#  Asymptotic Cohomology and Uniform Stability
for Lattices in Semisimple Groups

Lev Glebsky, Alexander Lubotzky, Nicolas Monod, Bharatram Rangarajan

###### Abstract

It is, by now, classical that lattices in higher rank semisimple groups have
various rigidity properties. In this work, we add another such rigidity
property to the list, namely uniform stability with respect to the family of
unitary operators on finite-dimensional Hilbert spaces equipped with
submultiplicative norms. Towards this goal, we first build an elaborate
cohomological theory capturing the obstruction to such stability, and show
that the vanishing of second cohomology implies uniform stability in this
setting. This cohomology can be roughly thought of as an asymptotic version of
bounded cohomology, and sheds light on a question raised in [Mon06] about a
possible connection between vanishing of second bounded cohomology and Ulam
stability. Along the way, we use this criterion to provide a short conceptual
(re)proof of the classical result of Kazhdan [Kaz82] that discrete amenable
groups are Ulam stable. We then use this machinery to establish our main
result, that lattices in a class of higher rank semisimple groups (which are
known to have vanishing bounded cohomology) are uniformly stable.

Dedicated to Robert J. Zimmer with admiration and affection

## Introduction

Consider a semisimple group $G=\prod^{k}_{i=1}\mathbf{G}_{i}(K_{i})$, where
for $1\leq i\leq k$, $K_{i}$ is a local field, and $\mathbf{G}_{i}$ is an
almost $K_{i}$-simple group. If the rank
$\sum^{k}_{i=1}rk_{K_{i}}(\mathbf{G}_{i})\geq 2$, such a $G$ is referred to as
a _higher rank_ semisimple group. The class of irreducible lattices $\Gamma$
in such groups $G$ (referred to as _higher rank lattices_) form an interesting
class of groups, which over the years, have been shown to satisfy many
rigidity properties, such as local rigidity, Mostow strong-rigidity, Margulis
super-rigidity (implying that they are arithmetic groups), Zimmer cocycle
rigidity, quasi-isometric rigidity, first-order rigidity, etc. (see [ES05],
[ALM19], [Mar91], [BFH20] and the references therein). A common feature of the
classical rigidity results is that such a higher rank lattice $\Gamma$ has
some clear family of representations, and all other representations are just
easy variants of them.
The goal of this paper is to demonstrate another type of rigidity phenomena of
these lattices. Before stating the exact formulation, let us recall that
Margulis super-rigidity, while usually not formulated this way, also gives a
full classification of all the finite dimensional unitary representations of a
higher rank lattice $\Gamma$ as above. Margulis super-rigidity implies that
all such irreducible representations come from a combination of those that
factor through finite quotients (and these are the only ones if $\Gamma$ is a
non-uniform lattice) and from the representations of $\Gamma$ appearing
naturally in its definition as an arithmetic group by Galois twisting (see
§$1.3$ in [Mar91]). The rigidity phenomenon we study here, which is called
_uniform stability_ , is the property that every unitary almost-representation
of $\Gamma$ is a small deformation of a unitary representation.

### Uniform Stability of Groups

Let $\Gamma$ be a discrete group and $(G,d_{G})$ be a metric group (where
$d_{G}$ is a bi-invariant metric on $G$). For $\epsilon>0$, a map
$\phi:\Gamma\to G$ is said to be an $\epsilon$-almost homomorphism (or
_$\epsilon$ -homomorphism_) if $d_{G}(\phi(xy),\phi(x)\phi(y))\leq\epsilon$
for every $x,y\in\Gamma$. The value
$\sup_{x,y\in\Gamma}d_{G}(\phi(xy),\phi(x)\phi(y))$ is called the _defect_ of
$\phi$.
Let $\mathcal{G}$ be a family of metric groups. We say that $\Gamma$ is
_uniformly stable_ with respect to $\mathcal{G}$ if for any $\epsilon>0$,
there exists $\delta=\delta(\epsilon)$ with $\lim_{\epsilon\to
0}\delta(\epsilon)=0$ such that for any $\epsilon$-homomorphism
$\phi:\Gamma\to G$ (for $G\in\mathcal{G}$), there exists a homomorphism
$\psi\in Hom(\Gamma,G)$ with
$\sup_{x\in\Gamma}\>d_{G}(\phi(x),\psi(x))\leq\delta$. In other words,
$\Gamma$ is uniformly stable with respect to $\mathcal{G}$ if any almost
homomorphism of $\Gamma$ to any group in the family $\mathcal{G}$ is close to
a (true) homomorphism.

Questions of this nature were first raised and studied in [Tur38], [vN29] and
[Ula60], and of particular interest is the case when $\mathcal{G}$ is the
family of unitary operators on Hilbert spaces and the metric is given by a
norm (on the space of bounded operators), as studied in [Kaz82] and [BOT13].
Note that in this work, we will be interested solely in uniform stability, as
opposed to _pointwise_ stability, as studied in [DCGLT20], [AP15] and the
references therein.

The notion of uniform stability with respect to unitary operators on Hilbert
spaces equipped with the operator norm is referred to in [BOT13] as _strong
Ulam stability_ , while if we restrict the family to unitary operators on
_finite-dimensional_ Hilbert spaces, it is referred to as _Ulam stability_. In
the pioneering work of Kazhdan [Kaz82] (and clarified further in [Sht13] and
[Joh88]), it is shown that

######  Theorem 0.0.1 ([Kaz82]).

Every (discrete) amenable group $\Gamma$ is Ulam stable (in fact, even
strongly Ulam stable).

It is worth noting that the only known examples of strongly Ulam stable groups
are amenable, and it is natural to ask if strong Ulam stability characterizes
amenability.
Let us mention at this point that one of the (innumerable) equivalent
characterizations of amenability is given in terms of the vanishing of bounded
cohomology with dual coefficients: $\Gamma$ is amenable iff
$\operatorname{H}_{b}^{n}(\Gamma,V)=0$ for every dual Banach $\Gamma$-module
$V$ and $n>0$. Here $\operatorname{H}_{b}^{n}(\Gamma,V)$ denotes the $n$-th
_bounded_ cohomology group of $\Gamma$ with coefficients in the Banach
$\Gamma$-module $V$. Kazhdan’s proof does not use this result explicitly but
does use a notion of $\epsilon$-cocycles and approximate cohomology in degree
$2$.

Ulam stability was further studied in [BOT13] where they show more examples
(and non-examples) of Ulam stable groups. It is shown there that if a group
contains a non-abelian free subgroup, then it is _not_ strongly Ulam stable.
In particular, this means that higher rank lattices are not strongly Ulam
stable. On the positive side, they show:

######  Theorem 0.0.2 ([BOT13]).

Let $\mathcal{O}$ be the ring of integers of a number field, $S$ a finite set
of primes, and $\mathcal{O}_{S}$ the corresponding localization. Then for
every $n\geq 3$, $SL(n,\mathcal{O}_{S})$ is Ulam stable.

The proof of this result in [BOT13] uses the fact that $SL(n,\mathcal{O}_{S})$
(for $n\geq 3$) is boundedly generated by elementary matrices, and makes no
reference to bounded cohomology (this result is further extended in [Gam11] in
the case of $n=2$ when $\mathcal{O}_{S}$ has infinitely many units). However,
note that for $\Gamma=SL(n,\mathcal{O}_{S})$,
$\operatorname{H}_{b}^{2}\left(\Gamma,V\right)=0$ for every dual separable
$\Gamma$-module $V$. In fact, it is shown in [BM99] that for every higher rank
lattice $\Gamma$ and any dual, separable Banach $\Gamma$-module $V$ with
$V^{\Gamma}=\\{0\\}$, $\operatorname{H}_{b}^{2}(\Gamma,V)=0$. All this hints
at a possible connection between bounded cohomology and Ulam stability, as
raised by Monod in his ICM talk [Mon06, Problem F], and serves as one of the
starting points for our current work.

