathscr{F}^{\prime\prime}\subseteq\mathscr{F}$
and we can assume that $\mathscr{F}^{\prime\prime}$ contains
$\mathscr{F}^{\prime}$. We apply again 2.10, this time to
$\Sigma^{\prime\prime}=\cup\mathscr{F}^{\prime\prime}$, noting that $\omega$
is also a cycle for this complex. The resulting chain map
$C_{q}(\Sigma^{\prime\prime})\to C_{q}(N\mathscr{F}^{\prime\prime})$ sends
$\omega$ to some cycle $\widehat{\eta}\in C_{q}(N\mathscr{F}^{\prime\prime})$.
We claim that $\widehat{\eta}-\eta=\partial\varphi$ for some $\varphi\in
C_{q+1}(N\mathscr{F}^{\prime\prime})$ with $\|\varphi\|\leq
c^{\prime}\|\omega\|$, where $c^{\prime}$ depends only on $q$, on $\Sigma$ and
on $N\mathscr{F}$. This claim will finish the proof that $\Sigma$ is uniformly
acyclic, since $\widehat{\eta}=\partial(\vartheta+\varphi)$ will then imply
that $\omega$ is a boundary (in $\Sigma^{\prime\prime}$) and since all
constants (including those from 2.10) depend on $q$, $\Sigma$ and
$\mathscr{F}$ only, not on $\omega$.

To justify the claim, we need to compare the two maps

$C_{q}(\Sigma^{\prime})\longrightarrow C_{q}(N\mathscr{F}^{\prime})\
\subseteq\ C_{q}(N\mathscr{F}^{\prime\prime})\kern 5.69054pt\text{and}\kern
5.69054ptC_{q}(\Sigma^{\prime})\ \subseteq\
C_{q}(\Sigma^{\prime\prime})\longrightarrow
C_{q}(N\mathscr{F}^{\prime\prime})$

which produce the cycles $\eta$, respectively $\widehat{\eta}$, from $\omega$.
It suffices to show that these maps are homotopic with uniform constants.

Consider the underlying two poset maps

$f^{\prime}\colon\Sigma^{\prime}\longrightarrow N\mathscr{F}^{\prime}\
\subseteq\ N\mathscr{F}^{\prime\prime}\kern 5.69054pt\text{and}\kern
5.69054ptf^{\prime\prime}\colon\Sigma^{\prime}\ \subseteq\
\Sigma^{\prime\prime}\longrightarrow N\mathscr{F}^{\prime\prime}$

which are constructed in the proof of 2.10. Strictly speaking, the first one
is the corestriction of the map $f$ constructed for $\Sigma^{\prime}$, the
second the restriction of the map $f$ for $\Sigma^{\prime\prime}$. Appealing
again to the uniform carrier lemma in the form of 2.11, it suffices to show
the following: for every $\sigma\in\Sigma^{\prime}$,
$f^{\prime}(\sigma)\subseteq f^{\prime\prime}(\sigma)$. This, however, is
apparent in the definition given for $f$ in the proof of 2.10.

The converse, which we will not need, is proved in exactly the same way.
Namely, given finite subcovers
$\mathscr{F}^{\prime}\subseteq\mathscr{F}^{\prime\prime}\subseteq\mathscr{F}$
and the corresponding subcomplexes
$\Sigma^{\prime}\subseteq\Sigma^{\prime\prime}$, it suffices to compare the
two poset maps

$g^{\prime}\colon N\mathscr{F}^{\prime}\longrightarrow\Sigma^{\prime}\
\subseteq\ \Sigma^{\prime\prime}\kern 5.69054pt\text{and}\kern
5.69054ptg^{\prime\prime}\colon N\mathscr{F}^{\prime}\ \subseteq\
N\mathscr{F}^{\prime\prime}\longrightarrow\Sigma^{\prime\prime}$

arising from the proof of 2.10. If the choice $\alpha\mapsto x_{\alpha}$ has
been fixed once and for all for every $\alpha\in\mathscr{F}$, then indeed
$g^{\prime}(\beta)\subseteq g^{\prime\prime}(\beta)$ holds for all $\beta\in
N\mathscr{F}^{\prime}$ and we conclude as above. ∎

##  3\. The flatmate complex of Euclidean buildings

###  3.A. Euclidean buildings

We shall adopt the viewpoint that a building is a complex endowed with a
system of apartments and refer to [AB08], [Ron09] and [Wei09] for background.
For simplicity, we only consider _irreducible_ buildings, which are therefore
simplicial (rather than polysimplicial) complexes. All arguments below adapt
to the non-irreducible case, but this setting is not needed for our vanishing
results because, as we shall recall in Section 4, the vanishing passes to
finite products of groups (and from finite index subgroups).

In the case of discrete irreducible _Euclidean_ buildings, each apartment is a
full subcomplex isomorphic to a triangulation of a Euclidean space, making the
synonym ‘flat’ especially congruous. We recall that the apartments are
combinatorially convex, which means by definition that every minimal gallery
connecting two chambers of an apartment remains in that apartment; see e.g.
Prop. 4.40 in [AB08].

######  Proposition 3.1.

Given a discrete irreducible Euclidean building, there exists an integer $k$
with the following property.

