# High-dimensional expansion and soficity of groups

Lukas Gohla  Lukas Gohla, TU Dresden, 01062 Dresden, Germany
<EMAIL_ADDRESS>and  Andreas Thom  Andreas Thom, TU Dresden, 01062
Dresden, Germany<EMAIL_ADDRESS>
###### Abstract.

For $d\geq 4$ and $p$ a sufficiently large prime, we construct a lattice
$\Gamma\leq{\rm PSp}_{2d}(\mathbb{Q}_{p}),$ such that its universal central
extension cannot be sofic if $\Gamma$ satisfies some weak form of stability in
permutations. In the proof, we make use of high-dimensional expansion
phenomena and, extending results of Lubotzky, we construct new examples of
cosystolic expanders over arbitrary finite abelian groups.

###### Contents

  1. 1 Introduction
  2. 2 Group cohomology and applications
  3. 3 Approximation and stability

##  1\. Introduction

Eversince the influential paper of Gromov [MR1694588] and subsequent work of
Weiss [MR1803462], the quest for a non-sofic group has inspired
mathematicians. In order to show that a group is _sofic_ , one has to provide
sufficiently rich almost representations of the group in permutations. The
competing notion of _stability_ requires that every sufficiently accurate
almost representation in permutations is close to an actual permutation
representation. Thus, it is easy to see that any group which is both sofic and
stable must be residually finite and, at least theoretically, this opens a
route to the construction of non-sofic groups, see [MR2500002]. Unfortunately,
stability is a rare phenomenon, even though there is some indication that
lattices in algebraic groups of rank at least $3$ share the right kind of
rigidity in order to ensure at least some form of stability. This was first
discovered in [MR4080477] and related to high-dimensional expansion, see also
[MR3966829], but the techniques are currently not able to produce stability
results for almost representations in permutations, see the discussion in
[chapman2023stability].

In this article we study groups $\Gamma$, particular torsionfree lattices in
the algebraic group ${\rm PSp}_{2d}(\mathbb{Q}_{p})$ for $d\geq 4$ and $p$
large. These groups are residually finite and hence sofic, but admit finite
central extensions $\tilde{\Gamma}$ which are not residually finite anymore.
This phenomenon was first discovered by Deligne [MR507760] in work on the
congruence subgroup problem. Our main result, Theorem 3.13, says that these
central extensions $\tilde{\Gamma}$ cannot be sofic if the group $\Gamma$ is
stable in a certain sense, see Section 3.2.

On a more conceptual level, which might be interesting in its own right, we
introduce cohomological invariants that obstruct containment of p.m.p. actions
of groups. These obstructions can be enhanced in the presence of cosystolic
inequalities to obstruct also weak containment. This is exactly the route our
proof takes: we show that for any sofic approximation of the finite central
extension $\tilde{\Gamma}$, the induced sofic approximation of the lattice
$\Gamma$ admits a limit action which is not weakly contained in finite actions
of the lattice. This contradicts a very weak form of stability that we
introduce and discuss in Section 3.2. We call a group _stable in finite
actions_ if any partition of a suffiently good sofic approximation can be
modelled (in the spirit of Kechris’ notion of weak containment) in a finite
action, see Definition 3.7. This seems much weaker than any other notion of
stability that has been studied so far.

All this is formulated in our main result, which is Theorem 3.13, and should
be compared with results of Bowen–Burton [MR4105530], see the discussion in
Section 3.3. It can also be compared with a result of Dogon in the hyperlinear
setting, see [dogon].

The paper is organized as follows. In Section 2, we recall basics on measured
Boolean algebras, weak containment of actions, metric group cohomology,
Bruhat-Tits buildings and high-dimensional expanders. In Section 3, we recall
the definitions of soficity and various notions of stability. We introduce a
new notion of stability, weaker than most notions that have been studied so
far, and prove our main theorem. Finally, we review in Section 3.4 the
construction of a group satisfying all conditions in our main theorem.

The results of this article also appeared in the doctoral thesis of the first-
named author, see [thesisgohla], with some text overlap.

##  2\. Group cohomology and applications

###  2.1. Measured Boolean algebras

Since our setup requires taking metric ultraproducts and we would like to
avoid working with non-standard measure spaces, we will quickly explain the
setup of measured Boolean algebras and metric rings. This way, we can focus on
the algebraic aspects of our reasoning. We will be rather brief in introducing
metric ultraproducts for measured Boolean algebras and their associated
abelian groups of $A$-valued functions for some finite abelian group $A$.

A measured Boolean algebra $({\rm P},\mu)$ is a Boolean algebra $({\rm
P},\cap,\cup,\neg,0,1)$ with a measure $\mu\colon{\rm P}\to[0,1]$, i.e., a map
such that

$\mu(X\cup Y)+\mu(X\cap Y)=\mu(X)+\mu(Y)\quad\mbox{and}\quad\mu(0)=0,\
\mu(1)=1.$

We say that $(\rm P,\mu)$ is _complete_ if ${\rm P}$ is complete with the
metric $d(X,Y)=\mu(X\cup Y)-\mu(X\cap Y).$ The measure $\mu$ is said to be
_faithful_ , if $\mu(X)=0$ only when $X=0$. We typically assume that $({\rm
P},\mu)$ is complete and that $\mu$ is faithful, even though we will not need
these conditions. A typical example for $({\rm P},\mu)$ to keep in mind is the
measure algebra of measurable subsets of a standard probability space up to
measure zero. In particular, this includes the power set of a finite set with
the normalized counting measure.

Let $A$ be a finite abelian group. We write ${\rm P}(A)$ for the abelian group
generated by symbols $X\otimes a$ for $X\in{\rm P}$ and $a\in A$, subject to
the identity

$X\otimes a+Y\otimes b=(X\setminus Y)\otimes a+(Y\setminus X)\otimes b+(X\cap
Y)\otimes(a+b),$

for all $X,Y\in{\rm P}$ and $a,b\in A.$ If $A$ is a finite commutative ring,
then ${\rm P}(A)$ carries a natural multiplication which is defined via the
formula $(X\otimes a)(Y\otimes b)=(X\cap Y)\otimes(ab).$ It is easy to see
that every element $x\in{\rm P}(A)$ can be written uniquely in the form

$x=\sum_{a\in A,a\neq 0}X_{a}\otimes a$

with $X_{a}\cap X_{b}=\varnothing$ if $a\neq b.$ We denote the natural
inclusion $A$ into ${\rm P}(A)$, given by $a\mapsto 1\otimes a$, by
$\theta^{\rm P}\colon A\to{\rm P}(A).$ Note that one can extend the measure
$\mu$ to ${\rm P}(A)$ via the formula:

$\mu(x)=\sum_{a\in A,a\neq 0}\mu(X_{a})\in[0,1].$

This turns ${\rm P}(A)$ into a metric abelian group via the formula
$d(x,y)=\mu(x-y).$

Let $I$ be a set and $({\rm P}_{j},\mu_{j})_{j\in I}$ be a family of measured
Boolean algebras indexed by $I$. Let $\mathcal{U}\in\beta I$ be a non-
principal ultrafilter. We set

${\rm P}_{\mathcal{U}}=\prod_{j\in I}{\rm P}_{j}/\sim$

with $(X_{j})_{j}\sim(Y_{j})_{j}$ if and only if
$\lim_{j\to{\mathcal{U}}}d_{i}(X_{j},Y_{j})=0.$ The equivalence class of
$(X_{j})_{j}$ is denoted by $[X_{j}]_{j}$. Note that ${\rm P}_{\mathcal{U}}$
is again a measured Boolean algebra with measure

$\mu_{\mathcal{U}}([X_{j}]_{j})=\lim_{j\to{\mathcal{U}}}\mu_{j}(X_{j}).$

The measured Boolean algebra $({\rm P}_{\mathcal{U}},\mu_{\mathcal{U}})$ is
called the metric ultraproduct. In a similar way, we can consider the metric
ultraproduct $({\rm P}(A)_{\mathcal{U}},\mu_{\mathcal{U}})$ of metric abelian
groups $({\rm P}_{j}(A),\mu)$ and note that ${\rm P}_{\mathcal{U}}(A)={\rm
P}(A)_{\mathcal{U}}$ in a natural way.

Let $\Gamma$ be a countable discrete group. We consider measured Boolean
algebras with a measure preserving $\Gamma$-action and in this case we speak
of a measured $\Gamma$-Boolean algebra. For a p.m.p. preserving
$\Gamma$-action $(X,\mu)$, we denote by ${\rm M}(X,\mu)$ its measure algebra,
which is naturally a measured $\Gamma$-Boolean algebra. If $A$ is a finite
abelian group, then ${\rm P}(A)$ is naturally a $\mathbb{Z}[\Gamma]$-module
whenever ${\rm P}$ is a measured $\Gamma$-Boolean algebra. In case of finite
measured $\Gamma$-Boolean algebras, we simply speak of _finite actions_. We
denote the family of finite transitive actions by $\mathcal{P}_{\rm tra}$ and
the family of finite actions by $\mathcal{P}_{\rm fin}.$

######  Definition 2.1.

Let $({\rm P},\mu)$ and $({\rm Q},\nu)$ be measured $\Gamma$-Boolean algebras
with measure preserving $\Gamma$-actions. We say that $({\rm P},\mu)$ is
contained in $({\rm Q},\nu)$ if there exists a measure-preserving and
$\Gamma$-equivariant homomorphism of Boolean algebras from $({\rm P},\mu)$ to
$({\rm Q},\nu)$.