## Main Results and Methods

In this paper, we generalize Theorem 0.0.1 and Theorem 0.0.2 to a wider class
of groups and metrics. We shall consider the question of uniform stability
with respect to the family $\mathfrak{U}$ of groups of unitary operators on
_finite-dimensional_ Hilbert spaces, with the metrics induced from
submultiplicative norms on matrices (which we shall denote uniform
$\mathfrak{U}$-stability). These include the $p$-Schatten norms for $1\leq
p\leq\infty$ (and in particular, uniform $\mathfrak{U}$-stability subsumes
Ulam stability). Furthermore, we shall show uniform $\mathfrak{U}$-stability
_with a linear estimate_ , which means that the distance of an almost
homomorphism from a homomorphism is linearly bounded by its defect, and all
our results are proved in this stronger notion of stability.

To this end, we build a new type of bounded cohomological theory that can
capture obstructions to uniform $\mathfrak{U}$-stability, so that uniform
$\mathfrak{U}$-stability follows as a consequence of the vanishing of the
second cohomology group in this theory. While we shall develop this in full
detail in §4, our technique involves the following two main steps:

  * •

Defect Diminishing: Expressing the problem of uniform stability as a
homomorphism lifting problem, we can treat it as a culmination of intermediate
lifts so that at each step, the lifting kernel is abelian. This is a uniform
variant of defect diminishing that was introduced in [DCGLT20] in the non-
uniform setting, and is applicable when the relevant norms in the target
groups are submultiplicative.

  * •

Asymptotic Cohomology: Such a homomorphism lifting problem with abelian kernel
naturally leads to a cohomological reformulation (as in [DCGLT20]). However,
unlike in the non-uniform setting where ordinary group cohomology comes up, in
our uniform setting we need to carefully construct a new cohomology theory
such that the vanishing of the second cohomology group in this model implies
(uniform) defect diminishing, and hence uniform stability.

The cohomological theory we construct is an asymptotic variant of the bounded
cohomology of the ultrapower ${}^{*}\Gamma$ with coefficients in a suitable
ultraproduct Banach space $\mathcal{W}$, but _restricted to the internal
objects in this universe_ , which we shall call the _asymptotic cohomology_ of
$\Gamma$ denoted $\operatorname{H}_{a}^{\bullet}(\Gamma,\mathcal{W})$.

######  Theorem 0.0.3.

Suppose $\operatorname{H}_{a}^{2}(\Gamma,\mathcal{W})=0$, then $\Gamma$ is
uniformly $\mathfrak{U}$-stable with a linear estimate.

This new cohomology theory bears some similarity to the theory of bounded
cohomology, and sometimes we can easily adapt arguments there to our model
(for instance, we can show that
$\operatorname{H}_{a}^{2}(\Gamma,\mathcal{W})=0$ for an amenable group
$\Gamma$, immediately implying that amenable groups are Ulam-stable as in
[Kaz82],[Sht13],[Joh88]), though other times serious difficulties arise (which
are responsible for the length of this paper).

The groups $\Gamma$ that we will be particularly interested in are lattices in
higher rank semisimple groups. Unlike some of the other rigidity results which
sometimes hold for some lattices in rank one simple groups, we also first show
the following result:

######  Proposition 0.0.4.

If $\Gamma$ is a lattice in a semisimple group of rank $1$, then $\Gamma$ is
not uniformly $\mathfrak{U}$-stable.

It is shown in [Fuj98] that for such a $\Gamma$,
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ is infinite dimensional. More
precisely, Fujiwara constructs (many) non-trivial quasi-homomorphisms
witnessing that the comparison map
$c:\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to H^{2}(\Gamma,\mathbf{R})$ is
not injective. By exponentiation, such quasimorphisms yield almost
homomorphisms to $U(1)$ that are not close to any homomorphism (we shall
discuss this in more detail in §1). In particular, a lattice of rank one is
not uniformly U-stable, and to hope for uniform $\mathfrak{U}$-stability, the
condition that rank of $\Gamma$ is at least $2$ is necessary.

For the main result of the paper, we need some definitions capturing
properties of the class of semisimple groups we will be interested in. For a
locally compact group $G$, we denote by
$\operatorname{H}_{b}^{\bullet}(G,\mathbf{R})$ the _continuous_ bounded
cohomology of $G$ with trivial coefficients.

  * •

A locally compact group $G$ is said to have the 2½property (of vanishing
bounded cohomology) if $\operatorname{H}_{b}^{2}(G,\mathbf{R})=0$ and
$\operatorname{H}_{b}^{3}(G,\mathbf{R})$ is Hausdorff.

  * •

Let $G$ be a non-compact simple Lie group, and fix a minimal parabolic
subgroup $P\leq G$. The group $G$ is said to have
Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$ if there exist two proper
parabolic subgroups $Q_{1}$ and $Q_{2}$ containing $P$, both having the
2½-property, such that $G$ is boundedly generated by the union $Q_{1}\cup
Q_{2}$. A semisimple group $G$ is said to have
Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$ if all its simple factors have
Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$.

Note that if a semisimple group has
Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$, then by definition, it must be
of rank at least $2$. But note that not all simple groups have the property
(for instance, $SL_{3}(\mathbf{R})$). However, in §6.3, we will show that many
classes of groups do have this property, for example, all simple groups (of
rank at least $2$) over $\mathbf{C}$ or over a non-archimedean field, and
$SL_{n}(\mathbf{R})$ for $n\geq 4$.
We can now state our main result:

######  Theorem 0.0.5.

Let $\Gamma$ be a lattice in a semisimple group $G$ that has
Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$. Then
$\operatorname{H}_{a}^{2}(\Gamma,\mathcal{W})=0$, so in particular, $\Gamma$
is uniformly $\mathfrak{U}$-stable.

The main result of our paper is thus concerned with showing that
$\operatorname{H}_{a}^{2}(\Gamma,\mathcal{W})=0$ for $\Gamma$ being a lattice
in a higher rank Lie group (satisfying certain conditions). For this, we take
inspiration from the results of [BM99] about the vanishing of bounded
cohomology for such lattices. More specifically, our approach is inspired by a
proof of [MS04] specifically in degree two, and a version of that result and
proof technique are outlined below.

######  Theorem 0.0.6 ([MS04]).

Let $G$ be a higher rank simple group, and $P\leq G$ be a minimal parabolic
subgroup. Suppose $G$ contains two proper parabolic subgroups $Q_{1}$ and
$Q_{2}$ such that $P\subseteq Q_{1}\cap Q_{2}$, $G$ is generated by $Q_{1}\cup
Q_{2}$, and
$\operatorname{H}_{b}^{2}(Q_{1},\mathbf{R})=\operatorname{H}_{b}^{2}(Q_{2},\mathbf{R})=0$.
Then for any lattice $\Gamma$ in $G$ and a dual separable Banach
$\Gamma$-module $W$, $\operatorname{H}_{b}^{2}(\Gamma,W)=0$.

The proof of Theorem 0.0.6 proceeds in several steps briefly sketched below,
where we also mention the corresponding steps and difficulties in the proof of
Theorem 0.0.5 even when $G$ is simple:

  * •

The first step is to use an Eckman-Shapiro induction to construct a dual,
separable, continuous Banach $G$-module $V$ so that
$\operatorname{H}_{b}^{2}(\Gamma,W)=\operatorname{H}_{b}^{2}(G,V)$, thus
reducing the problem to showing that $\operatorname{H}_{b}^{2}(G,V)=0$. A
similar inductive procedure, described in §5, allows us to construct an
ultraproduct Banach space $\mathcal{V}$ with an asymptotic action of $G$ so
that
$\operatorname{H}_{a}^{2}(\Gamma,\mathcal{W})=\operatorname{H}_{a}^{2}(G,\mathcal{V})$.
Note that in the setting of asymptotic cohomology, we actually work with
ultrapowers ${}^{*}\Gamma$ and ${}^{*}G$, and so ${}^{*}G/{}^{*}\Gamma$ is not
locally compact. However, the restriction to internal objects allows us to
carefully work out an induction procedure as needed. The induced module
$\mathcal{V}$ also has an internal continuity property (defined in §4.1) that
we establish in §5.