For every $n\in\mathbf{N}$ and every family of $n$ apartments
$F_{1},\ldots,F_{n}$, there exists a family of $kn$ apartments
$E_{1},\ldots,E_{kn}$ such that the union

$F_{1}\cup\cdots\cup F_{n}\cup E_{1}\cup\cdots\cup E_{kn}$

is contractible (as a simplicial complex).

###### Proof.

We define $k$ to be the number of chambers of the spherical Coxeter complex
associated to the building. Specifically, we realise it as the number of
chambers at infinity of any apartment.

Fix some chamber $c$ of the Euclidean building. Given any apartment $F$ and
any chamber $\xi$ of the apartment at infinity $\partial F$, choose some
apartment $E_{\xi}$ containing $c$ with $\xi\in\partial E_{\xi}$; for the
existence of such $E_{\xi}$, see e.g. Prop. 7.6 in [Wei09]. We claim that the
union of those $k$ apartments $E_{\xi}$ contains $F$.

Indeed, select a special vertex $y$ of $c$ and consider the sector $S_{\xi}$
based at $y$ and representing $\xi$. The combinatorial convexity of apartments
implies that $S_{\xi}$ is contained in $E_{\xi}$. However, it is known that
the union of the sectors $S_{\xi}$ contains $F$ as $\xi$ ranges over the
chambers of $\partial F$; this holds in the greater generality of possibly
non-discrete Euclidean buildings, see e.g. the proof of [Hit11, Lem. 6.3] or
of [BSH14, Prop. 7.3]. This justifies the claim.

Returning to the statement of the proposition, we define the family $E_{j}$ as
the collection of all $E_{\xi}$ chosen as above for each
$F=F_{1},\ldots,F_{n}$. By the claim, the union in the statement of the
proposition reduces to the union of all $E_{j}$. That union is combinatorially
starlike with respect to $c$, which by definition means the following: any
minimal gallery from $c$ to any chamber in the union remains in this union.
Indeed, this holds by combinatorial convexity of the apartments since every
$E_{j}$ contains $c$. In only remains to note that, in a Euclidean building,
combinatorially starlike chamber subcomplexes are contractible by exactly the
same shellability argument as used for the Solomon–Tits theorem [AB08, Thm.
4.127] to obtain the contractibility of the Euclidean building itself, compare
[AB08, Ex. 4.130]. ∎

###  3.B. The flatmate complex

We begin with one more general construction of simplicial complexes. Let $X$
be a set and $\mathscr{F}$ a cover of $X$. Given $F\in\mathscr{F}$, the full
simplex $\Delta F$ is a (full) subcomplex of $\Delta X$. We can therefore
define a subcomplex $\Delta_{\mathscr{F}}X$ of $\Delta X$ by

$\Delta_{\mathscr{F}}X=\bigcup_{F\in\mathscr{F}}\Delta F.$

We now apply the nerve principle of 2.12 to these complexes.

######  Corollary 3.2.

Let $X\neq\varnothing$ be a set and let $\mathscr{F}$ be a cover of $X$ by
non-empty subsets.

If $N\mathscr{F}$ is uniformly acyclic, then so is $\Delta_{\mathscr{F}}X$.

###### Proof.

By definition, the complex $\Delta_{\mathscr{F}}X$ is covered by the family of
subcomplexes $\Delta F$ as $F$ ranges over $\mathscr{F}$. For any collection
$F_{1},\ldots,F_{n}$ of elements of $\mathscr{F}$, we have

$\Delta\big{(}F_{1}\cap\ldots\cap
F_{n}\big{)}=\Delta(F_{1})\cap\ldots\cap\Delta(F_{n}).$

Thus the nerve of this cover of $\Delta_{\mathscr{F}}X$ is canonically
isomorphic to the nerve $N\mathscr{F}$ of the cover of $X$. We are therefore
indeed in the situation of 2.12. ∎

We now specialise to buildings and flats.

Consider a building $\mathscr{B}$ with apartment system $\mathscr{A}$. The
apartment system provides in particular a cover $\mathscr{F}$ of the set $X$
of vertices of $\mathscr{B}$; formally,

$\mathscr{F}=\big{\\{}F=\operatorname{Vert}(A):A\in\mathscr{A}\big{\\}}\kern
8.53581pt\text{covers}\kern 8.53581ptX=\operatorname{Vert}(\mathscr{B}).$

The following is the simplicial form of the algebraic definition in terms of
chain groups that we proposed with Bucher in [BM19, Scholium].

######  Definition 3.3.

The flatmate complex of the building $\mathscr{B}$ is the simplicial complex
$\Delta_{\mathscr{F}}X$ as defined above. We also refer to it, in Tits’s
tongue, as the cokotcomplex of $\mathscr{B}$.

We can now answer the problem suggested in [BM19], as announced in F. We
restate it here since the notation has now been introduced:

######  Theorem 3.4.

The flatmate complex of any discrete irreducible Euclidean building is
uniformly acyclic.

The remaining ingredient for the proof of this theorem is as follows.

######  Proposition 3.5.

Let $\mathscr{B}$ be a discrete irreducible Euclidean building with apartment
system $\mathscr{A}$. Then the nerve $N\mathscr{A}$ is uniformly acyclic.