Clearly, any metric ultraproduct of measured $\Gamma$-Boolean algebras
inherits a natural measure preserving $\Gamma$-action from the factors. We
sometimes simply speak of the _limit action_ if the context is clear. Using
the notion of metric ultraproduct we can introduce the fundamental notion of
_weak containment_.

######  Definition 2.2.

We say that a measured $\Gamma$-Boolean algebra $({\rm P},\mu)$ is _weakly
contained_ in a family of measured $\Gamma$-Boolean algebras if $({\rm
P},\mu)$ is contained in a metric ultraproduct of measured $\Gamma$-Boolean
algebras from that family.

This definition of weak containment corresponds to the notion of weak
containment of measure preserving actions on probability spaces, see [kechris,
MR4138908].

###  2.2. Metric group cohomology

We refer to Brown [MR1324339] for the basics on group cohomology. Let us just
recall some basic notation. Let $B\Gamma$ be a classifying space for $\Gamma$,
i.e., the $\Gamma$-quotient of a contractible simplicial complex $E\Gamma$
with a free and simplicial $\Gamma$-action. Let $M$ be a
$\mathbb{Z}\Gamma$-module. We denote by ${\rm
C}^{*}(B\Gamma,M)\coloneqq\hom_{\mathbb{Z}\Gamma}({\rm C}_{*}(E\Gamma),M)$ the
cochain complex with differentials

$\delta_{i}\colon{\rm C}^{i}(B\Gamma,M)\to{\rm C}^{i+1}(B\Gamma,M).$

We set as usual

${\rm Z}^{i}(B\Gamma,M)=\ker\delta_{i},\quad{\rm B}^{i}(B\Gamma,M)={\rm im}\
\\!\delta_{i-1}$

and define the group cohomology with coefficients in $M$ as

${\rm H}^{i}(B\Gamma,M)=\frac{{\rm Z}^{i}(B\Gamma,M)}{{\rm
B}^{i}(B\Gamma,M)}.$

Let ${\rm P}$ be a measured $\Gamma$-Boolean algebra, $A$ be a finite abelian
group and $n\in\mathbb{N}$. Then, there exists a natural homomorphism

$\theta^{\rm P}_{*}\colon{\rm H}^{i}(B\Gamma,A)\to{\rm H}^{i}(B\Gamma,{\rm
P}(A)),$

where ${\rm P}(A)$ and $\theta^{\rm P}$ as in the previous section. Let
$\Gamma$ be of finite type and $B\Gamma$ a finite model for its classifying
space and let $B\Gamma(i)$ denote the set of $i$-simplices. Then, fixing a
faithful probability measure $\nu_{i}$ on $B\Gamma(i)$, we can endow ${\rm
C}^{i}(B\Gamma,{\rm P}(A))$ with a natural metric that measures non-triviality
of the co-chain, i.e. we define a length function by the formula

$\|c\|_{\mu,\nu}\coloneqq\sum_{x\in B\Gamma(i)}\mu(c(x))\nu_{i}(x),$

where we view $c\in{\rm C}^{i}(B\Gamma,{\rm
P}(A))=\hom_{\mathbb{Z}\Gamma}({\rm C}_{i}(E\Gamma),{\rm P}(A))$ simply as a
function $c\colon B\Gamma(i)\to{\rm P}(A)$ after choosing a section of the map
$E\Gamma(i)\to B\Gamma(i)$. Note that the length is independent of this choice
by $\Gamma$-invariance of the $\mu.$ Note also that $\delta_{i}\colon{\rm
C}^{i}(B\Gamma,M)\to{\rm C}^{i+1}(B\Gamma,M)$ is Lipschitz with respect to the
length functions introduced above with a Lipschitz constant just depending on
the cell structure of $B\Gamma$ and the measures $\nu_{i}$ and $\nu_{i+1}$.

There is a natural quotient pseudo-metric on ${\rm H}^{i}(B\Gamma,{\rm P}(A))$
which we also denote by

${|\kern-0.86108pt|\kern-0.86108pt|.|\kern-0.86108pt|\kern-0.86108pt|}_{\mu,\nu}\colon{\rm
H}^{i}(B\Gamma,{\rm P}(A))\to[0,1].$

Note that this pseudo-metric depends on the choice of finite model for
$B\Gamma$ and the measure $\nu_{i}$, but it is unique up to bi-Lipschitz
equivalence of pseudo-metrics. (In order to avoid the ambiguity, we fix a
finite model for $B\Gamma$ and choices for the measures $\nu_{i}$ for the rest
of the text and suppress the dependence on the $\nu_{i}$’s in the notation.)
In particular

${\rm NH}^{i}(B\Gamma,{\rm P}(A)):=\\{\alpha\in{\rm H}^{i}(B\Gamma,{\rm
P}(A))\mid{|\kern-0.86108pt|\kern-0.86108pt|\alpha|\kern-0.86108pt|\kern-0.86108pt|}=0\\}$

is well-defined and independent of a choice of a finite model. We also define
the reduced cohomology

${\rm H}_{\rm red}^{i}(B\Gamma,{\rm P}(A))=\frac{{\rm H}^{i}(B\Gamma,{\rm
P}(A))}{{\rm NH}^{i}(B\Gamma,{\rm P}(A))}.$

There are two natural choices for the measures on $B\Gamma(i)$ that we will
briefly discuss.

######  Definition 2.3.

Let $X$ be a finite simplicial complex of dimension $d$ and $X(i)$ be the set
of $i$-simplices. We define

$\mu(\sigma)\coloneqq\frac{|\\{\tau\in
X(d)\mid\sigma\subset\tau\\}|}{\binom{d+1}{i+1}\cdot|X(d)|}.$

This is the most common choice in the literature. Note however, that this
particular choice has, a priori, the problem of $i$-simplices which are
ignored in case an $i$-simplex is not contained in any $d$-dimensional cell.
In the sequel, we will always assume that all simplicial complexes are finite
dimensional and _pure_ , i.e. every $i$-simplex is contained in at least one
simplex of maximal dimension. This is true for buildings, which are the only
relevant examples which we discuss.

The other common choice for the weights differs just by a normalization factor
which depends on the dimension.

######  Definition 2.4.

Let $X$ be a finite simplicial complex of dimension $d$ and $X(i)$ be the set
of $i$-simplices. We define

$m(\sigma)\coloneqq(d-i)!\cdot|\\{\tau\in X(d)\mid\sigma\subseteq\tau\\}|.$

This choice comes with the additional property that the weight of a
$k$-dimensional simplex is given as the sum of all weights of the
$(k+1)$-dimensional simplices it is contained in.

###  2.3. Shapiro’s lemma in the metric context

In this section we want to tie together the metric and homological aspects of
co-induction. Let $B\Gamma$ be a finite classifying space for $\Gamma$ and
$\Lambda\leq\Gamma$ be a subgroup of finite index. A natural model for
$B\Lambda$ is the finite covering $E\Gamma/\Lambda$ and we will fix this model
for $B\Lambda$. Given any probability measure on $B\Gamma(i)$, there is a
natural probability measure on $B\Lambda(n)$, which arises as the normalized
equidistribution on the fibers of the covering map $B\Lambda\to B\Gamma.$ The
well-known Shapiro’s lemma says that the cohomology of $\Lambda$ with
coefficients in a $\mathbb{Z}\Lambda$-module $M$ can be computed in terms of
the cohomology of $\Gamma$ with respect to the co-induced module
$\hom_{\Lambda}(\mathbb{Z}\Gamma,M).$ If $M=A$ is just an abelian group with
trivial $\Lambda$-action, then $\hom_{\Lambda}(\mathbb{Z}\Gamma,M)={\rm
map}(\Gamma/\Lambda,A)={\rm P}^{\Gamma/\Lambda}(A),$ where ${\rm
P}^{\Gamma/\Lambda}$ denotes the Boolean algebra of subset of
$\Gamma/\Lambda$. The following lemma states that the Shapiro isomorphism is
in fact isometric when we consider ${\rm P}^{\Gamma/\Lambda}$ with the
normalized counting measure and with the choices of measures on cells as
discussed above.

######  Lemma 2.5 (Shapiro).

Let $\Lambda\leq\Gamma$ be a subgroup. There is a natural isometric
isomorphism

${\rm H}^{i}(B\Lambda,A)\stackrel{{\scriptstyle\kappa}}{{\to}}{\rm
H}^{i}(B\Gamma,{\rm P}^{\Gamma/\Lambda}(A))$

such that

$\kappa([c]|_{\Lambda})=\theta^{\Gamma/\Lambda}_{*}([c])\in{\rm
H}^{i}(B\Gamma,{\rm P}^{\Gamma/\Lambda}(A))$

for all $[c]\in{\rm H}^{i}(B\Gamma,A)$.