  * •

Since $P$ is amenable, the bounded cohomology
$\operatorname{H}_{b}^{\bullet}(G,V)$ can be computed as the cohomology of the
complex

${0}$${V^{G}}$${L^{\infty}(G/P,V)^{G}}$${L^{\infty}((G/P)^{2},V)^{G}}$${L^{\infty}((G/P)^{3},V)^{G}}$${\dots}$$\scriptstyle{\epsilon}$$\scriptstyle{d^{0}}$$\scriptstyle{d^{1}}$$\scriptstyle{d^{2}}$

Furthermore, for a parabolic subgroup $P\leq Q\leq G$, the bounded cohomology
$\operatorname{H}_{b}^{\bullet}(Q,V)$ can be computed as the cohomology of the
complex

${0}$${V^{Q}}$${L^{\infty}(G/P,V)^{Q}}$${L^{\infty}((G/P)^{2},V)^{Q}}$${L^{\infty}((G/P)^{3},V)^{Q}}$${\dots}$$\scriptstyle{\epsilon}$$\scriptstyle{d^{0}}$$\scriptstyle{d^{1}}$$\scriptstyle{d^{2}}$

These steps too can be reworked in the asymptotic setting, analogous to the
procedure in bounded cohomology theory, again thanks to the restriction on
internality, and this is described in §4.

  * •

The motivation behind the preceding step is that we have at our disposal the
following double ergodicity theorem: let $V$ be a continuous $G$-module and
$\alpha\in L^{\infty}\left((G/P)^{2},V\right)^{G}$, then $\alpha$ is
essentially constant. This theorem follows from the Mautner’s lemma: let $V$
be a continuous $G$-module and $N\leq G$ be a non-compact subgroup, then
$V^{N}=V^{G}$. Both these results are particularly useful in the context of
$\operatorname{H}_{b}^{2}(G,V)$.
In our setting, there are particular difficulties in obtaining an analoguous
Maunter lemma due to the asymptotic nature of our model. We overcome them for
our specific Banach module $\mathcal{V}$ by applying a suitable correction to
exact cocycles using structure results for the semisimple group $G$; this is
worked out in §6.

  * •

For the parabolic subgroups $Q_{1}$ and $Q_{2}$ as in the hypothesis, an
inflation-restriction sequence argument implies that
$\operatorname{H}_{b}^{2}(Q_{i},V)=\operatorname{H}_{b}^{2}(Q_{i}/N_{i},V^{N_{i}})$
for $N_{i}$ being the (amenable) radical of $Q_{i}$ for $i\in\\{1,2\\}$. By
Mautner’s lemma and the hypothesis that
$\operatorname{H}_{b}^{2}(Q_{i},\mathbf{R})=0$, one concludes that
$\operatorname{H}_{b}^{2}(Q_{i},V)=0$.
The analogous hypothesis in our asymptotic setting is
Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$, where the conditions that
$\operatorname{H}_{b}^{2}(Q_{i},\mathbf{R})=0$ and
$\operatorname{H}_{b}^{3}(Q_{i},\mathbf{R})$ is Hausdorff together are used to
conclude that $\operatorname{H}_{a}^{2}(Q_{i},\mathcal{V})=0$ in §6.1.

  * •

Let $\omega\in L^{\infty}\left((G/P)^{3},V\right)^{G}$ be a bounded
$2$-cocycle for $G$. Since
$\operatorname{H}_{b}^{2}(Q_{1},V)=\operatorname{H}_{b}^{2}(Q_{2},V)=0$, there
exist $\alpha_{1}\in L^{\infty}\left((G/P)^{2},V\right)^{Q_{1}}$ and
$\alpha_{2}\in L^{\infty}\left((G/P)^{2},V)\right)^{Q_{2}}$ such that
$\omega=d\alpha_{1}=d\alpha_{2}$. In particular, $\alpha_{1}-\alpha_{2}$ is a
$1$-cocycle for $Q_{1}\cap Q_{2}$. Since $P\leq Q_{1}\cap Q_{2}$,
$\alpha_{1}-\alpha_{2}$ is a $1$-cocycle for $P$ as well. But since
$\operatorname{H}_{b}^{1}(P,V)=0$, one can show using the double ergodicity
theorem, that $\alpha_{1}=\alpha_{2}$ ($=\alpha$, say), implying that $\alpha$
is equivariant with respect to both $Q_{1}$ and $Q_{2}$, and hence is
$G$-equivariant. Thus $\omega=d\alpha$ for $\alpha\in
L^{\infty}\left((G/P)^{2},V\right)^{G}$, proving that
$\operatorname{H}_{b}^{2}(G,V)=0$. This step too goes through in our setting
once we have our asymptotic variant of the ergodicity theorem used in the
classical case.

While in this paper, we focus on using the theory of asymptotic cohomology to
prove uniform $\mathfrak{U}$-stability for lattices in semisimple groups, the
framework and tools developed here have also been used in subsequent work
[FFR23] to prove uniform $\mathfrak{U}$-stability for other classes of groups
such as lamplighter groups $\Gamma\wr\Lambda$ where $\Lambda$ is infinite and
amenable, as well as several groups of dynamical origin such as Thompson’s
group $F$. The techniques there too are analogous to corresponding vanishing
results of bounded cohomology in [Mon22], yet again highlighting the
connections between the theories of bounded cohomology and asymptotic
cohomology.

### Outline of the Paper

We begin with the much simple setting of uniform stability with respect to the
fixed group $U(1)$ (equipped with the absolute value metric) in §1. In this
case, we can reduce the question of uniform $U(1)$-stability of $\Gamma$ to
the injectivity of the comparison map
$c:\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to H^{2}(\Gamma,\mathbf{R})$.
After recalling how classical results from [Fuj98] imply that lattices in Lie
groups of rank $1$ are not uniformly $U(1)$-stable, we then show that lattices
in higher rank Lie groups are uniformly $U(1)$-stable. While the connection
between uniform $U(1)$-stability of a group $\Gamma$ and the second bounded
cohomology $\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ is classical, it
motivates the idea of using the logarithm map on an almost homomorphism to
construct a bounded $2$-cocycle of $\Gamma$ in a Banach space, which we
develop in §3.2 for a more general setting.

In §2, we begin by defining the basic notions in full detail and rigor in
§2.1. In particular we focus on interpreting stability in terms of sequences
of maps and asymptotic homomorphisms, which we then refine further in §2.2 in
the language of non-standard analysis. This formulation will allow us to
reinterpret the question of uniform stability as a _homomorphism lifting
problem_ on the lines of the approach used in [DCGLT20] [AP15]. While such a
lifting problem motivates the attempt at constructing a cohomology, an
obstacle here is that the kernel of the extension is not abelian. This issue
is resolved in [DCGLT20] by considering lifts in small increments so that the
kernel at each step is abelian. This idea, known as _defect diminishing_ , is
explored in §2.3, and can be shown to imply uniform stability.

In §3 we begin by focusing on a particular family of metric groups for which
defect diminishing corresponds to a homomorphism lifting problem with an
abelian kernel. This is the family of unitary groups equipped with
submultiplicative norms, discussed in §3.1. In [DCGLT20], defect diminishing
combined with ordinary group cohomology with coefficients in the abelian
kernel (which turns out to be a Banach $\Gamma$-module) is sufficient to study
non-uniform stability. But in our setting, the uniformity condition involves
subtleties that necessitate transfering to the internal Lie algebra with an
internal _asymptotic_ -action of ${}^{*}\Gamma$ (the ultrapower of $\Gamma$),
and the formulation of an internal and asymptotic bounded cohomology with
cofficients in that internal space, denoted $\mathcal{W}$. This is motivated
in §3.2, and we conclude the section by demonstrating the machinery built so
far in (re)proving the result of Kazhdan [Kaz82] that discrete amenable groups
are Ulam stable.