###### Proof.

We shall argue that the simplicial complex $\Sigma=N\mathscr{A}$ satisfies the
assumption of 2.1 for the function $\varphi(n)=(k+1)n$, where $k$ is the
apartment size of the associated spherical building as in Section 3.A. To that
end, note that for any non-empty subset $\mathscr{A}_{0}\subseteq\mathscr{A}$,
the nerve $N\mathscr{A}_{0}$ is a subcomplex of $\Sigma$. In view of 3.1, we
only need to justify the following claim:

Given any non-empty finite subset $\mathscr{A}_{0}$ of $\mathscr{A}$, we
consider the simplicial subcomplex $\mathscr{B}_{0}$ of $\mathscr{B}$ covered
by $\mathscr{A}_{0}$. The claim is that if the simplicial complex
$\mathscr{B}_{0}$ is contractible, then so is the nerve $N\mathscr{A}_{0}$.

To justify the claim, we appeal to the nerve lemma in _ordinary_ simplicial
homology. This is sometimes called the Borsuk nerve lemma; in precisely the
setting of abstract simplicial complexes, two proofs can be found in [Bjö81,
Lem. 1.1]. That lemma asserts that $\mathscr{B}_{0}$ and $N\mathscr{A}_{0}$
have the same homotopy type provided that every non-empty intersection of
subcomplexes taken from the family $\mathscr{A}_{0}$ is contractible. Such an
intersection is a subcomplex of the building $\mathscr{B}$, namely a non-empty
intersection of apartments. The combinatorial convexity of apartments implies
that this intersection is contractible and therefore the claim is established.
∎

###### End of proof of 3.4 (i.e. F).

The nerve $N\mathscr{F}$ coincides with the nerve $N\mathscr{A}$ of the cover
of the building $\mathscr{B}$ by its apartments. Thus 3.5 states that
$N\mathscr{F}$ is uniformly acyclic. Therefore, 3.2 implies indeed that the
flatmate complex $\Delta_{\mathscr{F}}X$ is uniformly acyclic. ∎

##  4\. Vanishing theorems

We first recall some terminology. A locally compact group $G$ is boundedly
acyclic (as a topological group) if its continuous bounded cohomology with
real coefficients $\mathrm{H}_{\mathrm{cb}}^{n}(G)$ vanishes in every degree
$n>0$.

A topological group is amenable if every jointly continuous affine $G$-action
on any non-empty convex compact set (in any Hausdorff locally convex
topological vector space) admits a fixed point. This holds notably when $G$ is
compact or soluble (e.g. abelian), and is preserved by group extensions.

A subgroup $H<G$ of the topological group $G$ is co-amenable in $G$ if $G$ has
the above fixed-point property for the subclass of those convex compact sets
having an $H$-fixed point. This holds for instance if $H$ has finite index, or
if $H$ is normal with $G/H$ amenable.

We shall use the well-known general principles summarised in the proposition
below to reduce the proof of A from general algebraic groups to the simple
case.

######  Proposition 4.1.

A locally compact group $G$ is boundedly acyclic in each of the following
cases:

  1. (i)

$G$ admits a co-amenable closed subgroup which is boundedly acyclic;

  2. (ii)

$G$ admits an amenable normal closed subgroup $N\lhd G$ with $G/N$ boundedly
acyclic;

  3. (iii)

$G$ is the direct product of finitely many boundedly acyclic groups;

  4. (iv)

$G$ is the quotient of a boundedly acyclic group by an amenable normal closed
subgroup.

###### Proof.

For (i), see [Mon01, Prop. 8.6.6]. For (ii) and (iv), see [Mon01, Cor. 8.5.2].
For (iii), combine [Mon01, Prop. 12.2.1] with [Mon01, Prop. 12.2.2(ii)]. These
references make a second countability assumption which is not necessary for
real coefficients (but in our case all algebraic groups are second countable
anyways). ∎

###  4.A. Automorphism groups of buildings

Recall that a group of building automorphisms is strongly transitive if it
acts transitively on the set of pairs consisting of a chamber and an apartment
containing it.***It is ironic that the term Tits chose for his marvelous
concept of building is _immeuble_ — literally: _that which cannot be moved_ —
whereas he demonstrated how deeply buildings are entwined with their rich
transformation groups. The fact that B should follow from 3.4 was introduced
in [BM19]. We shall nonetheless give all details of the argument since the
language is somewhat different here.

###### Proof of B.

Let $G$ be a locally compact group with a strongly transitive proper action by
automorphisms on a locally finite Euclidean building $\mathscr{B}$ with
apartment system $\mathscr{A}$. In order to prove the vanishing of the bounded
cohomology $\mathrm{H}_{\mathrm{cb}}^{n}(G)$ for all $n>0$, we can assume that
$\mathscr{B}$ is irreducible and hence fits the assumptions of 3.4. Indeed,
the automorphism group of a product admits the product of the automorphism
groups of the factors as a finite index subgroup.

We therefore consider the flatmate complex $\Delta_{\mathscr{F}}X$ on
$X=\operatorname{Vert}\mathscr{B}$ defined in Section 3.B and note that $G$
acts on it by simplicial automorphisms. The bounded simplicial cochains of
this flatmate complex yield an augmented cochain complex

$\textstyle{0}$$\textstyle{\mathbf{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\ell^{\infty}\big{(}(\Delta_{\mathscr{F}}X)_{0}\big{)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{{\ell^{\infty}\big{(}(\Delta_{\mathscr{F}}X)_{1}\big{)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$

In terms of vertices, a $q$-cochain
$f\in\ell^{\infty}\big{(}(\Delta_{\mathscr{F}}X)_{q}\big{)}$ is a bounded
alternating function on $(q+1)$-tuples of vertices of the building, where each
tuple is restricted to lie in some apartment. (This is the viewpoint adopted
in [BM19].)