This lemma has particularly interesting consequences when there are infinitely
many subgroups of finite-index and the metric properties of cohomology groups
and groups of cocycles and coboundaries can be controlled uniformly.

###  2.4. Bruhat-Tits buildings

Let us continue by recalling a particular family of classifying spaces. For
background on linear algebraic groups and basic notions we refer to [borel].
The standard reference for the theory of buildings is [AB].

######  Definition 2.6.

Let $G$ be a reductive algebraic group over $\mathbb{Q}_{p}$. Then there
exists a simplicial complex $\mathcal{B}(\mathbf{G})$ such that
$\mathbf{G}(\mathbb{Q}_{p})$ acts continuously on $\mathcal{B}(\mathbf{G})$ by
simplicial automorphisms and

  1. $(i)$

the action of $\mathbf{G}(\mathbb{Q}_{p})$ is proper,

  2. $(ii)$

$\mathcal{B}(\mathbf{G})$ is contractible,

  3. $(iii)$

$\mathcal{B}(\mathbf{G})$ is of dimension ${\rm rk}_{\mathbb{Q}_{p}}G<\infty$
and

  4. $(iv)$

$\mathcal{B}(\mathbf{G})$ is locally finite, i.e. all proper links are finite.

$\mathcal{B}(\mathbf{G})$ will be called the _Bruhat-Tits building of
$\mathbf{G}$_.

The following lemma is well-known.

######  Lemma 2.7.

Let $\Gamma$ be a torsion-free lattice of $\mathbf{G}(\mathbb{Q}_{p})$. Then
$\mathcal{B}(\mathbf{G})/\Gamma$ is a finite classifying space of $\Gamma$.

Bruhat-Tits buildings are sources of all kind of rigidity phenomena as will be
discussed in the next section.

###  2.5. Cosystolic inequalities and expansion

In this section, we follow the exposition of Lubotzky [MR3966743], see also
[MR3536553, kaufman2018cosystolic, LMM]. It is natural to wonder in what sense
the metric and the algebraic structure on the cohomology groups discussed in
Section 2.2 are compatible. One immediate requirement is that we would want
that ${\rm H}^{i}(B\Gamma,{\rm P}(A))$ is Hausdorff, i.e. the pseudo-metric is
in fact a metric. However, it turns out that it is of even greater importance
to require that the induced metric is discrete. Another subtle point is the
comparison of the quotient pseudo-metric and the subspace metric on the
subgroup

${\rm im}\ \delta_{i-1}\subset{\rm C}^{i}(B\Gamma,{\rm P}(A)).$

######  Definition 2.8.

Let $\varepsilon,\mu>0$ and let $\Gamma$ be a group of finite type and fix a
finite model for $B\Gamma$. Let $\mathcal{P}$ be a family of measured
$\Gamma$-Boolean algebras and $A$ be a finite abelian group. We say that
$\Gamma$ satisfies

  1. $(i)$

a _cosystolic inequality_ in dimension $i$ with respect to $\mathcal{P}$ and
$A$, if there exists $\varepsilon>0$, such that

${|\kern-0.86108pt|\kern-0.86108pt|\alpha|\kern-0.86108pt|\kern-0.86108pt|}_{i}\geq\varepsilon$

for all non-zero $\alpha\in{\rm H}^{i}(B\Gamma,{\rm P}(A))$ and all $({\rm
P},\mu)\in\mathcal{P},$ and

  2. $(ii)$

_expansion_ in dimension $i$ with respect to $\mathcal{P}$ and $A$ if there
exists $\varepsilon>0$, such that

$\|\delta_{i}(c)\|_{i+1}\geq\varepsilon\inf_{z\in{\rm Z}^{i}(B\Gamma,{\rm
P}(A))}\|c+z\|_{i},$

for all $c\in{\rm C}^{i}(B\Gamma,{\rm P}(A))$ and all ${\rm P}\in\mathcal{P}.$

Expansion is visible only in the inner workings of the co-chain complex and
says that the quotient pseudo-metric and the subspace metric are Lipschitz
equivalent on ${\rm im}\ \\!\delta_{i}$. Note that by the discussion in
Section 2.2 these properties are independent of the choice of the finite model
$B\Gamma$, so in fact they are properties of $\Gamma.$

######  Definition 2.9.

We say that a $d$-dimensional finite complex with fundamental group $\Gamma$
is a cosystolic expander with respect to $\mathcal{P}$ and $A$, if it
satisfies a cosystolic inequality and expansion in dimensions $0\leq i\leq
d-1.$ Moreover, we call a cosystolic expander a coboundary expander, if in
addition its cohomology vanishes in dimensions $1\leq i\leq d-1$

A typical result says that the $(d-1)$-skeleton of a finite quotient of a
$d$-dimensional Bruhat-Tits building is (under certain conditions) a
cosystolic expander in a uniform way, i.e., we need that $\mu,\varepsilon$
above do not depend on the particular quotient. We conclude that the
fundamental group of such a quotient satisfies cosystolic inequalities and
expansion in dimensions $0\leq i\leq d-2$ with respect to the family of finite
actions. More precisely, we will obtain the following result:

######  Theorem 2.10.

Let $d\geq 4$ and $p$ large enough. Let $\Gamma$ be a torsionfree lattice in
${\rm PSp}_{2d}(\mathbb{Q}_{p})$ and let $A$ be a finite abelian group. Then,
$\Gamma$ satisfies a cosystolic inequality and expansion with respect to the
family of finite transitive $\Gamma$-actions and $A$ in dimensions $0\leq
i\leq d-2.$

The goal of this section is to prove the following theorem, which implies the
previous theorem by the discussion above and Lemma 2.5 – it is itself a
generalization of [MR3966743]*Thm. 3.8 to arbitrary abelian groups.

######  Theorem 2.11.

For $2\leq d\in\mathbb{N}$ there exist $\varepsilon=\varepsilon(d)>0$ and
$q_{0}=q_{0}(d)$ such that if ${\mathbb{K}}$ is a local non-archimedean field
of fixed residue degree $q>q_{0}$, $\mathbf{G}$ a simple ${\mathbb{K}}$-group
of ${\mathbb{K}}$-rank $d$. Let $A$ be a finite abelian group and let
$\Lambda$ a torsion-free lattice of $\mathbf{G}(\mathbb{K})$ that acts with
injectivity radius at least $3$ on the Bruhat-Tits building
$\mathcal{B}(\mathbf{G})$.

Then, the $(d-1)$-skeleton $Y$ of the finite quotient
$\mathcal{B}(\mathbf{G})/\Lambda$ is a $(d-1)$-dimensional
$(\varepsilon,\varepsilon)$-cosystolic expander over $A$.

Let’s first spell out how Theorem 2.10 can be derived.

###### Proof of Theorem 2.10.

Let $\Gamma$ be as above and $\Gamma\curvearrowright\Gamma/\Lambda$ be a
finite transitive action. Then the cochain complex computing ${\rm
H}^{i}(\Gamma,{\rm P}^{\Gamma/\Lambda}(A))$ is isometrically isomorphic to the
the simplical cochain complex ${\rm
C}^{*}(\mathcal{B}(\mathbf{G})/\Lambda,A)$. Thus, if the $(d-1)$-skeleton of
$\mathcal{B}(\mathbf{G})/\Lambda$ satisfies cosystolic and expansion
inequalities this translates into the corresponding inequalities for group
cohomology with coefficients in ${\rm P}^{\Gamma/\Lambda}(A)$. Since the
estimates of Theorem 2.11 are independent of $\Lambda$, we obtain uniform
estimates for all finite transitive actions. ∎

The result is essentially contained in the literature, even though it took us
some effort to find the right results and understand how to combine them. We
will prove Theorem 2.11 as a consequence of the following result of
Kaufman–Mass.

######  Theorem 2.12 (Theorem 3.3 and 3.4, [kaufman2018cosystolic]).

Let $d,Q\in\mathbb{N}$ and $X$ be a finite $d$-dimensional simplicial complex
of $Q$-bounded degree, $A$ a finite abelian group and $\beta\in(0,1)$. There
exists a $\mu=\mu(d,\beta)>0$ such that if

  1. $(i)$

for every $\emptyset\neq\sigma\in X$ the link $X_{\sigma}$ is a
$\beta$-coboundary expander over $A$ and

  2. $(ii)$

for every $\sigma\in X$ the link $X_{\sigma}$ is a $\mu$-skeleton expander,

then the $(d-1)$-skeleton of $X$ is a $(\min\\{Q^{-2},\mu\\},\mu)$-cosystolic
expander over $A$.