§4 begins with the rigorous definitions of an internal Banach spaces and
asymptotic ${}^{*}G$-modules for a locally compact group $G$ (defining our
notions in the category of topological groups is necessary in order to deal
with lattices in Lie groups, which shall involve an Eckmann-Shapiro induction
of cohomologies explored in §5) in §4.1. In §4.2, we formally define
$\operatorname{H}_{a}^{\bullet}(G,\mathcal{V})$ using an internal
$L^{\infty}$-spaces, and study some functorial properties and different
cochain complexes that can be used to compute
$\operatorname{H}_{a}^{\bullet}(G,\mathcal{V})$ in §4.3. Many of the
techniques used here have parallels in the theory of continuous bounded
cohomology as in [Mon01].

In §5, we restrict our attention to a lattice $\Gamma$ in a Lie group $G$, and
begin by studying an intermediate structure
$\mathcal{L}_{b}^{\infty}(G,\mathcal{W})^{\sim{}^{*}\Gamma}$ that is not quite
the induction module $\mathcal{V}=\mathcal{L}^{\infty}(D,\mathcal{W})$ in our
machinery, but comes with an internal (true) ${}^{*}G$-action (as opposed to
an action upto infinitesimals). This structure leads to useful results proved
in §5.1, and the actual induction module and an Eckmann-Shapiro induction
procedure are discussed in §5.2. We conclude the section with a continuity
property of our module $\mathcal{V}$, which we use to define contracting
elements to lay the groundwork for a Mautner’s lemma to be proved in the next
section.

Finally, in §6, we come to the higher rank semisimple groups of interest, and
begin by discussing an analogue of the Mautner’s lemma in our setting, along
with double ergodicity lemmas in §6.1, and use these to prove that
$\operatorname{H}_{a}^{2}(Q,\mathcal{V})=0$ for $Q\leq G$ being a proper
parabolic subgroup. All these techniques come together in §6.2 where we prove
that $\operatorname{H}_{a}^{2}(G,\mathcal{V})=0$ for semisimple groups $G$
that have Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$, thus implying uniform
stability with a linear estimate for lattices in such groups. In §6.3, we list
out classes of simple groups $G$ that have
Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$, and conclude in §7 with some
discussion on the limitations of our method and related open questions.

### Acknowledgements

The second author acknowledges with gratitude the hospitality and support of
the Fields Institute (Toronto) and the Institute for Advanced Study
(Princeton) where part of this work was carried on, as well as a grant by the
European Research Council (ERC) under the European Union’s Horizon 2020 (grant
agreement No $882751$). The results presented here are part of the fourth
author’s PhD thesis, also supported by the same grant. The authors would like
to thank Andrei Rapinchuk for his guidance in the proof of Proposition 6.1.1.

This paper is dedicated to Bob Zimmer for his 75th birthday in honor of his
remarkable achievements and influence on the study of the rigidity of lattices
in semisimple Lie groups. That influence is also apparent in the current
paper.

###### Contents

  1. 1 $U(1)$-Stability of Groups
  2. 2 Preliminaries and Basic Constructions
    1. 2.1 Uniform Stability and Asymptotic Homomorphisms
    2. 2.2 Ultraproducts and Internal Maps
    3. 2.3 Internal Liftings and Defect Diminishing
  3. 3 A Cohomological Interpretation of Stability
    1. 3.1 Lifting with an Abelian Kernel
    2. 3.2 Linearization and the Lie Algebra
    3. 3.3 Uniform Stability of Amenable groups
  4. 4 Asymptotic Cohomology of Groups
    1. 4.1 Basic Definitions and Some Cohomological Algebra
    2. 4.2 The $L^{\infty}$-cohomology and $\operatorname{H}_{a}^{\bullet}(G,\mathcal{V})$
    3. 4.3 Amenable Actions and Cohomology of Subgroups
  5. 5 The Induction Module
    1. 5.1 The ${}^{*}G$-action on $\mathcal{L}^{\infty}_{b}({}^{*}G,\mathcal{W})^{\sim{}^{*}\Gamma}$
    2. 5.2 $\mathcal{L}^{\infty}({}^{*}D,\mathcal{W})$ and the Eckmann-Shapiro Induction
    3. 5.3 Internal Contraction and Fixed Points
  6. 6 Vanishing of $\operatorname{H}_{a}^{2}(\Gamma,\mathcal{W})$
    1. 6.1 Asymptotic Mautner Property and Ergodicity
    2. 6.2 Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$ and the Main Theorem
    3. 6.3 Groups with Property-$G(\mathcal{Q}_{1},\mathcal{Q}_{2})$
  7. 7 Conclusions and Discussion

##  1 $U(1)$-Stability of Groups

We begin with a simpler setting, namely uniform stability of a discrete group
$\Gamma$ with respect to the abelian group $U(1)$. This section can be read
independently of the rest.

######  Definition 1.0.1.

For $\epsilon>0$, a map $\phi:\Gamma\to U(1)$ is said to be an
$\epsilon$-homomorphism if

$\sup_{x,y\in\Gamma}\lvert\phi(xy)-\phi(x)\phi(y)\rvert\leq\epsilon$

The value $\sup_{x,y\in\Gamma}\lvert\phi(xy)-\phi(x)\phi(y)\rvert$ is called
the _defect_ of $\phi$.

######  Definition 1.0.2.

A group $\Gamma$ is said to be uniformly $U(1)$-stable if for every
$\delta>0$, there exists $\epsilon>0$ such that if $\phi:\Gamma\to U(1)$ is an
$\epsilon$-homomorphism, there exists a homomorphism $\psi:\Gamma\to U(1)$
such that $\sup_{x\in\Gamma}\lvert\phi(x)-\psi(x)\rvert<\delta$.

A closely related notion is that of a quasimorphism. A _quasimorphism_ is a
map $f:\Gamma\to\mathbf{R}$ such that there exists $D\geq 0$ such that for
every $x,y\in\Gamma$,

$\lvert f(x)+f(y)-f(xy)\rvert\leq D$

Let $QM(\Gamma)$ denote the space of all quasimorphisms of $\Gamma$. A trivial
example of a quasimorphism is obtained by perturbing a homomorphism by a
bounded function, and such quasimorphisms form the subspace
$Hom(\Gamma,\mathbf{R})\oplus C_{b}(\Gamma,\mathbf{R})$ of $QM(\Gamma)$ (where
$C_{b}(\Gamma,\mathbf{R})$ denotes the space of all bounded functions from
$\Gamma$ to $\mathbf{R}$) A quasimorphism that is not at a bounded distance
from any homomorphism is called a _non-trivial quasimorphism_. In this
setting, a question analogous to uniform stability is whether every
quasimorphism is at a bounded distance from a homomorphism. That is, is
$QM(\Gamma)=Hom(\Gamma,\mathbf{R})\oplus C_{b}(\Gamma,\mathbf{R})$?
It is known that every quasimorphism class contains a unique _homogenous_
quasimorphism (a quasimorphism $f$ is said to be homogenous if for every
$g\in\Gamma$ and $n\in\mathbf{N}$, $f(g^{n})=nf(g)$). Suppose $f$ is a
homogenous quasimorphism of $\Gamma$ that is not a homomorphism. Then its
exponent $\mu\coloneq e^{2\pi i\epsilon f}:\Gamma\to U(1)$ is a function whose
defect can be made arbitrarily small (by choosing $\epsilon\to 0$), but whose
distance from homomorphisms is bounded below by a positive constant.

######  Proposition 1.0.3 ([BOT13] [MS06]).

If $\Gamma$ admits a non-trivial quasimorphism, then $\Gamma$ is not uniformly
$U(1)$-stable.

Quasimorphisms are also closely related to group cohomology (this is well-
known and classical, see [Fri17],[Mon06] for references). Observe that given a
quasimorphism $f:\Gamma\to\mathbf{R}$, the map

$df:\Gamma\times\Gamma\to\mathbf{R}$ $df(x,y)=f(x)+f(y)-f(xy)$

is a $2$-coboundary for $\Gamma$ in $\mathbf{R}$, while also being a bounded
function that satisfies the $2$-cocycle condition (and hence a _bounded
$2$-cocycle_). This leads to the following characterization of quasimorphisms
classes ([Fri17], [Mon06]):

######  Proposition 1.0.4.

The kernel, denoted $E\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$, of the
comparison map $c:\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to
H^{2}(\Gamma,\mathbf{R})$ is isomorphic to the space of quasimorphisms modulo
the suabspace of trivial quasimorphisms.