3.4 above implies that this cochain complex is acyclic since it is the norm
dual of the normed chain complex, which is boundedly acyclic by 3.4 (this is
the duality of vanishing introduced by [MM85]). On the other hand, this dual
cochain complex is a $G$-complex of Banach $G$-modules and thus it is a
resolution of $\mathbf{R}$ in the sense of continuous bounded cohomology
[Mon01].

The properness assumption implies that for each $q\geq 0$ the $G$-action on
any set of $(p+1)$-tuples is also proper; this guarantees that each of the
modules $\ell^{\infty}\big{(}(\Delta_{\mathscr{F}}X)_{q}\big{)}$ is relatively
injective in the sense of continuous bounded cohomology, see [Mon01, Thm.
4.5.2]. It follows that the continuous bounded cohomology of $G$ is
canonically realised by the subcomplex of $G$-invariant functions in
$\ell^{\infty}\big{(}(\Delta_{\mathscr{F}}X)_{q}\big{)}$, see e.g. [Mon01,
Thm. 7.2.1].

We continue along the lines that we proposed with Bucher in [BM19]. Choose
some apartment $E\in\mathscr{A}$ and denote by $H<G$ its (set-wise) stabiliser
in $G$. The restriction to tuples in $E$ is a chain map

$\ell^{\infty}\big{(}(\Delta_{\mathscr{F}}X)_{q}\big{)}^{G}\longrightarrow\ell^{\infty}\big{(}(\operatorname{Vert}E)^{q+1}\big{)}^{H}$

The claim is that this map is bijective, thus establishing an isomorphism
between the continuous bounded cohomology of $G$ and of $H$. This will
complete the proof because $H$ is an amenable group and is therefore boundedly
acyclic.

The injectivity of the claim follows from the transitivity of $G$ on
$\mathscr{A}$ and the definition of the flatmate complex
$\Delta_{\mathscr{F}}X$. Surjectivity is the only time where we are using the
strong transitivity, as follows.

Pick $f$ in $\ell^{\infty}((\operatorname{Vert}E)^{q+1})^{H}$ and let $x$ be
any $(q+1)$-tuple of vertices in any apartment (i.e. $x$ represents any
simplex of the flatmate complex). We know that $gx\subseteq E$ for some $g\in
G$ and we want to extend $f$ to this $x$ by setting $f(x)=f(gx)$. To show that
this is well-defined and that the resulting function $f$ on
$(\Delta_{\mathscr{F}}X)_{q}$ is $G$-invariant, the only point to verify is
that any other $g^{\prime}\in G$ with $g^{\prime}x\subseteq E$ satisfies
$f(g^{\prime}x)=f(gx)$.

Consider the two apartments $g^{-1}E$ and ${g^{\prime}}^{-1}E$; both contain
$x$. By strong transitivity, there exists $q\in G$ which maps $g^{-1}E$ to
${g^{\prime}}^{-1}E$ and such that $q$ fixes pointwise the intersection
$g^{-1}E\cap{g^{\prime}}^{-1}E$, see e.g. [AB08, Prop. 6.6]. In particular,
$q$ fixes every vertex in $x$. Now $h=g^{\prime}qg^{-1}$ is an element of $H$
and $hgx=g^{\prime}qx=g^{\prime}x$. Since $f$ was supposed $H$-invariant on
tuples in $E$, it follows $f(gx)=f(g^{\prime}x)$ as desired. ∎

######  Remark 4.2.

In view of potential generalisations, we point out that the above argument
used only the bounded acyclicity, rather than the amenability, of the
apartment stabiliser $H$.

###  4.B. Algebraic groups

We now turn to the proof of A and consider an arbitrary algebraic group
$\mathbf{G}$ over a non-Archimedean local field $k$. Our goal is to show that
the locally compact group $G=\mathbf{G}(k)$ is boundedly acyclic. We recall
here that $G$ is endowed with the canonical Hausdorff “strong” topology
[Mum99, I§10].

We use general structure theory to reduce $\mathbf{G}$ to a simpler class of
groups, the class of simple groups. We need however to keep track of
$G=\mathbf{G}(k)$ in view of the discrepancy between quotients of algebraic
groups and of the corresponding $k$-points.

There is no loss of generality in assuming $\mathbf{G}$ connected since
passing to $\mathbf{G}^{0}$ will replace $G$ by a finite index closed
subgroup, which is fine by 4.1(i).

We first recall that there is a canonical affinisation quotient, i.e. a
faithfully flat morphism $\psi\colon\mathbf{G}\to\mathbf{G}_{\mathrm{aff}}$ to
an affine group
$\mathbf{G}_{\mathrm{aff}}=\mathrm{Spec}\mathscr{O}(\mathbf{G})$ and that the
kernel $\operatorname{Ker}\psi$ is contained in the center of $\mathbf{G}$.
This is a general form of a theorem of Rosenlicht [Ros56] established in
[DG70, III.3.8]: combine Thm. 8.2 and Cor. 8.3 therein under our assumption
$\mathbf{G}=\mathbf{G}^{0}$. It can also be read in [Bri17, Thm. 1].