###### Proof of Theorem 2.11.

We need to check that the conditions of Theorem 2.12 are satisfied and the
bounds are independent of $\Gamma$. First of all, note that $X$ is of
$Q$-bounded degree independent of the choice of $\Gamma$. By [AB]*Proposition
4.9 every link of a building is still a building and we already know that
every proper link is finite. By [kaufman2018cosystolic]*Theorem 4.1 every
finite building is a coboundary expander over any abelian group. This shows
that every proper link $X_{\sigma}$ is a $\beta_{d-|\sigma|}$-coboundary
expander over any abelian group. Hence, Condition $(i)$ is satisfied.

The verification of Condition $(ii)$ is split into two parts. First we
consider proper links and then the remaining degenerate case.

Every reductive algebraic group $\mathbf{G}$ admits a BN-pair and by
[AB]*Theorem 6.56 $\mathbf{G}(\mathbb{K})$ acts strongly transitively on its
Bruhat-Tits building. Now, [MR3536553]*Lemma 5.17 states that a suitable
stabilizer also acts strongly transitively on all links of said building.
Hence [MR3536553]*Lemma 5.14 states that all proper links are regular
complexes and therefore they are $\lambda(X_{\sigma})$-skeleton expanders by
[MR3536553]*Corollary 4.7. Since there are only finitely many isomorphisms
classes of links in $\mathcal{B}(\mathbf{G})$, this allows us to to find a
bound on the skeleton expansion of proper links in $X$ independently of the
specific $\Gamma$ we chose.

It remains to show skeleton expansion for $X_{\emptyset}=X$ and here the
argument gets slightly more complicated. Again, the proof relies on a local-
global method. As we have mentioned before every proper link in $X$ appears as
a proper link in $\mathcal{B}(\mathbf{G})$. Furthermore we know that every
link of $\mathcal{B}(\mathbf{G})$ is again a building. This means two things
for us:

  1. $(a)$

Every link of $X$ is connected.

  2. $(b)$

By [kaufman2018cosystolic]*Theorem 4.1 spherical buildings are coboundary
expanders. In particular, this shows that the $1$-dimensional links of $X$ are
expander graphs with a Cheeger constant which is bounded by a constant that is
independent of the specific link.

Therefore $X$ is a _$\lambda$ -local spectral expander_ in the notation of
[MR3755725] and the assumptions of [MR3755725]*Corollary 6.1 are satisfied. In
particular, then the non-trivial spectrum of the upper Laplacian
$\Delta^{+}\coloneqq\Delta^{+}_{0}$ with respect to the inner product $\langle
f,g\rangle\coloneqq\sum_{\sigma\in\Sigma(k)}\frac{m(\sigma)}{k!}\cdot
f(\sigma)\cdot g(\sigma)$ on ${\rm C}^{k}(X,\mathbb{R})$ and with $m$ from
Remark 2.4 is contained in $[a_{\alpha},\infty)$ with $a_{\alpha}>0$ and
$\lim_{\alpha\rightarrow k-1}a_{\alpha}=1$. This implies that $X$ is a
$\mu$-skeleton expander as required. ∎

###  2.6. An application to weak containment

Let $\kappa\colon{\rm P}\to{\rm Q}$ be a measure-preserving,
$\Gamma$-equivariant homomorphism of Boolean algebras. Let $A$ be a finite
abelian group. Note that the induced map $\kappa_{*}\colon{\rm
H}^{i}(B\Gamma,{\rm P}(A))\to{\rm H}^{i}(B\Gamma,{\rm Q}(A))$ is contractive.
This simple observation allows us to find a cohomological obstruction to weak
containment of a measured $\Gamma$-Boolean algebra provided a suitable
cosystolic inequality holds. This is explained in the next lemma and its
corollary.

######  Lemma 2.13.

Let $\Gamma$ be of finite type, $\mathcal{P}$ be a family of measured
$\Gamma$-Boolean algebras and $A$ be an abelian group. Let $({\rm
P}_{j},\mu_{j})_{j\in I}$ be an indexed set of elements of $\mathcal{P}$ and
${\mathcal{U}}\in\beta I$ be a non-principal ultrafilter. Then, the natural
map

${\rm H}_{\rm red}^{i}(\Gamma,{\rm P}_{j}(A))_{\mathcal{U}}\to{\rm H}_{\rm
red}^{i}(\Gamma,{\rm P}_{\mathcal{U}}(A))$

is an isometric injection and it is surjective if $\Gamma$ satisfies expansion
in dimension $i$ with respect to the family $\mathcal{P}$ and $A.$ In
particular, we have

${|\kern-0.86108pt|\kern-0.86108pt|\theta^{{\rm
P}_{\mathcal{U}}}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{\mathcal{U}}}=\lim_{j\to{\mathcal{U}}}{|\kern-0.86108pt|\kern-0.86108pt|\theta^{{\rm
P}_{j}}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{j}}$

for all $\alpha\in{\rm H}^{i}(\Gamma,A)$.

###### Proof.

Let $\alpha_{j}\in{\rm H}^{i}(\Gamma,{\rm P}_{j}(A))$ and $\delta>0$ for $j\in
I$. Denote the image of in ${\rm H}^{i}(\Gamma,{\rm P}_{\mathcal{U}}(A))$ by
$\alpha_{\mathcal{U}}$. The inequality

${|\kern-0.86108pt|\kern-0.86108pt|\alpha_{\mathcal{U}}|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{\mathcal{U}}}\leq\lim_{j\to{\mathcal{U}}}{|\kern-0.86108pt|\kern-0.86108pt|\alpha_{j}|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{j}}$

is obvious. Indeed, let $\delta>0$ be arbitrary and let $\beta_{j}\colon
B\Gamma(i)\to{\rm P}_{j}(A)$ be cocycle representatives of $\alpha_{j}$ such
that
${|\kern-0.86108pt|\kern-0.86108pt|\alpha_{j}|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{j}}\geq\|\beta\|_{j}-\delta.$
Then, $\beta=[\beta_{j}]_{j}$ is by definition a cocycle representative of
$\alpha_{\mathcal{U}}$ and

${|\kern-0.86108pt|\kern-0.86108pt|\alpha_{\mathcal{U}}|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{\mathcal{U}}}\leq\|\beta\|_{\mu_{\mathcal{U}}}=\lim_{j\to\mathcal{U}}\|\beta_{j}\|_{\mu_{j}}\leq\lim_{j\to\mathcal{U}}{|\kern-0.86108pt|\kern-0.86108pt|\alpha_{j}|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{j}}+\delta.$

Since $\delta>0$ was arbitrary, this proves the first inequality.

Let now $\beta^{\prime}\colon B\Gamma(i)\to{\rm P}_{\mathcal{U}}(A)$ be a
cocycle representative of $\alpha_{\mathcal{U}}$, such that
${|\kern-0.86108pt|\kern-0.86108pt|\alpha_{\mathcal{U}}|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{\mathcal{U}}}\geq\|\beta^{\prime}\|_{\mu_{\mathcal{U}}}-\delta.$
Note that $\delta_{i}(\beta^{\prime})=0$ and
$\beta^{\prime}-\beta=\delta_{i-1}(\zeta)$ for some
$\zeta=[\zeta_{j}]_{j}\colon B\Gamma(i-1)\to{\rm P}_{\mathcal{U}}(A).$ We set
$\beta^{\prime}_{j}:=\beta_{j}-\delta_{i-1}(\zeta_{j})$ and note that
$[\beta^{\prime}_{j}]_{j}=\beta^{\prime}$. The elements $\beta^{\prime}_{j}$
are clearly cocycles that represent $\alpha_{j}$. Hence

$\lim_{j\to\mathcal{U}}{|\kern-0.86108pt|\kern-0.86108pt|\alpha_{j}|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{j}}\leq\lim_{j\to\mathcal{U}}\|\beta_{j}^{\prime}\|_{\mu_{j}}=\|\beta^{\prime}_{\mathcal{U}}\|_{\mu_{\mathcal{U}}}\leq{|\kern-0.86108pt|\kern-0.86108pt|\alpha_{\mathcal{U}}|\kern-0.86108pt|\kern-0.86108pt|}+\delta.$

This proves the other inequality and shows that the natural map is indeed an
isometric embedding.