$E\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\cong\frac{QM(\Gamma)}{C_{b}(\Gamma,\mathbf{R})\oplus
Hom(\Gamma,\mathbf{R})}$

Hence to show that $\Gamma$ is not $U(1)$-stable, it is sufficient to show
that the comparison map $c:\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to
H^{2}(\Gamma,\mathbf{R})$ is not injective. In [Fuj98], it is shown that for a
lattice in a rank one semisimple Lie group, its comparison map has non-zero
kernel, and hence

######  Theorem 1.0.5.

Let $H$ be a semisimple group (as in the introduction) of rank one, and let
$\Gamma$ be a lattice in $H$. Then $\Gamma$ is not uniformly $U(1)$-stable.

The rest of this section is devoted to showing that higher rank lattices are
$U(1)$-stable. For this goal the bounded cohomology plays a central role. For
simplicity, we endow $U(1)$ with the distance coming from seeing it as
$\mathbf{R}/\mathbf{Z}$ (so we just have to apply a trigonometric formula if
we prefer the norm distance).
Recall the long exact sequences associated to
$\mathbf{Z}\to\mathbf{R}\to\mathbf{R}/\mathbf{Z}$ for $H^{n}$ and
$\operatorname{H}_{b}^{n}$. Together with the comparison maps, we get a
commutative diagram:

${\cdots}$${H^{1}(\Gamma,\mathbf{R}/\mathbf{Z})}$${\operatorname{H}_{b}^{2}(\Gamma,\mathbf{Z})}$${\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})}$${H^{2}(\Gamma,\mathbf{R}/\mathbf{Z})}$${\cdots}$${\cdots}$${H^{1}(\Gamma,\mathbf{R}/\mathbf{Z})}$${H^{2}(\Gamma,\mathbf{Z})}$${H^{2}(\Gamma,\mathbf{R})}$${H^{2}(\Gamma,\mathbf{R}/\mathbf{Z})}$${\cdots}$

Consider the subset $K\subseteq\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$
given by

$K=\mathrm{Ker}\Big{(}\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to
H^{2}(\Gamma,\mathbf{R}/\mathbf{Z})\Big{)}=\mathrm{Image}\Big{(}\operatorname{H}_{b}^{2}(\Gamma,\mathbf{Z})\to\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\Big{)}$

Recall that $\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ is a Banach space;
we can therefore consider the norm of elements in $K$.

######  Proposition 1.0.6.

The following are equivalent for a group $\Gamma$.

  1. 1.

(Lipschitz $U(1)$ stability): Every $\epsilon$-representation to $U(1)$ with
$\epsilon$ small enough is at distance less than $\epsilon$ from a
representation.

  2. 2.

(Linear $U(1)$ stability): There is a constant $c$ such that every
$\epsilon$-representation to $U(1)$ is at distance less than $c\epsilon$ from
a representation.

  3. 3.

($U(1)$ stability): For each $\delta>0$ there is $\epsilon>0$ so that every
$\epsilon$-representation to $U(1)$ is at distance less than $\delta$ from a
representation.

  4. 4.

($U(1)$ $1/3$-stability): There is $\delta<1/3$ such that every
$\epsilon$-representation to $U(1)$ with $\epsilon$ small enough is at
distance less than $\delta$ from a representation.

  5. 5.

(Cohomology gap): Non-zero elements of $K$ have norm bounded below by a
positive constant.

######  Remark 1.0.7.

The kernel of the comparison map
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to H^{2}(\Gamma,\mathbf{R})$ is
contained in $K$, see above diagram. Since this kernel is a vector space,
point (5) fails as soon as this kernel is non-zero. This explains why
quasimorphisms imply non-stability: it is a special case of the above as
quasimorphisms live in $K$.

###### Proof.

Trivially (1)$\Rightarrow$(2)$\Rightarrow$(3)$\Rightarrow$(4).

We prove (4)$\Rightarrow$(5) for the positive constant $\epsilon$ as in (4)
which, without loss of generality, can be made to satisfy
$\epsilon<1-3\delta$. Consider a class $[\omega]$ in $K$ with
$\|[\omega]\|<\epsilon$. We can choose the representative
$\omega\in\ell^{\infty}(\Gamma^{2})$ such that $\|\omega\|_{\infty}<\epsilon$.
Since $[\omega]$ is in the kernel to $H^{2}(\Gamma,\mathbf{R}/\mathbf{Z})$,
there is $f\colon\Gamma\to\mathbf{R}$ such that $\omega\equiv df$ modulo
$\mathbf{Z}$. The map $\pi=\exp(2\pi if)$ is an $\epsilon$-representation.
Thus $\pi$ is at distance less than $\delta$ of an actual representation,
which means that there is $b\colon\Gamma\to[-\delta,\delta]$ with
$d(f+b)\equiv 0$ modulo $\mathbf{Z}$. This means that $\omega+db$ is integer-
valued. On the other hand, $\|db\|_{\infty}$ is at most $3\delta$. Thus, since
$\epsilon+3\delta<1$ we deduce that $\omega+db$ actually vanishes and thus
$[\omega]$ is zero in $K$.

We now prove (5)$\Rightarrow$(1). We show it for $0<\epsilon<1/4$ smaller than
the constant given by (5). For any $\pi\colon\Gamma\to U(1)$, choose
$f\colon\Gamma\to\mathbf{R}$ such that $\pi=\exp(2\pi if)$. If $\pi$ is an
$\epsilon$-representation, then there is
$\omega\colon\Gamma^{2}\to[-\epsilon,\epsilon]$ such that $\omega\equiv df$
modulo $\mathbf{Z}$. In particular, $d\omega\equiv 0$ modulo $\mathbf{Z}$,
i.e. $d\omega$ is integer-valued. Given that $\|d\omega\|_{\infty}\leq
4\epsilon$, we have in fact $d\omega=0$ and hence we obtain a class $[\omega]$
in $K$ of norm less than $\epsilon$. Therefore, we can assume that $[\omega]$
is trivial, which means $\omega=db$ for some $b\in\ell^{\infty}(\Gamma)$.

In fact the operator $d$ on $\ell^{\infty}(\Gamma)$ has norm one: this was
first observed by [Mit84, p. 468] in a special case; the short proof is given
in general in (the proof of) Corollary 2.7 in [MM85]. We can therefore choose
$b$ such that $\|b\|_{\infty}\leq\epsilon$. Now $\exp(2\pi i(f-b))$ is a
representation at distance less than $\epsilon$ of $\pi$. ∎

######  Theorem 1.0.8.

Let $\Gamma$ be a group such that
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ is finite-dimensional and
injects into $H^{2}(\Gamma,\mathbf{R})$. Then $\Gamma$ is uniformly
$U(1)$-stable.

######  Lemma 1.0.9.

Let $A$ be an abelian group and let $B$ be a subgroup of
$Hom_{\mathbf{Z}}(A,\mathbf{Z})$. If the image of $B$ in
$Hom_{\mathbf{Z}}(A,\mathbf{R})$ spans a subspace of finite
$\mathbf{R}$-dimension $d$, then $B$ is free abelian of rank $d$.