In particular, $\operatorname{Ker}\psi$ is commutative and
$\mathbf{G}_{\mathrm{aff}}$, being affine, is a linear algebraic group [Wat79,
§3.4].

The quotient of $\mathbf{G}_{\mathrm{aff}}$ by its radical is a connected
semi-simple group $\mathbf{S}$. Thus $\mathbf{S}$ is the almost-direct product
of some number $r$ of (quasi-)simple connected factors $\mathbf{G}_{i}$,
$i=1,\ldots r$. We reorder the factors so that $\mathbf{G}_{i}$ is
$k$-isotropic exactly when $i\leq s$ for some $0\leq s\leq r$.

Recall that $\mathbf{G}_{i}(k)^{+}$ denotes the normal subgroup of
$\mathbf{G}_{i}(k)$ generated by the $k$-points of the $k$-split unipotent
subgroups of $\mathbf{G}_{i}$. This group is introduced in detail by
Borel–Tits [BT73, §6]; several equivalent definitions are given in [BT73,
6.2]. In many cases (including local fields of characteristic zero),
$\mathbf{G}_{i}(k)^{+}=\mathbf{G}_{i}(k)$ holds for isotropic groups; in the
general case, we shall use that the quotient
$\mathbf{G}_{i}(k)/\mathbf{G}_{i}(k)^{+}$ is compact [BT73, 6.14].

Consider first $i\leq s$. Then Bruhat–Tits theory [BT72, BT84] (as above we
refer to [AB08], [Ron09] and [Wei09] for background) shows that the locally
compact group $\mathbf{G}_{i}(k)$ acts strongly transitively on an irreducible
locally finite Euclidean building. The resulting action of
$\mathbf{G}_{i}(k)^{+}$ is still strongly transitive; this follows e.g. from
the decomposition given in [BT73, 6.11(i)]. Moreover this action is proper
since the center of $\mathbf{G}_{i}(k)$ is finite. Therefore, B implies that
$\mathbf{G}_{i}(k)^{+}$ is boundedly acyclic for all $i\leq s$. If $i>s$, then
by convention $\mathbf{G}_{i}(k)^{+}$ is trivial. In conclusion, 4.1(iii)
allows us to obtain that the product $\prod_{i=1}^{r}\mathbf{G}_{i}(k)^{+}$ is
boundedly acyclic. It follows by 4.1(iv) that $\mathbf{S}(k)^{+}$ is also
boundedly acyclic because $\mathbf{S}(k)^{+}$ is the almost-direct product of
the $\mathbf{G}_{i}(k)^{+}$, see [BT73, 6.2(iii)].

At this point we consider the continuous group homomorphism

$f\colon
G=\mathbf{G}(k)\longrightarrow\mathbf{G}_{\mathrm{aff}}(k)\longrightarrow\mathbf{S}(k)=\mathbf{G}_{1}(k)\cdots\mathbf{G}_{r}(k).$

We claim that the image $f(G)$ in $\mathbf{S}(k)$ contains
$\mathbf{S}(k)^{+}$. Indeed this image is a Zariski-dense normal subgroup;
therefore it must contain each $\mathbf{G}_{i}(k)^{+}$ since the latter is
abstractly simple modulo its center by the main result of [Tit64]. The claim
follows.

We deduce that the pre-image $G^{+}<G$ of $\mathbf{S}(k)^{+}$ is a normal
cocompact subgroup of $G$. In particular it is co-amenable and therefore, by
4.1(i), it suffices to show that $G^{+}$ is boundedly acyclic. Since we
already know that $\mathbf{S}(k)^{+}$ is boundedly acyclic, this follows from
4.1(ii) if we justify that the kernel of $f|_{G^{+}}$ is amenable. By
construction, this kernel is contained in a central extension of the group of
$k$-points of the radical of $\mathbf{G}_{\mathrm{aff}}$; therefore it is
soluble and this completes the proof.∎

######  Remark 4.3.

$\operatorname{Ker}\psi$ and the radical of $\mathbf{G}_{\mathrm{aff}}$ could
have been combined into one “radical of $\mathbf{G}$” in the above reductions,
but the author is more comfortable separating the two steps since the
classical structure theory is often stated in the context of _linear_
algebraic groups [Bor91, Hum75, Spr98].

We still need to justify C, for which we claim no originality. The idea is
taken from the introduction of [BM02b], but translated to the non-Archimedean
context.

Let $G=\mathbf{SL}_{2}(\mathbf{Q}_{p})$ and let $\Gamma$ be a (finite rank
non-abelian) free group realised as a cocompact lattice in $G$. Then
$\mathrm{H}_{\mathrm{b}}^{2}(\Gamma)$ is non-trivial: this was already
established by Johnson in [Joh72, Prop. 2.8] and rediscovered by Brooks
[Bro81]. The coefficient induction of bounded cohomology [Mon01, §10.1]
implies that $\mathrm{H}_{\mathrm{cb}}^{2}(G,L^{\infty}(G/\Gamma))$ is also
non-trivial. Since we are in degree two, a “double ergodicity with
coefficients” argument shows that
$\mathrm{H}_{\mathrm{cb}}^{2}(G,L^{2}(G/\Gamma))$ is non-vanishing: this was
established in [BM99], see also [BM02a, Cor. 11].