It remains to show that the map is surjective in presence of expansion. Let
$\alpha_{\mathcal{U}}\in{\rm H}^{i}(\Gamma,{\rm P}_{\mathcal{U}}(A))$ be
represented by $\beta\colon B\Gamma(i)\to{\rm P}_{\mathcal{U}}(A).$ Further,
choose functions $\beta_{j}\colon B\Gamma(i)\to P_{j}(A)$, such that
$[\beta_{j}]_{j}=\beta$. From (ii) in Definition 2.8, we conclude that there
exists $z_{j}\in{\rm Z}^{i}(B\Gamma,P_{j}(A))$ with

$\lim_{j\to{\mathcal{U}}}\|\delta_{i}(\beta_{j})\|\geq\varepsilon\lim_{j\to{\mathcal{U}}}\|\beta_{j}-z_{j}\|.$

We obtain

$0=\|\delta_{i}(\beta)\|_{\mu_{\mathcal{U}}}=\lim_{j\to{\mathcal{U}}}\|\delta_{i}(\beta_{j})\|_{\mu_{j}}\geq\varepsilon\lim_{j\to{\mathcal{U}}}\|\beta_{j}-z_{j}\|_{\mu_{j}}.$

Since the image of $[z_{j}]_{j}$ in ${\rm H}^{i}(\Gamma,{\rm
P}_{\mathcal{U}}(A))$ is exactly $\alpha_{\mathcal{U}}$, we conclude that
surjectivity of the map as claimed. ∎

######  Corollary 2.14.

Let $\Gamma$ be a group of finite type, let $({\rm P},\mu)$ be a measured
$\Gamma$-Boolean algebra and let $\mathcal{P}$ be a family of measured
$\Gamma$-Boolean algebra. If

${|\kern-0.86108pt|\kern-0.86108pt|\theta^{\rm
P}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\mu}<\inf\\{{|\kern-0.86108pt|\kern-0.86108pt|\theta^{{\rm
Q}}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\nu}\mid({\rm
Q},\nu)\in\mathcal{P}\\}$

for some $\alpha\in{\rm H}^{i}(\Gamma,A),$ then $({\rm P},\mu)$ is not weakly
contained in the family $\mathcal{P}$.

###### Proof.

This a consequence of Lemma 2.13 which is applicable because of Condition
$(i)$. Indeed, if $({\rm P},\mu)$ is contained in some ultra-product $({\rm
P}_{\mathcal{U}},\mu_{\mathcal{U}})$ for some ultrafilter
${\mathcal{U}}\in\beta I$ and $({\rm P}_{j},\mu_{j})\in\mathcal{P}$ for $j\in
I$, then the natural map

${\rm H}^{i}(\Gamma,{\rm P}(A))\to{\rm H}^{i}(\Gamma,{\rm
P}_{\mathcal{U}}(A))$

is contractive and hence

${|\kern-0.86108pt|\kern-0.86108pt|\theta^{\rm
P}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\mu}\geq{|\kern-0.86108pt|\kern-0.86108pt|\theta^{{\rm
P}_{\mathcal{U}}}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{\mathcal{U}}}=\lim_{j\to{\mathcal{U}}}{|\kern-0.86108pt|\kern-0.86108pt|\theta^{{\rm
P}_{j}}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\mu_{j}}\geq\inf\\{{|\kern-0.86108pt|\kern-0.86108pt|\theta^{{\rm
Q}}_{*}(\alpha)|\kern-0.86108pt|\kern-0.86108pt|}_{\nu}\mid({\rm
Q},\nu)\in\mathcal{P}\\}.$

However, this contradicts our assumption. ∎

###  2.7. An application to residually finite groups

Let $\Gamma$ be a residually finite group. Let $A$ be a finite abelian group
and let $\alpha\in{\rm Z}^{2}(\Gamma,A)$ be a 2-cocycle that classifies a
central extension

$1\to A\to\tilde{\Gamma}\to\Gamma\to 1.$

The following lemma provides a cohomological consequence of the situation that
$\tilde{\Gamma}$ is not residually finite.

######  Lemma 2.15.

Let $\Gamma$ be a residually finite group, let $\Lambda\leq\Gamma$ be a
subgroup of finite index and let $A$ be a finite abelian group. If
$\alpha\in{\rm H}^{2}(\Gamma,A)$ classifies a central extension
$\tilde{\Gamma}$ which is not residually finite, then

$0\neq\theta^{\Gamma/\Lambda}_{*}(\alpha)\in{\rm H}^{i}(\Gamma,{\rm
P}^{\Gamma/\Lambda}(A)).$

###### Proof.

By Lemma 2.5, we have
$\kappa(\alpha|_{\Lambda})=\theta^{\Gamma/\Lambda}_{*}(\alpha)$, so that
$\alpha|_{\Lambda}$ must vanish if $\theta^{\Gamma/\Lambda}_{*}(\alpha)=0.$
But then, the inverse image of $\Lambda$ in $\tilde{\Gamma}$ is isomorphic to
$\Lambda\times A$. In particular, $\tilde{\Gamma}$ contains a residually
finite subgroup of finite index. This implies that $\tilde{\Gamma}$ is also
residually finite in contradiction to our assumption. ∎

The preceding lemma has an interesting consequence in presence of a suitable
cosystolic inequality in dimension $2$. We denote the profinite completion of
a residually finite group $\Gamma$ by $\hat{\Gamma}.$ Recall that it carries a
natural Haar measure.

######  Theorem 2.16.

Let $A$ be a finite abelian group. Let $\Gamma$ be a residually finite group
which satisfies a co-systolic inequality in dimension $2$ for all finite
transitive actions with respect to $A$. If $\alpha\in{\rm H}^{2}(\Gamma,A)$
classifies a central extension $\tilde{\Gamma}$ which is not residually
finite, then

$0\neq\theta^{\rm P}_{*}(\alpha)\in{\rm H}^{2}(\Gamma,{\rm P}(A))$

whenever the measured $\Gamma$-Boolean algebra ${\rm P}$ is weakly contained
in the family of finite actions. In particular, this holds for ${\rm P}={\rm
M}(\hat{\Gamma}).$

###### Proof.

This is a consequence of Lemma 2.15 and Corollary 2.14. Indeed, by the
cosystolic inequality, there is a uniform lower bound for the length of the
cohomology class $\theta^{\rm P}_{*}(\alpha)\in{\rm H}^{2}(\Gamma,{\rm P}(A))$
for all finite transitive actions. Since any finite action is a disjoint union
of transitive actions, this implies the same lower bound on the length of
$\theta^{\rm P}_{*}(\alpha)$ for _all_ finite actions. Now, Corollary 2.14
yields a lower bound on the length of $\theta^{\rm P}_{*}(\alpha)$. ∎

##  3\. Approximation and stability

###  3.1. Sofic approximations

For background on sofic groups and sofic approximations we refer to the
excellent survey by Pestov [pestov]. Let $\Gamma$ be a finitely presented
group. Fix a surjection $\pi\colon{\rm F}\to\Gamma$, where ${\rm F}$ is a
finitely generated free group and let $R\subseteq\ker(\pi)$ be a finite set
that generates $\ker(\pi)$ as a normal subgroup. We set

$\ell_{n}(\sigma)=\frac{1}{n}|\\{i\mid\sigma(i)\neq
i\\}|,\quad\forall\sigma\in{\rm Sym}(n)$

and define the normalized Hamming metric as
$d_{n}(\sigma,\tau)=\ell_{n}(\sigma\tau^{-1})$ for all $\sigma,\tau\in{\rm
Sym}(n).$

######  Definition 3.1.

A sofic representation of the group $\Gamma$ is a sequence of homomorphisms
$\sigma_{n}\colon{\rm F}\to{\rm Sym}(k_{n})$ such that $\ell_{k_{n}}(r)\to 0$
for all $r\in R$ and $\ell_{k_{n}}(g)\to 1$ for all $g\notin\ker(\pi).$

Note that each $\sigma_{n}$ gives rise to a measured ${\rm F}$-Boolean algebra
${\rm P}_{n}$. Moreover, after fixing an ultrafilter
${\mathcal{U}}\in\beta\mathbb{N}$, each sofic approximation gives rise to a
natural _limit action_ given by the metric ultraproduct of measured ${\rm
F}$-Boolean algebras. Note that the limit action of a sofic approximation of
$\Gamma$ is naturally a measured $\Gamma$-Boolean algebra.

######  Lemma 3.2.

Let $\tilde{\Gamma}$ be a group with a finite normal subgroup $A$ and let
$\Gamma\coloneqq\tilde{\Gamma}/A$ be the quotient. Every sofic approximation
$(\sigma_{n}\colon{\rm F}\to{\rm Sym}(k_{n}))_{n}$ of $\tilde{\Gamma}$ is
equivalent to a sofic approximation of the form $(\tilde{\sigma}_{n}\colon{\rm
F}\to{\rm Sym}(m_{n}\times A))_{n}$, where $A$ acts regularly on the right
factor. In particular, it induces a sofic approximation
$(\sigma^{\prime}_{n}\colon{\rm F}\to{\rm Sym}(m_{n}))_{n}$ of the group
$\Gamma$.

Using the previous lemma makes it easy to study the corresponding limit
actions.

######  Lemma 3.3.

Let ${\mathcal{U}}\in\beta\mathbb{N}$ be a non-principal ultrafilter and let
${\rm Q}_{p}$ be the limit $\tilde{\Gamma}$-action of a sofic approximation
$(\sigma_{n}\colon{\rm F}\to{\rm Sym}(k_{n}))_{n}$ of $\tilde{\Gamma}$ as
above. Then the limit $\Gamma$-action ${\rm P}_{\mathcal{U}}$ of the sofic
approximation $(\sigma^{\prime}_{n}\colon{\rm F}\to{\rm Sym}(m_{n}))_{n}$ of
$\Gamma$ as above is given by the subalgebra of $A$-invariants, i.e., ${\rm
P}_{\mathcal{U}}=({\rm Q}_{p})^{A}.$

Thus, the limit action of a sofic approximation of $\Gamma$, that is induced
from a sofic approximation of $\tilde{\Gamma}$, is of a particular kind.
Indeed, it arises as the $A$-invariants of a measured $\tilde{\Gamma}$-Boolean
algebra, and we will make crucial use of this fact. We will now restrict to
central extensions and show that for any such action, the cohomological
classes obtained by pushing forward the classifying cocycle of the central
extension vanish.