###### Proof of Lemma 1.0.9.

We can consider $B$ as a subgroup spanning a space of finite
$\mathbf{Q}$-dimension $d$ in $Hom_{\mathbf{Z}}(A,\mathbf{Q})$. Viewing $A$ as
a quotient of a free abelian group on some set $X$, the group
$Hom_{\mathbf{Z}}(A,\mathbf{Z})$ is contained in $\mathbf{Z}^{X}$. While
$\mathbf{Z}^{X}$ is not free abelian in general, in Specker (Satz I p. 133 in
[Spe50]) it is proved that countable subgroups of $\mathbf{Z}^{X}$ are free
abelian . To be more precise, Specker proved it for $X$ countable but his
proof works in general; alternatively, the statement immediately reduces to
the case $X$ countable by taking a subset of $X$ separating the points of $B$.
Our $B$, being a subgroup of a finite dimensional $\mathbf{Q}$-space, is
countable and hence free (from the above mentioned result from [Spe50]).
Spanning a $d$-dimensional $\mathbf{Q}$-vector space, its rank must be also
$d$. ∎

###### Proof of Theorem 1.0.8.

It suffices to show that $\Gamma$ satisfies the cohomology gap — i.e.,
condition (5) of Proposition 1.0.6. Let $\tilde{B}$ denote the image of
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{Z})$ in $H^{2}(\Gamma,\mathbf{Z})$.
Thus the image of $\tilde{B}$ in $H^{2}(\Gamma,\mathbf{R})$ is precisely the
image of $K$.

${\operatorname{H}_{b}^{2}(\Gamma,\mathbf{Z})}$${K\subseteq\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})}$${\tilde{B}\subseteq
H^{2}(\Gamma,\mathbf{Z})}$${H^{2}(\Gamma,\mathbf{R})}$

Let $d<\infty$ be the dimension of the space spanned by $K$ in
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$. To prove the cohomology gap
(5), we first claim that $K$ is free abelian of rank $d$. This claim implies
that $K$ is discrete in the finite dimensional space
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ since it spans a space of
dimension $d$ (we use here the comparison with the standard lattice
$\mathbf{Z}^{d}$ in $\mathbf{R}^{d}$ and the fact that linear maps are
continuous in finite dimensions). Then, discreteness implies the desired gap.
To prove the claim that $K$ is free abelian of rank $d$, we can work with
$H^{2}(\Gamma,\mathbf{R})$ since $\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$
injects there. The universal coefficient theorem gives the following
commutative diagram with exact rows.

${0}$${Ext_{\mathbf{Z}}^{1}(H_{1}(\Gamma,\mathbf{Z}),\mathbf{Z})}$${H^{2}(\Gamma,\mathbf{Z})}$${Hom_{\mathbf{Z}}(H_{2}(\Gamma,\mathbf{Z}),\mathbf{Z})}$${0}$${0}$${H^{2}(\Gamma,\mathbf{R})}$${Hom_{\mathbf{Z}}(H_{2}(\Gamma,\mathbf{Z}),\mathbf{R})}$${0}$

Therefore it suffices to show that the image $B$ of $\tilde{B}$ in
$Hom_{\mathbf{Z}}(H_{2}(\Gamma,\mathbf{Z}),\mathbf{Z})$ is free abelian of
rank $d$, which follows from Lemma 1.0.9 applied to
$A=H_{2}(\Gamma,\mathbf{Z})$. ∎

Since finitely presented groups have finite-dimensional $H^{2}$, we obtain the
following clean necessary and sufficient condition:

######  Corollary 1.0.10.

Let $\Gamma$ be a finitely presented group. Then $\Gamma$ is uniformly
$U(1)$-stable if and only if every quasi-morphism of $\Gamma$ is at bounded
distance from a homomorphism.

###### Proof.

As explained above if $QM(\Gamma)\neq 0$ (i.e., not every quasi-morphism of
$\Gamma$ is at bounded distance from a homomorphism) then $\Gamma$ is not
$U(1)$-stable. On the other hand, if $QM(\Gamma)=0$, then
$Ker(\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to
H^{2}(\Gamma,\mathbf{R}))=0$, i.e.,
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ injects into
$H^{2}(\Gamma,\mathbf{R})$. If in addition $\Gamma$ is finitely presented
$H^{2}(\Gamma,\mathbf{R})$ is of finite dimension. Thus all the assumptions of
Theorem 1.0.8 are satisfied and $\Gamma$ is uniformly $U(1)$-stable. ∎

Now, back to the case in which $\Gamma$ is a lattice in a higher rank group.
For such $\Gamma$, Burger and Monod [BM99] showed that
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ injects into
$H^{2}(\Gamma,\mathbf{R})$. Such $\Gamma$ is usually finitely presented (and
then we can apply the last corollary) except when $\Gamma$ is a lattice in
(rank 2) semisimple Lie group of positive characteristic. But in this case
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ is anyway zero, by another
result of Burger and Monod [BM99], so Theorem 1.0.8 applies and we can deduce:

######  Theorem 1.0.11.

If $\Gamma$ is a higher rank lattice then it is uniformly $U(1)$-stable.

The main result of this paper will be a far reaching extension of the above
theorem (with some additional assumptions on the lattice $\Gamma$) to a larger
family of metric groups (namely any unitary group $U(n)$ equipped with any
submultiplicative matrix norm $\|\cdot\|$). In this general case, bounded
cohomology theory would not suffice, so we will motivate and build a more
appropriate cohomology theory in the upcoming sections.

We conclude this section with an observation in the case of groups of
Hermitian type:

######  Proposition 1.0.12.

Let $G$ be a split Lie group of Hermitian type (for example,
$Sp(2m,\mathbf{R})$), with universal central extension $\tilde{G}$. Let
$\Gamma$ be a cocompact lattice in $G$, and $\tilde{\Gamma}$ be its preimage
in $\tilde{G}$. Then $\tilde{\Gamma}$ is not uniformly $U(1)$-stable (and
hence, not uniformly $\mathfrak{U}$-stable).

###### Proof.

Consider the non-trivial $2$-cocycle $\alpha\in H^{2}(\Gamma,\mathbf{Z})$
corresponding to the central extension $\tilde{\Gamma}$ of $\Gamma$. From
[GW78], we know that $\alpha$ is actually an element of
$\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})$ that does not vanish in
$H^{2}(\Gamma,\mathbf{R})$. In particular, $\alpha$ is not contained in the
kernel of the comparison map $c:\operatorname{H}_{b}^{2}(\Gamma,\mathbf{R})\to
H^{2}(\Gamma,\mathbf{R})$.
Since the extension $\tilde{\Gamma}$ of $\Gamma$ is central (in particular,
has amenable kernel), $\alpha$ has a pullback
$\tilde{\alpha}\in\operatorname{H}_{b}^{2}(\tilde{\Gamma},\mathbf{R})$ which
is a non-trivial bounded $2$-cocycle. However, $\tilde{\alpha}$ is trivial in
cohomology, hence $\tilde{\alpha}$ is an element of the kernel of the
comparison map $c:\operatorname{H}_{b}^{2}(\tilde{\Gamma},\mathbf{R})\to
H^{2}(\tilde{\Gamma},\mathbf{R})$. Thus, from Propositions Proposition 1.0.3
and Proposition 1.0.4, we conclude that $\tilde{\Gamma}$ is not uniformly
$U(1)$-stable. ∎

######  Remark 1.0.13.

A non-trivial quasimorphism in the above proof of Proposition 1.0.12 can be
explicitly described: let $j:\Gamma\to\tilde{\Gamma}$ be a section
corresponding to the cocycle $\alpha\in H^{2}(\Gamma,\mathbf{Z})$ so that as a
set, $\tilde{\Gamma}=\mathbf{Z}\times j(\Gamma)$. Consider the map
$\phi:\tilde{\Gamma}\to\mathbf{R}$ defined to be $\phi(m,j(\gamma))\coloneqq
m$. One can check that this is a non-trivial quasimorphism of
$\tilde{\Gamma}$, and note that this map is trivial on $\Gamma$ and we do
know, from Theorem 1.0.11, that $\Gamma$ is indeed uniformly $U(1)$-stable if
it has rank at least $2$, even if $\tilde{\Gamma}$ is not.

##  2 Preliminaries and Basic Constructions

In this section, we establish the connection between uniform stability and a
homomorphism lifting problem, which is then further explored in §3 for
discrete groups, and in §4 for topological groups.
In the first §2.1, we define our central notion of uniform stability, and
describe it using sequences of maps. This then allows for a reformulation of
the notion of stability as a homomorphism lifting problem using the language
of ultrafilters in §2.2. Finally, in §2.3, we introduce the idea of defect
diminishing, which is a relaxation of the lifting problem with abelian
kernels. This property can naturally be related to a cohomological problem
that is then studied in the subsequent §3.