Consider now a direct integral decomposition of $L^{2}(G/\Gamma)$ into
irreducible continuous unitary representations of $G$, which is even a
Hilbertian sum decomposition in this case [DE14, Thm. 9.2.2]. Appealing again
to double ergodicity with coefficients, we conclude that some of these
representations must have non-trivial $\mathrm{H}_{\mathrm{cb}}^{2}$ (see e.g.
Thm. 3.3 and Cor. 3.4 in [MS06]; these are stated for discrete groups but hold
verbatim in the locally compact case).∎

######  Remark 4.4.

The first paragraph of the above argument can be replaced with a geometric
construction of a cocycle with coefficients in a multiple of the regular
representation of $G$, as explained in [MS03, §2] and [MS04, §4]. Then the
second paragraph holds using a direct integral decomposition (the Plancherel
decomposition).

###  4.C. Arithmetic groups

We retain the notation of Dbis and we refer to [Mar91, §I.3] for background on
$S$-arithmetic groups, notably the following few facts:

Given $v\in S$ we denote by $K_{v}$ the corresponding completion and recall
that $\mathbf{G}(K_{v})$ is non-compact precisely when $\mathbf{G}$ is
$K_{v}$-isotropic, i.e. when $\mathrm{rank}_{K_{v}}(\mathbf{G})>0$. Let
$S_{1}\subseteq S$ be the collection of those isotropic valuations in $S$; we
can assume $S_{1}\neq\emptyset$ since otherwise the statement is void. The
assumption $S_{0}\subseteq S$ implies that $\Gamma$ is a lattice in
$\prod_{v\in S_{1}}\mathbf{G}(K_{v})$ and this lattice is irreducible by the
strong approximation theorem. We also note at this point that D is indeed a
particular case of Dbis.

Applying [Mon10, Cor. 1.4], we deduce that the restriction map from the full
product of Archimedean as well as non-Archimedean groups

$\mathrm{H}_{\mathrm{cb}}^{n}\big{(}\prod_{v\in
S_{1}}\mathbf{G}(K_{v})\big{)}\longrightarrow\mathrm{H}_{\mathrm{b}}^{n}(\Gamma)$

is an isomorphism for all $n<2\sum_{v\in S}\mathrm{rank}_{K_{v}}(\mathbf{G})$.
Next we observe that the restriction map from the Lie group considered in Dbis
is induced by the composition

$\Gamma\longrightarrow\prod_{v\in
S_{1}}\mathbf{G}(K_{v})\longrightarrow\prod_{v\in S_{0}}\mathbf{G}(K_{v})=L,$

where the right arrow is the projection. Therefore, what is needed is to prove
that the inflation map corresponding to that projection is an isomorphism. A
implies that this is true in all degrees, using the Hochschild–Serre sequence;
specifically, the statements of Prop. 12.2.1 and Prop. 12.2.2(ii) in [Mon01].∎