######  Theorem 3.4.

Let $\tilde{\Gamma}$ be a central extension of $\Gamma$ by the finite abelian
group $A$ and let $\alpha\in{\rm H}^{2}(\Gamma,A)$ be the cohomology class
that classifies this extension. Let $({\rm Q},\mu)$ be an $A$-free measured
$\tilde{\Gamma}$-Boolean algebra and let ${\rm P}\coloneqq{\rm Q}^{A}$ be the
subalgebra of $A$-invariants. Then,

$0=\theta^{\rm P}_{*}(\alpha)\in{\rm{\rm H}^{2}}(\Gamma,{\rm P}(A)).$

###### Proof.

Consider the natural transformation of the low degree part of the Lyndon-
Hochschild-Serre spectral sequence:

${{\rm H}^{1}(A,A)^{\Gamma}}$${{\rm H}^{2}(\Gamma,A)}$${{\rm
H}^{2}(\tilde{\Gamma},A)}$${{\rm H}^{1}(A,{\rm Q}(A))^{\Gamma}}$${{\rm
H}^{2}(\Gamma,{\rm P}(A))}$${{\rm H}^{2}(\tilde{\Gamma},{\rm
Q}(A))}$$\scriptstyle{\theta^{P}_{*}}$

with ${\rm P}={\rm Q}^{A}$ by definition. Note that $\alpha\in{\rm
H}^{2}(\Gamma,A)$ maps to zero in ${\rm H}^{2}(\tilde{\Gamma},A)$. Hence,
$\theta^{\rm P}_{*}(\alpha)$ maps to zero in ${\rm H}^{2}(\tilde{\Gamma},{\rm
Q}(A))$. The claim follows if we can show that ${\rm H}^{1}(A,{\rm
Q}(A))^{\Gamma}=0.$ However, ${\rm Q}(A)={\rm
hom}_{\mathbb{Z}}(\mathbb{Z}[A],{\rm P}(A))$ as $\mathbb{Z}[A]$-module, since
${\rm Q}$ is assumed to be $A$-free. Hence, ${\rm H}^{1}(A,{\rm Q}(A))=0$ and
this finishes the proof. ∎

######  Remark 3.5.

The vanishing of $\theta^{\rm P}_{*}(\alpha)$ for the limit action of a sofic
approximation of $\Gamma$ which is induced by a sofic approximation
$\tilde{\Gamma}$ can be proved directly along the approximation. We provide a
sketch of the proof here: Let $S$ be a finite generating set of $\Gamma$ and
let ${\rm F}_{S}$ be the free group on $S$. Pick a set-theoretic lift
$\rho\colon\Gamma\to\tilde{\Gamma}$ and consider $\tilde{\Gamma}$ as generated
by $\rho(S)\cup A$. Then the induced sofic approximation
$(\sigma^{\prime}_{n}\colon{\rm F}_{S}\to{\rm Sym}(m_{n}))_{n}$, when viewed
as a sequence of $S$-edge-labeled graphs, comes endowed with an additional
$A$-labelling on edges. This $A$-labeling measures the failure of $\rho$ being
a homomorphism. In other terms, we obtain a map $\beta_{n}\colon S\to{\rm
map}(\\{1,\dots,m_{n}\\},A)$ measuring non-multiplicativity on the good part
of the sofic approximation. On the other side $\alpha\in{\rm H}^{2}(\Gamma,A)$
is represented by a $2$-cocycle $\alpha^{\prime}\colon R\to A$, where $R$ is a
set of generating relations of $\Gamma$ and $\alpha(r)$ is defined to be
$\tilde{\pi}(r)\in A,$ where
$\tilde{\pi}\colon{\mathrm{F}}_{S}\to\tilde{\Gamma}$ is the extension of
$\rho|_{S}$ to a homomorphism on ${\rm F}_{S}$. It is easy to see that
$\delta_{1}(\beta_{n})$ almost represents the image of $\alpha^{\prime}$ in
${\rm map}(R,{\rm map}(\\{1,\dots,m_{n}\\},A))$. Indeed,
$\delta_{1}(\beta_{n})(r)$ is computed by adding up the errors along a
relation $r\in R$. On the good part of the sofic approximation, this is just a
way of computing $\alpha^{\prime}(r).$ Taking the limit, we obtain
$\theta^{\rm P}_{*}(\alpha)=0.$ This finishes the sketch of the argument.

###  3.2. Stability

By now there is a zoo of notions of stability of almost representations in
permutations; including being _stable_ [MR2500002], _weakly stable_
[MR3350728], _flexibly stable_ , _weakly flexibly stable_ [MR4027744], _very
flexibly stable_ , and _weakly very flexibly stable_ [MR4134896]. Additional
confusion arises since the notion of very flexible stability is weaker than
flexible stability itself.

In retrospect and in view of the results of [MR4027744], the unflexible
notions of stability are somewhat pathological and too rigid in order to be
useful beyond the amenable case, see [MR3999445]. The difference between weak
and non-weak notions refers to the questions if one wants to assume from the
start that the almost representations are almost free. Since we are only
interested in sofic approximations this requirement is automatic in our
setting.

Anyhow, in view of this discussion it needs a profound justification and
linguistic skillfuless to introduce yet another notion of stability. Let’s
start with an observation:

######  Lemma 3.6.

A group $\Gamma$ is weakly flexibly stable if and only if the limit action of
any sofic approximation is $\Gamma$-equivariantly internally isomorphic to a
metric ultraproduct of finite $\Gamma$-actions.

###### Proof.

This is a consequence of the results in [MR919287]. ∎

The requirement that the isomorphism be internal is non-trivial, but leads us
in a different direction and we will ignore this point.

######  Definition 3.7.

We say that a group $\Gamma$ is _stable in finite actions_ if the limit action
of any sofic approximation is weakly contained in the family of finite
$\Gamma$-actions. Equivalently, the limit action of any sofic approximation of
$\Gamma$ is contained in a metric ultraproduct of finite $\Gamma$-actions.

It is clear from Lemma 3.6 that any weakly flexibly stable group $\Gamma$ is
stable in finite actions. Let us also note that any sofic group which is
stable in finite actions must be residually finite. Using the definition of
weak containment and following the discussion in [MR4434538], let us now
characterize and give a criterion that implies stability in finite actions in
more familiar terms. Spelling everything out, we obtain the following result:

######  Lemma 3.8.

Let $\Gamma$ be a countable discrete group. Consider the following conditions:

  1. $(i)$

The group $\Gamma$ be stable in finite actions.

  2. $(ii)$

For all $K\subset\Gamma$ finite, $d\in\mathbb{N}$, and $\varepsilon>0$, there
exists $\delta>0$ and $F\subset\Gamma$ finite, such that for every
$n\in\mathbb{N}$, every map $\varphi\colon\Gamma\to{\rm Sym}(n)$ satisfying

$d_{n}(\varphi(gh),\varphi(g)\varphi(h))<\delta\quad\forall g,h\in F$

and

$d_{n}(1_{n},\varphi(g))>1-\delta\quad\forall g\in
F\setminus\\{1_{\Gamma}\\},$

and any partition $\\{1,\dots,n\\}=A_{1}\cup\cdots\cup A_{d}$, there exists a
finite set $X$, a homomorphism $\psi\colon\Gamma\to{\rm Sym}(X)$, and a
partition $X=B_{1}\cup\cdots\cup B_{d}$ such that

$\left|\frac{|A_{i}\cap\varphi(g)A_{j}|}{n}-\frac{|B_{i}\cap\psi(g)B_{j}|}{|X|}\right|<\varepsilon,\quad\forall
i,j\in\\{1,\dots,d\\},g\in K.$

  3. $(iii)$

For all $K\subset\Gamma$ finite and $\varepsilon>0$, there exists $\delta>0$
and $F\subset\Gamma$ finite, such that for every $n\in\mathbb{N}$ and every
map $\varphi\colon\Gamma\to{\rm Sym}(n)$ satisfying

$d_{n}(\varphi(gh),\varphi(g)\varphi(h))<\delta\quad\forall g,h\in F$

and

$d_{n}(1_{n},\varphi(g))>1-\delta\quad\forall g\in
F\setminus\\{1_{\Gamma}\\},$

there exists $m\in\mathbb{N}$, a set $X$ of cardinality in
$[nm,(1+\varepsilon)nm]$, a subset $X_{0}\subseteq X$ of cardinality $nm$, a
$m$-$1$-map $\pi\colon X_{0}\to\\{1,\dots,n\\},$ and a homomorphism
$\psi\colon\Gamma\to{\rm Sym}(X)$ such that

$|\\{x\in
X_{0}\mid\pi(\psi(g)x)\neq\varphi(g)\pi(x)\\}|<\varepsilon|X_{0}|,\quad\forall
g\in K.$

Then, the following implications hold:

$(i)\Leftrightarrow(ii)\Leftarrow(iii).$

Condition $(iii)$ is a way of understanding a sufficient condition for
stability in finite actions in terms of the notion of _branched cover_ (of an
almost homomorphism) as studied by Ioana [MR4434538]*Definition 1.2. In
particular, a group is stable in finite actions if every sofic approximation
has a branched cover which is equivalent to a sequence of finite actions, see
[MR4434538]*Remark 1.4. As pointed out to us by Adrian Ioana, it is likely
that, as a consequence of results in [MR4434538], Condition $(iii)$ for
Kazhdan groups might also imply flexible stability. See Remark 3.16 for
examples of groups which are stable in finite actions and potential non-
examples. As a positive result, we can record the following inheritance
property:

######  Theorem 3.9.

Let $\Gamma=\Lambda\rtimes H$, such that:

  1. $(i)$

The group $\Lambda$ is stable in finite actions.