###  2.1 Uniform Stability and Asymptotic Homomorphisms

Let $\Gamma$ be a countable discrete group, and let $(G,d_{G})$ be a metric
group (that is, a group $G$ equipped with a bi-invariant metric $d_{G}$). We
use the metric to define the (uniform) distance between maps from $\Gamma$ to
$G$ as follows: for $f_{1},f_{2}:\Gamma\to G$, the distance between $f_{1}$
and $f_{2}$, denoted $dist_{\Gamma,G}(f_{1},f_{2})$, as

$dist_{\Gamma,G}(f_{1},f_{2})\coloneq\sup_{x\in\Gamma}d_{G}\left(f_{1}(x),f_{2}(x)\right)$

This allows us to define the distance of a function $f:\Gamma\to G$ from a
homomorphism as follows:

######  Definition 2.1.1.

The homomorphism distance of a function $f:\Gamma\to G$, denoted
$D_{\Gamma,G}(f)$, is defined as

$D_{\Gamma,G}(f)\coloneq\inf\\{dist_{\Gamma,G}(f,\psi):\psi\in
Hom(\Gamma,G)\\}$

The function $f$ is said to be $\delta$-close to a homomorphism if
$D_{\Gamma,G}(f)\leq\delta$.

There is another invariant of a function $f$ that also quantifies its distance
from being a homomorphism.

######  Definition 2.1.2.

For a function $f:\Gamma\to G$, we define its (uniform) defect
$def_{\Gamma,G}(f)$ as

$def_{\Gamma,G}(f)\coloneq\sup_{x,y\in\Gamma}d_{G}\left(f(xy),f(x)f(y)\right)$

The function $f$ is said to be an $\epsilon$-homomorphism if
$def_{\Gamma,G}(f)\leq\epsilon$.

Note that a priori both $def_{\Gamma,G}(f)$ and $D_{\Gamma,G}(f)$ could be
$\infty$. It is easy to show (by the triangle inequality) that
$def_{\Gamma,G}(f)\leq 3D_{\Gamma,G}(f)$ for any function $f$, so if $f$ is
close to a homomorphism, then it has small defect. Uniform stability is the
question of whether the converse is true: is any function with small defect
necessarily close to a homomorphism?
Uniform stability is usually studied with respect to not just one metric group
$(G,d_{G})$ but a family $\mathcal{G}$ of metric groups. In this case, we can
define

$F_{\Gamma,\mathcal{G}}:[0,\infty]\to[0,\infty]$
$F_{\Gamma,\mathcal{G}}(\epsilon)\coloneq\sup_{G\in\mathcal{G}}\sup_{f}\\{D_{\Gamma,G}(f):def_{\Gamma,G}(f)\leq\epsilon\\}$

######  Definition 2.1.3.

The group $\Gamma$ is said to be uniformly $\mathcal{G}$-stable if

$\lim\limits_{\epsilon\to 0^{+}}F_{\Gamma,\mathcal{G}}(\epsilon)=0$

A further refinement of the above definition involves a quanitification of the
above convergence:

######  Definition 2.1.4.

The group $\Gamma$ is uniformly $\mathcal{G}$-stable _with a linear estimate_
if $\exists\epsilon_{0}>0$ and $\exists M\geq 0$ such that
$\forall\epsilon<\epsilon_{0}$ and $\forall G\in\mathcal{G}$, every
$\epsilon$-homomorphism $\phi:\Gamma\to G$ is $M\epsilon$-close to a
homomorphism.

It is helpful to rephrase these notions also in terms of sequences of maps,
especially since we shall further refine this view in §2.2 and §2.3. Consider
a sequence of functions $\\{\phi_{n}:\Gamma\to G_{n}\\}_{n\in\mathbf{N}}$
where $G_{n}\in\mathcal{G}$ for every $n\in\mathbf{N}$.

  * •

A sequence $\\{\phi_{n}:\Gamma\to G_{n}\\}_{n\in\mathbf{N}}$ is said to be a
(uniform) asymptotic homomorphism of $\Gamma$ to $\mathcal{G}$ if
$\lim\limits_{n\to\infty}def_{\Gamma,G_{n}}(\phi_{n})=0$.

  * •

A sequence $\\{\phi_{n}:\Gamma\to G_{n}\\}_{n\in\mathbf{N}}$ is said to be
(uniformly) asymptotically close to a homomorphism if
$\lim\limits_{n\to\infty}D_{\Gamma,G_{n}}(\phi_{n})=0$.

It is easy to see that a group $\Gamma$ is uniformly $\mathcal{G}$-stable iff
every asymptotic homomorphism of $\Gamma$ to $\mathcal{G}$ is asymptotically
close to a homomorphism.
Uniform $\mathcal{G}$-stability with a linear estimate too can be rephrased in
terms of sequences of maps. Recall the Laundau big-O notation: for sequences
$\\{x_{n}\\}_{n\in\mathbf{N}}$ and $\\{y_{n}\\}_{n\in\mathbf{N}}$ of positive
real numbers, we denote $x_{n}=O(y_{n})$ if there exists a constant $C\geq 0$
and $N\in\mathbf{N}$ such that for all $n\geq N$, $x_{n}\leq Cy_{n}$. We
denote by $x_{n}=o(y_{n})$ if there exists a sequence
$\\{\epsilon_{n}\\}_{n\in\mathbf{N}}$ with $\lim_{n\to\infty}\epsilon_{n}=0$
and such that $x_{n}=\epsilon_{n}y_{n}$. Firstly, note that if $\Gamma$ is
uniformly $\mathcal{G}$-stable with a linear estimate, then for any asymptotic
homomorphism $\\{\phi_{n}:\Gamma\to G_{n}\\}_{n\in\mathbf{N}}$ of $\Gamma$ to
$\mathcal{G}$,
$D_{\Gamma,G_{n}}(\phi_{n})=O\left(def_{\Gamma,G_{n}}(\phi_{n})\right)$. The
following lemma shows that the converse is also true:

######  Lemma 2.1.5.

The group $\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate
iff for every asymptotic homomorphism $\\{\phi_{n}:\Gamma\to
G_{n}\\}_{n\in\mathbf{N}}$ of $\Gamma$ to $\mathcal{G}$,

$D_{\Gamma,G_{n}}(\phi_{n})=O\left(def_{\Gamma,G_{n}}(\phi_{n})\right)$

###### Proof.

Suppose $\Gamma$ is not uniformly $\mathcal{G}$-stable with a linear estimate.
Then for any $M>0$ and any $\epsilon>0$, there exists a map $\phi:\Gamma\to G$
(for some $G\in\mathcal{G}$) such that $def_{\Gamma,G}(\phi)\leq\epsilon$ and
$D_{\Gamma,G}(\phi)>M\epsilon$. Now consider a sequence
$\\{M_{n}\\}_{n\in\mathbf{N}}$ with $\lim_{n\to\infty}M_{n}=\infty$ and a
sequence $\\{\epsilon_{n}\\}_{n\in\mathbf{N}}$ with
$\lim_{n\to\infty}\epsilon_{n}=0$. For each $n\in\mathbf{N}$, let
$\phi_{n}:\Gamma\to G_{n}$ be the map with
$def_{\Gamma,G_{n}}(\phi_{n})\leq\epsilon_{n}$ but
$D_{\Gamma,G_{n}}(\phi_{n})>M_{n}\epsilon_{n}$. Then
$\\{\phi_{n}\\}_{n\in\mathbf{N}}$ is an asymptotic homomorphism of $\Gamma$ to
$\mathcal{G}$ such that $D_{\Gamma,G_{n}}(\phi_{n})$ is clearly not
$O\left(def_{\Gamma,G_{n}}(\phi_{n})\right)$. ∎

We shall now conclude this section by informally introducing the following
relaxation of the hypothesis of Lemma 2.1.5, which we shall call _asymptotic
defect diminishing_. The idea is to look for improvements to the asymptotic
homomorphism, rather than a true homomorphism.