## References

  * [AB08] Peter Abramenko and Kenneth Stephen Brown, _Buildings_ , Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008, Theory and applications.
  * [BBI18] Michelle Bucher, Marc Burger, and Alessandra Iozzi, _The bounded Borel class and 3-manifold groups_ , Duke Math. J. 167 (2018), no. 17, 3129–3169.
  * [Beh69] Helmut Behr, _Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern_ , Invent. Math. 7 (1969), 1–32.
  * [Bjö81] Anders Björner, _Homotopy type of posets and lattice complementation_ , J. Combin. Theory Ser. A 30 (1981), no. 1, 90–100.
  * [BM99] Marc Burger and Nicolas Monod, _Bounded cohomology of lattices in higher rank Lie groups_ , J. Eur. Math. Soc. (JEMS) 1 (1999), no. 2, 199–235.
  * [BM02a] Marc Burger and Nicolas Monod, _Continuous bounded cohomology and applications to rigidity theory_ , Geom. Funct. Anal. 12 (2002), no. 2, 219–280.
  * [BM02b] Marc Burger and Nicolas Monod, _On and around the bounded cohomology of ${\rm SL}_{2}$_, Rigidity in dynamics and geometry (Cambridge, 2000), Springer, Berlin, 2002, pp. 19–37.
  * [BM19] Michelle Bucher and Nicolas Monod, _The bounded cohomology of $\rm SL_{2}$ over local fields and $S$-integers_, Int. Math. Res. Not. IMRN (2019), no. 6, 1601–1611.
  * [Bor63] Armand Borel, _Some finiteness properties of adele groups over number fields_ , Publ. Math., Inst. Hautes Étud. Sci. 16 (1963), 101–126.
  * [Bor91] Armand Borel, _Linear algebraic groups_ , second ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991.
  * [Bri17] Michel Brion, _Some structure theorems for algebraic groups_ , Algebraic groups: structure and actions, Proc. Sympos. Pure Math., vol. 94, Amer. Math. Soc., Providence, RI, 2017, pp. 53–126.
  * [Bro81] Robert Brooks, _Some remarks on bounded cohomology_ , Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., vol. 97, Princeton Univ. Press, 1981, pp. 53–63.
  * [BS76] Armand Borel and Jean-Pierre Serre, _Cohomologie d’immeubles et de groupes $S$-arithmétiques_, Topology 15 (1976), no. 3, 211–232.
  * [BSH14] Curtis Dwight Bennett and Petra Schwer (Hitzelberger), _On axiomatic definitions of non-discrete affine buildings_ , Adv. Geom. 14 (2014), no. 3, 381–412, With an appendix by Koen Struyve.
  * [BT72] François Bruhat and Jacques Tits, _Groupes réductifs sur un corps local_ , Inst. Hautes Études Sci. Publ. Math. (1972), no. 41, 5–251.
  * [BT73] Armand Borel and Jacques Tits, _Homomorphismes “abstraits” de groupes algébriques simples_ , Ann. of Math. (2) 97 (1973), 499–571.
  * [BT84] François Bruhat and Jacques Tits, _Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée_ , Inst. Hautes Études Sci. Publ. Math. (1984), no. 60, 197–376.
  * [Buc04] Michelle Bucher, _Characteristic classes and bounded cohomology_ , Ph.D. thesis, ETHZ Diss. Nr. 15636, 2004.
  * [Buc07] Michelle Bucher, _Finiteness properties of characteristic classes of flat bundles_ , Enseign. Math. (2) 53 (2007), no. 1-2, 33–66.
  * [BW80] Armand Borel and Nolan Russell Wallach, _Continuous cohomology, discrete subgroups, and representations of reductive groups_ , Princeton University Press, 1980.
  * [Cas74] William Allen Casselman, _On a $p$-adic vanishing theorem of Garland_, Bull. Amer. Math. Soc. 80 (1974), 1001–1004.
  * [CFFLM23] Caterina Campagnolo, Francesco Fournier-Facio, Yash Lodha, and Marco Moraschini, _An algebraic criterion for the vanishing of bounded cohomology_ , preprint, arXiv:2311.16259v3, 2023.
  * [CW74] William Allen Casselman and David Wigner, _Continuous cohomology and a conjecture of Serre’s_ , Invent. Math. 25 (1974), 199–211.
  * [DE14] Anton Deitmar and Siegfried Echterhoff, _Principles of harmonic analysis_ , second ed., Universitext, Springer, Cham, 2014.
  * [DG70] Michel Demazure and Pierre Gabriel, _Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs_ , Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970, Avec un appendice Corps de classes local par Michiel Hazewinkel.
  * [DlCM23] Carlos De la Cruz Mengual, _The degree-three bounded cohomology of complex Lie groups of classical type_ , Preprint, arXiv:2304.00607v2, 2023.
  * [DlCMH23] Carlos De la Cruz Mengual and Tobias Hartnick, _Stabilization of bounded cohomology for classical groups_ , Preprint, arXiv:2201.03879, 2023.
  * [DlCMH24] Carlos De la Cruz Mengual and Tobias Hartnick, _A Quillen stability criterion for bounded cohomology_ , Preprint, arXiv:2307.12808, 2024.
  * [Dup79] Johan L. Dupont, _Bounds for characteristic numbers of flat bundles_ , Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, 1978), Springer, Berlin, 1979, pp. 109–119.
  * [FFLM23] Francesco Fournier-Facio, Clara Löh, and Marco Moraschini, _Bounded cohomology and binate groups_ , J. Aust. Math. Soc. 115 (2023), no. 2, 204–239.
  * [FFMN24] Francesco Fournier-Facio, Nicolas Monod, and Sam Nariman, _The bounded cohomology of transformation groups of Euclidean spaces and discs_ , with an appendix by A. Kupers. Preprint, arXiv:2405.20395v1, 2024.
  * [Fri17] Roberto Frigerio, _Bounded cohomology of discrete groups_ , Mathematical Surveys and Monographs, vol. 227, American Mathematical Society, Providence, RI, 2017.
  * [Gar73] Howard Garland, _$p$ -adic curvature and the cohomology of discrete subgroups of $p$-adic groups_, Ann. of Math. (2) 97 (1973), 375–423.
  * [Gar75] Howard Garland, _On the cohomology of discrete subgroups of $p$-adic groups_, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 449–453.
  * [Gau01] Carl Friedrich Gauß, _Disquisitiones Arithmeticae_ , Gerhard Fleischer, Leipzig, 1801.