  2. $(ii)$

Every finite index normal subgroup of $\Lambda$ contains a finite index
subgroup, which is normalized by $H$. In particular, this holds if $\Lambda$
is finitely generated.

  3. $(iii)$

The group $H$ is residually finite and amenable.

Then, the group $\Gamma$ is stable in finite actions.

###### Proof.

One can follow almost verbatim the argument on page 488 in [MR2931406], while
noting that the restriction of the limit action of a sofic approximation of
$\Gamma$ is the limit action of the induced sofic approximation of $\Lambda$,
and that Problem ${\rm A}.4$ on page 505 of the same paper has been solved in
[MR3047068]. ∎

Thus, as a consequence of the preceding theorem,
$\mathbb{F}_{2}\times\mathbb{Z}$ is an example of a group which is stable in
finite actions, but not flexibly stable, see [MR4134896].

######  Remark 3.10.

Theorem 3.9 should be compared to [MR2931406]*Conjecture 4.14 which was proved
as [MR3047068]*Theorem 1.4. There, it is not assumed that the extension of $H$
by $\Lambda$ is a crossed product. However, the argument on [MR2931406]*page
488 seems to be incomplete without further assumptions. Indeed, in the
argument, it is assumed implicitly that a finite extension of a residually
finite amenable group is again residually finite. While this is true for
crossed products, counterexamples to the general case have been produced in
[erschler].

Another line of argument can be extracted from the proof of
[MR2077037]*Theorem 2.8 by Lubotzky–Shalom. Recall, a group is termed LERF
(for locally extended residually finite) if every finitely generated subgroup
is closed in the profinite topology

######  Theorem 3.11.

Let $\Gamma$ be a group and $\Lambda\leq\Gamma$ be a normal subgroup, such
that:

  1. $(i)$

The group $\Lambda$ is stable in finite actions.

  2. $(ii)$

Every finitely generated subgroup of $\Lambda$ is closed in the profinite
topology of $\Gamma$. In particular, this holds if $\Gamma$ is LERF.

  3. $(iii)$

The group $\Gamma/\Lambda$ is amenable.

Then, the group $\Gamma$ is stable in finite actions.

###### Proof.

Let $\Gamma\curvearrowright({\rm P},\mu)$ be the limit action of a sofic
approximation of $\Gamma$. The, by $(i)$ the restriction to $\Lambda$ is
weakly contained in finite $\Lambda$-actions. Thus, since the quotient
$\Gamma/\Lambda$ is amenable by $(iii)$, the original action is weakly
contained in the co-induction of finite $\Lambda$-actions using
[MR3047068]*Theorem 1.1. However, by $(ii)$ and [MR2931406]*Lemma 4.15, any
such action is weakly contained in the family of finite $\Gamma$-actions. ∎

Note that Theorem 3.11 implies that surface groups are stable in finite
actions. This is also a consequence of results in by Lazarovich–Levit–Minsky
[minsky] and the remark after Definition 3.7.

######  Question 3.12.

Is there a residually finite group, which is not stable in finite actions?

We believe the answer to this question should be positive; but it seems hard
to come up with a concrete example, see also Remark 3.16.

###  3.3. Proof of the main result

Let us now state our main result precisely, prove it, and discuss some
potential consequences.

######  Theorem 3.13.

Let $\Gamma$ be a countable discrete group and $A$ be a finite abelian group.
Assume the following conditions:

  1. $(i)$

$\Gamma$ is residually finite and of finite type,

  2. $(ii)$

$\Gamma$ satisfies a co-systolic inequality for all finite transitive actions
with coefficients in $A$ and dimension $2$,

  3. $(iii)$

there exists a central extension $\tilde{\Gamma}$ of $\Gamma$ by $A$ which is
not residually finite.

Then, the following implication holds: If $\Gamma$ is stable in finite
actions, then the group $\tilde{\Gamma}$ is not sofic.

This result should be compared with a result of Bowen-Burton [MR4105530], who
proved that if ${\rm PSL}_{d}(\mathbb{Z})$, for $d\geq 5$, is flexibly stable,
then there exists a non-sofic group. The advantage of our result is a
weakening of the assumption of flexible stability and also that the resulting
candidate of a non-sofic group is a bit more concrete. Nevertheless, we will
spend the next section discussing the existence of a group that satisfies
Conditions $(i)$-$(iii)$ in Theorem 3.13.

###### Proof of Theorem 3.13.

Let $\alpha\in{\rm H}^{2}(\Gamma,A)$ be the cohomology class classifying the
central extension $\tilde{\Gamma}$. Assume that $\tilde{\Gamma}$ is sofic and
let ${\rm Q}$ be the associated limit $\tilde{\Gamma}$-action and ${\rm
P}\coloneqq{\rm Q}^{A}$ be the limit action of the associated sofic
approximation of $\Gamma$, see Lemma 3.3. Then, on the one hand $\theta^{\rm
P}_{*}(\alpha)=0$ by Theorem 3.4. On the other side, if $\Gamma$ is stable in
finite actions, then ${\rm P}$ is weakly contained in finite actions and hence
$\theta^{\rm P}_{*}(\alpha)\neq 0$ by Theorem 2.16. This is a contradition and
finishes the proof. ∎

######  Remark 3.14.

We want to comment on the vanishing and non-vanishing of
$\theta_{*}^{P}(\alpha)$ for particular measured $\Gamma$-Boolean algebras.
Recall that for a p.m.p. preserving $\Gamma$-action $(X,\mu)$, we denote by
${\rm M}(X,\mu)$ its measure algebra. By a result of Abért–Weiss [abertweiss],
the Bernoulli action
$\Gamma\curvearrowright(\\{0,1\\}^{\Gamma},\nu^{\otimes\Gamma})$ is weakly
contained in any other free p.m.p. action. In particular, since $\Gamma$ is
residually finite, it is weakly contained in the pro-finite action
$\Gamma\curvearrowright(\hat{\Gamma},\mu_{\rm Haar})$. Hence,

$\theta^{{\rm M}(\\{0,1\\}^{\Gamma},\nu^{\otimes\Gamma})}_{*}(\alpha)\neq 0$

by Theorem 2.16. On the other side, we have $\theta_{*}^{{\rm
M}(X,\mu)}(\alpha)=0$ for $\Gamma\curvearrowright
X=\\{0,1\\}^{\tilde{\Gamma}}/A$ with its natural probability measure and
$\Gamma$-action by Theorem 3.4. In particular, $\Gamma\curvearrowright
X=\\{0,1\\}^{\tilde{\Gamma}}/A$ is not weakly contained in the family of
finite actions of $\Gamma.$

There is also a notion of soficity of a p.m.p. $\Gamma$-action on a standard
probability space, see [MR2566316]*Definition 1.3, which amounts to weak
containment in the limit action of a sofic approximation of the group
$\Gamma$. In view of the previous remark, we obtain the following corollary.

######  Corollary 3.15.

If $\Gamma$ satisfies Conditions $(i)$-$(iii)$ of Theorem 3.13 and is stable
in finite actions, then the p.m.p. action
$\Gamma\curvearrowright\\{0,1\\}^{\tilde{\Gamma}}/A$ is not sofic.

######  Remark 3.16.

There are only few groups $\Gamma$ for which _all_ p.m.p. $\Gamma$-actions are
weakly contained in the family of finite actions. Any such group is clearly
stable in finite actions. These include free groups, surface groups,
hyperbolic 3-manifold groups [MR3047068] and residually finite amenable groups
but likely no groups with Kazhdan’s property (T), see the discussion in
[MR2931406]*p. 465. In particular, it is known that ${\rm SL}_{n}(\mathbb{Z})$
fails to have this property for $n\geq 3$, see [MR2077037]. Thus, potentially,
if any Kazhdan group can be shown to be stable in finite actions, then there
exists a non-sofic p.m.p. action. Another test case for stability in finite
actions is the group $\mathbb{F}_{2}\times\mathbb{F}_{2}$. By the recent
refutation of the Connes’ Embedding Conjecture [mipstar], there do exist
p.m.p. actions of $\mathbb{F}_{2}\times\mathbb{F}_{2}$ that are not weakly
contained in finite actions. Again, any such action would be non-sofic if
$\mathbb{F}_{2}\times\mathbb{F}_{2}$ was stable in finite actions.