  * [GLMR23] Lev Glebsky, Alexander Lubotzky, Nicolas Monod, and Bharatram Rangarajan, _Asymptotic cohomology and uniform stability for lattices in semisimple groups_ , 2023, Preprint, arXiv:2301.00476v4.
  * [Gro82] Michaïl Gromov, _Volume and bounded cohomology_ , Inst. Hautes Études Sci. Publ. Math. (1982), no. 56, 5–99 (1983).
  * [Har69] Günter Harder, _Minkowskische Reduktionstheorie über Funktionenkörpern_ , Invent. Math. 7 (1969), 33–54.
  * [Har77] Günter Harder, _Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern_ , Invent. Math. 42 (1977), 135–175.
  * [Hit11] Petra Hitzelberger, _Non-discrete affine buildings and convexity_ , Adv. Math. 227 (2011), no. 1, 210–244.
  * [HO12] Tobias Hartnick and Andreas Ott, _Surjectivity of the comparison map in bounded cohomology for Hermitian Lie groups_ , Int. Math. Res. Not. IMRN (2012), no. 9, 2068–2093.
  * [HO15] Tobias Hartnick and Andreas Ott, _Bounded cohomology via partial differential equations, I_ , Geom. Topol. 19 (2015), no. 6, 3603–3643.
  * [Hum75] James Edward Humphreys, _Linear algebraic groups_ , Graduate Texts in Mathematics, vol. No. 21, Springer-Verlag, New York-Heidelberg, 1975.
  * [Iva20] Nikolai V. Ivanov, _Notes on the bounded cohomology theory_ , Preprint, arXiv:1708.05150v3, 2020.
  * [Joh72] Barry E. Johnson, _Cohomology in Banach algebras_ , AMS, 1972, Mem. Am. Math. Soc. 127.
  * [Köt66] Gottfried Köthe, _Hebbare lokalkonvexe Räume_ , Math. Ann. 165 (1966), 181–195.
  * [KS23] Thorben Kastenholz and Robin Janik Sroka, _Simplicial bounded cohomology and stability_ , 2023, Preprint, arXiv:2309.05024v1.
  * [Löh17] Clara Löh, _A note on bounded-cohomological dimension of discrete groups_ , J. Math. Soc. Japan 69 (2017), no. 2, 715–734.
  * [Mar91] Gregori Aleksandrovich Margulis, _Discrete subgroups of semisimple Lie groups_ , Springer-Verlag, Berlin, 1991.
  * [Mil58] John Milnor, _On the existence of a connection with curvature zero_ , Comment. Math. Helv. 32 (1958), 215–223.
  * [MM85] Shigenori Matsumoto and Shigeyuki Morita, _Bounded cohomology of certain groups of homeomorphisms_ , Proc. Amer. Math. Soc. 94 (1985), no. 3, 539–544.
  * [MN23] Nicolas Monod and Sam Nariman, _Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups_ , Invent. Math. 232 (2023), no. 3, 1439–1475.
  * [Mon01] Nicolas Monod, _Continuous bounded cohomology of locally compact groups_ , Lecture Notes in Mathematics 1758, Springer, Berlin, 2001.
  * [Mon04] Nicolas Monod, _Stabilization for $\ {\rm SL}_{n}$ in bounded cohomology_, Discrete geometric analysis, Contemp. Math., vol. 347, Amer. Math. Soc., Providence, RI, 2004, pp. 191–202.
  * [Mon06] Nicolas Monod, _An invitation to bounded cohomology_ , Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society, 1183–1211, 2006.
  * [Mon10] Nicolas Monod, _On the bounded cohomology of semi-simple groups, $S$-arithmetic groups and products_, J. reine angew. Math. [Crelle’s J.] 640 (2010), 167–202.
  * [Mon22] Nicolas Monod, _Lamplighters and the bounded cohomology of Thompson’s group_ , Geom. Funct. Anal. 32 (2022), no. 3, 662–675.
  * [MS03] Nicolas Monod and Yehuda Shalom, _Negative curvature from a cohomological viewpoint and cocycle superrigidity_ , C. R. Acad. Sci. Paris Sér. I Math. 337 (2003), no. 10, 635–638.
  * [MS04] Nicolas Monod and Yehuda Shalom, _Cocycle superrigidity and bounded cohomology for negatively curved spaces_ , Journal of Differential Geometry 67 (2004), 395–455.
  * [MS06] Nicolas Monod and Yehuda Shalom, _Orbit equivalence rigidity and bounded cohomology_ , Ann. of Math. (2) 164 (2006), no. 3, 825–878.
  * [Mum99] David Mumford, _The red book of varieties and schemes_ , expanded ed., Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, Berlin, 1999, Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico Arbarello.
  * [Mun84] James Raymond Munkres, _Elements of algebraic topology_ , Addison-Wesley Publishing Company, Menlo Park, CA, 1984.
  * [Pie18] Hester Pieters, _The boundary model for the continuous cohomology of ${\rm Isom}^{+}(\mathbb{H}^{n})$_, Groups Geom. Dyn. 12 (2018), no. 4, 1239–1263.
  * [Qui78] Daniel Quillen, _Homotopy properties of the poset of nontrivial $p$-subgroups of a group_, Adv. in Math. 28 (1978), no. 2, 101–128.
  * [Ron09] Mark Ronan, _Lectures on buildings_ , University of Chicago Press, Chicago, IL, 2009, Updated and revised.
  * [Ros56] Maxwell Rosenlicht, _Some basic theorems on algebraic groups_ , Amer. J. Math. 78 (1956), 401–443.
  * [Seg68] Graeme Segal, _Classifying spaces and spectral sequences_ , Inst. Hautes Études Sci. Publ. Math. (1968), no. 34, 105–112.
  * [Ser71] Jean-Pierre Serre, _Cohomologie des groupes discrets_ , Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 399, Springer, Berlin, 1971, pp. 337–350. Lecture Notes in Math., Vol. 244.
  * [Ser77] Jean-Pierre Serre, _Arbres, amalgames, ${\mathrm{SL}_{2}}$_, Société Mathématique de France, Paris, 1977, Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46.
  * [Spr98] Tonny Albert Springer, _Linear algebraic groups_ , second ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998.
  * [Tit64] Jacques Tits, _Algebraic and abstract simple groups_ , Ann. of Math. (2) 80 (1964), 313–329.
  * [Wat79] William Charles Waterhouse, _Introduction to affine group schemes_ , Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979\.
  * [Wei09] Richard Mark Weiss, _The structure of affine buildings_ , Annals of Mathematics Studies, vol. 168, Princeton University Press, Princeton, NJ, 2009\.