###  3.4. A candidate for a non-sofic group

In this section, we will recall the construction of torsionfree lattices in
${\rm PSp}_{2d}(\mathbb{Q}_{p})$ for $d\geq 4$ and $p$ large that satisfy all
conditions in Theorem 3.13.

Condition $(i)$ is satisfied by Lemma 2.7 and Mal’cevs well-known theorem
stating that any finitely generated subgroup of a linear group is residually
finite. Condition $(ii)$ is satisfied by Theorem 2.10 for $p$ large enough. It
remains to discuss Condition $(iii)$, where we borrow the discussion from
[MR4080477]*Section 5.2. We also refer to [MR4080477] for further references.

Let $D$ be the quaternion algebra over $\mathbb{Z}$ and $R$ be a commutative,
unital ring. Then we can define the _quaternion ring_ $D_{R}$ over $R$ as

$D_{R}\coloneqq R\langle i,j,k\rangle/\langle
i^{2}=j^{2}=k^{2}=ijk=1_{R}\rangle.$

There is an involution on $D_{R}$

$\tau_{R}\colon D_{R}\rightarrow D_{R},(r_{0}+r_{1}\cdot i+r_{2}\cdot
j+r_{3}\cdot k)\mapsto(r_{0}-r_{1}\cdot i-r_{2}\cdot j-r_{3}\cdot k)$

and therefore we can define the canonical hermitian sesquilinearform

$h_{n}\colon D_{R}^{n}\times D_{R}^{n}\rightarrow
D_{R},((x_{1},\ldots,x_{n}),(y_{1},\ldots,y_{n}))\mapsto\sum_{i=1}^{n}x_{i}\cdot\tau_{R}(y_{i}).$

We define the algebraic group

$\displaystyle\mathbf{G}(R)$ $\displaystyle\coloneqq{\rm SU}_{n}(D_{R},h_{n})$
$\displaystyle\coloneqq\\{A\in D_{R}^{n\times n}\mid\det(A)=1,\forall x,y\in
D_{R}^{n}:h_{n}(x,y)=h_{n}(A\cdot x,A\cdot y)\\},$

i.e. the $n\times n$-matrices with entries in $D_{R}$ of determinant $1$ and
whose associated linear maps preserve the form $h_{n}$.

Note that $\mathbf{G}$ is an absolutely almost simple, simply connected
$\mathbb{Q}$-algebraic group of type $C_{n}$. We refer an interested reader to
[MR1278263] for general algebraic groups and [MR4080477] for this particular
one. First of all, note that $\mathbf{G}(\mathbb{Q}_{p})$ is isomorphic to
${\rm Sp}_{2n}(\mathbb{Q}_{p})$ and this group is not compact. Recall also
that

(1)
$\mathbf{G}(\mathbb{Z}[\nicefrac{{1}}{{p}}])\leq\mathbf{G}(\mathbb{R})\times\mathbf{G}(\mathbb{Q}_{p})$

is a lattice, see [MR1278263]*Chapter 5.4. Moreover, since
$\mathbf{G}(\mathbb{R})$ is isomorphic to ${\rm U}(2n)\cap{\rm
Sp}_{2n}(\mathbb{C})$ and hence compact, it follows that
$\mathbf{G}(\mathbb{Z}[\nicefrac{{1}}{{p}}])\leq\mathbf{G}(\mathbb{Q}_{p})$ is
also a lattice embedding. Our choice of $\Gamma$ is a torsion-free subgroup of
$\mathbf{G}(\mathbb{Z}[\nicefrac{{1}}{{p}}])$ of finite index. Note that any
such $\Gamma$ is also a torsionfree lattice in ${\rm
PSp}_{2d}(\mathbb{Q}_{p}),$ since the center of ${\rm
Sp}_{2d}(\mathbb{Q}_{p})$ is finite.

We continue with a discussion of the universal central extension of $\Gamma$,
which was first studied by Deligne [MR507760] and later Prasad, who showed
universality [MR2020658].

######  Theorem 3.17.

For every prime $p\geq 5$ there is a unique central extension

(2)
$0\longrightarrow\mathbb{Z}/(p-1)\cdot\mathbb{Z}\overset{\iota}{\longrightarrow}\mathbf{G}(\mathbb{Q}_{p})^{\prime}\overset{\pi}{\longrightarrow}\mathbf{G}(\mathbb{Q}_{p})\longrightarrow
0$

which is universal in the sense that whenever $A$ is a discrete abelian group
and there exists a central extension of topological groups

$0\longrightarrow A\longrightarrow
E\longrightarrow\mathbf{G}(\mathbb{Q}_{p})\longrightarrow 0,$

then there are continuous group homomorphisms such that the following diagram
commutes

$\leavevmode\hbox to253.07pt{\vbox
to54.29pt{\pgfpicture\makeatletter\hbox{\hskip
126.53352pt\lower-27.19705pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-126.53352pt}{-27.09721pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
6.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
6.80554pt\hfil&\hfil\hskip 32.05551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.05554pt\hfil&\hfil\hskip 32.28467pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.97916pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${E}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
8.2847pt\hfil&\hfil\hskip 41.41568pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.11017pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathbf{G}(\mathbb{Q}_{p})}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
17.41571pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
6.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil&\hfil\hskip 53.32106pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.01555pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathbb{Z}/(p-1)\cdot\mathbb{Z}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
29.32109pt\hfil&\hfil\hskip 42.18567pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.88016pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathbf{G}(\mathbb{Q}_{p})^{\prime}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
18.1857pt\hfil&\hfil\hskip 41.41568pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-13.11017pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathbf{G}(\mathbb{Q}_{p})}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
17.41571pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-112.72244pt}{12.2778pt}\pgfsys@lineto{-68.25688pt}{12.2778pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-68.0569pt}{12.2778pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-51.34584pt}{12.2778pt}\pgfsys@lineto{3.02072pt}{12.2778pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.2207pt}{12.2778pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-59.60138pt}{5.91809pt}\pgfsys@lineto{-59.60138pt}{-15.33754pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-59.60138pt}{-15.53752pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{20.39008pt}{12.2778pt}\pgfsys@lineto{53.49109pt}{12.2778pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{53.69107pt}{12.2778pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{11.90538pt}{5.91809pt}\pgfsys@lineto{11.90538pt}{-13.782pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{11.90538pt}{-13.98198pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{89.12247pt}{12.2778pt}\pgfsys@lineto{112.32248pt}{12.2778pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{112.52246pt}{12.2778pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.33746pt}\pgfsys@invoke{
}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{71.50676pt}{3.41809pt}\pgfsys@lineto{71.50676pt}{-15.7375pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.5375pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{71.50676pt}{3.41809pt}\pgfsys@lineto{71.50676pt}{-15.7375pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-112.72244pt}{-24.59721pt}\pgfsys@lineto{-89.52243pt}{-24.59721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-89.32245pt}{-24.59721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-30.08029pt}{-24.59721pt}\pgfsys@lineto{-6.88028pt}{-24.59721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.6803pt}{-24.59721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-19.51907pt}{-22.24445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\iota}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{30.29108pt}{-24.59721pt}\pgfsys@lineto{53.49109pt}{-24.59721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{53.69107pt}{-24.59721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{40.09598pt}{-22.24445pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\pi}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{89.12247pt}{-24.59721pt}\pgfsys@lineto{112.32248pt}{-24.59721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{112.52246pt}{-24.59721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
The pull-back of this central extension to the lattice
$\Gamma\leq\mathbf{G}(\mathbb{Q}_{p})$ is denoted by
$\tilde{\Gamma}\coloneqq\pi^{-1}(\Gamma)$. The final ingredient is the
following theorem, which provides Condition $(iii)$ of Theorem 3.13.

######  Theorem 3.18 (following Deligne, see [MR4080477]).

Let $p\geq 5$ and $A$ the cyclic group of order $p-1$. For the torsion-free
lattice $\Gamma\leq{\rm PSp}_{2d}(\mathbb{Q}_{p})$ discussed above, the
central extension $\tilde{\Gamma}$

$0\longrightarrow
A\longrightarrow\tilde{\Gamma}\longrightarrow\Gamma\rightarrow 0$

is not residually finite.

Thus, taking $p$ large enough in order to allow for the application of Theorem
2.10, we see that all conditions of Theorem 3.13 are met.

## Acknowledgments

The ideas for this paper stem from 2018 and were communicated back then among
various experts and were presented at a conference in Warwick in March 2019.
The second named author thanks Alon Dogon for corrections and Adrian Ioana for
a discussion on Section 3.2. We both thank Vadim Alekseev for inspiring
discussions. The results of this paper also appear in the doctoral thesis of
the first named author, see [thesisgohla].

## References