######  Definition 2.1.6.

The group $\Gamma$ is said to have the asymptotic defect diminishing property
with respect to the family $\mathcal{G}$ if for any asymptotic homomorphism
$\\{\phi_{n}:\Gamma\to G_{n}\\}_{n\in\mathbf{N}}$, there exists an asymptotic
homomorphism $\\{\psi_{n}:\Gamma\to G_{n}\\}_{n\in\mathbf{N}}$ such that

  * •

The defect
$def_{\Gamma,G_{n}}(\psi_{n})=o\left(def_{\Gamma,G_{n}}(\phi_{n})\right)$.

  * •

The distance
$dist_{\Gamma,G_{n}}(\phi_{n},\psi_{n})=O\left(def_{\Gamma,G_{n}}(\phi_{n})\right)$.

The following results motivate this notion, which we shall formally study in
more detail in §2.2 in an ultraproduct setting.

######  Lemma 2.1.7.

Suppose $\Gamma$ has the asymptotic defect diminishing property with respect
to $\mathcal{G}$. Then there exists $\epsilon_{0}>0$ and $M>0$ such that for
$\epsilon<\epsilon_{0}$ and any $\epsilon$-homomorphism $\phi:\Gamma\to G$,
there exists a map $\psi:\Gamma\to G$ with defect
$def_{\Gamma,G}(\psi)<\frac{1}{2}def_{\Gamma,G}(\phi)$ and
$dist_{\Gamma,G}(\psi)<Mdef_{\Gamma,G}(\phi)$.

###### Proof.

We shall prove this by contradiction, so suppose for every $\epsilon>0$ and
every $M>0$, there exists an $\epsilon$-homomorphism $\phi:\Gamma\to G$ such
that for any $\psi:\Gamma\to G$, either
$def_{\Gamma,G}(\psi)\geq\frac{1}{2}def_{\Gamma,G}(\phi)$ or
$dist_{\Gamma,G}(\psi)\geq Mdef_{\Gamma,G_{n}}(\phi_{n})$.
Consider a sequence $\\{M_{n}\\}_{n\in\mathbf{N}}$ with
$\lim_{n\to\infty}M_{n}=\infty$ and a sequence
$\\{\epsilon_{n}\\}_{n\in\mathbf{N}}$ with $\lim_{n\to\infty}\epsilon_{n}=0$.
For each $n\in\mathbf{N}$, let $\phi_{n}:\Gamma\to G_{n}$ (with
$G_{n}\in\mathcal{G}$) be a map with
$def_{\Gamma,G_{n}}(\phi_{n})\leq\epsilon_{n}$ such that for any map
$\psi:\Gamma\to G_{n}$, either
$def_{\Gamma,G_{n}}(\psi_{n})\geq\frac{1}{2}def_{\Gamma,G_{n}}(\phi_{n})$ or
$dist_{\Gamma,G_{n}}(\psi_{n})\geq M_{n}def_{\Gamma,G_{n}}(\phi_{n})$. Then
the asymptotic homomorphism $\\{\phi_{n}\\}_{n\in\mathbf{N}}$ as constructed
proves that that $\Gamma$ does not have the asymptotic defect diminishing
property with respect to $\mathcal{G}$. ∎

######  Theorem 2.1.8.

Suppose $\mathcal{G}$ is such that every $G\in\mathcal{G}$ is a complete
metric space (with respect to its metric $d_{G}$). Then group $\Gamma$ is
uniformly $\mathcal{G}$-stable with a linear estimate iff $\Gamma$ has the
asymptotic defect diminishing property with respect to $\mathcal{G}$

###### Proof.

Suppose $\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate,
then the implication is immediate. Conversely, suppose $\Gamma$ has the
asymptotic defect diminishing property with respect to $\mathcal{G}$. From the
previous lemma, this means that there exists $\epsilon_{0}>0$ and $M>0$ such
that for any $\epsilon$-homomorphism $\phi:\Gamma\to G$ (with
$\epsilon<\epsilon_{0}$), there exists a map $\psi:\Gamma\to G$ with defect
$def_{\Gamma,G}(\psi)<\frac{1}{2}def_{\Gamma,G}(\phi)$ and
$dist_{\Gamma,G}(\psi)<Mdef_{\Gamma,G}(\phi)$. Set $\phi^{(0)}\coloneq\phi$
and $\phi^{(1)}\coloneq\psi$. Applying Lemma 2.1.7 inductively on $\phi^{(i)}$
to get $\phi^{(i+1)}$, we obtain a sequence of maps $\\{\phi^{(j)}:\Gamma\to
G\\}_{j\in\mathbf{N}}$ where for each $j\in\mathbf{N}$, $\phi^{(j)}$ has
defect at most $\epsilon/2^{j}$ and is of distance at most $M\epsilon/2^{j}$
from $\phi^{(j-1)}$. This gives us a Cauchy sequence of maps from $\Gamma\to
G$. Since $G$ is a complete, the sequence $\\{\phi^{(j)}$ has a limit
$\phi^{\infty}:\Gamma\to G$ with defect $def(\phi^{\infty})=0$ (hence it is a
homomorphism). As for its distance from $\phi$,

$dist_{\Gamma,G}(\phi,\phi^{\infty})\leq\sum\limits_{j=0}^{\infty}\frac{M\epsilon}{2^{j}}=2M\epsilon$

Hence $\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate. ∎

###  2.2 Ultraproducts and Internal Maps

We can quantify the asymptotic rates succintly using ultraproducts. We shall
now briefly review some concepts from the theory of ultraproducts and non-
standard analysis that would be of relevance to our constructions (for more
details, refer [Gol12] and [ACH12]).
Let $\mathcal{U}$ be a non-principal ultrafilter on $\mathbf{N}$, which is
fixed throughout. A subset $S\subseteq\mathbf{N}$ is said to be _large_ if
$S\in\mathcal{U}$.

######  Definition 2.2.1.

The (algebraic) ultraproduct $\prod_{\mathcal{U}}X_{n}$ (or alternately,
$\\{X_{n}\\}_{\mathcal{U}}$) of an indexed collection
$\\{X_{n}\\}_{n\in\mathbf{N}}$ of sets is defined to be

$\prod_{\mathcal{U}}X_{n}\coloneq\prod_{n\in\mathbf{N}}X_{n}/\sim$

where for
$\\{x_{n}\\}_{n\in\mathbf{N}},\\{y_{n}\\}_{n\in\mathbf{N}}\in\prod_{n\in\mathbf{N}}X_{n}$,
$\\{x_{n}\\}_{n\in\mathbf{N}}\sim\\{y_{n}\\}_{n\in\mathbf{N}}$ if
$\\{n:x_{n}=y_{n}\\}\in\mathcal{U}$.

In other words, we identify two sequences
$\\{x_{n}\\}_{n\in\mathbf{N}},\\{y_{n}\\}_{n\in\mathbf{N}}\in\prod_{n\in\mathbf{N}}X_{n}$
if they agree on a large set of indices. The image of a sequence
$\\{x_{n}\\}_{n\in\mathbf{N}}\in\prod_{n\in\mathbf{N}}X_{n}$ under this
equivalence relation shall be denoted $\\{x_{n}\\}_{\mathcal{U}}$. Conversely,
given an element of $\prod_{\mathcal{U}}X_{n}$, we shall always regard it as
$\\{x_{n}\\}_{\mathcal{U}}$ for some sequence
$\\{x_{n}\\}_{n\in\mathbf{N}}\in\prod_{n\in\mathbf{N}}X_{n}$.
If $X_{n}=X$ for every $n\in\mathbf{N}$, then $\prod_{\mathcal{U}}X$ is called
the (algebraic) _ultrapower_ of $X$, denoted ${}^{*}X$. Note that $X$ can be
embedded in ${}^{*}X$ via a diagonal embedding (for $x\in X$,
$x\mapsto\\{x\\}_{\mathcal{U}}\in{}^{*}X$).
Ultraproducts can be made to inherit algebraic structures of their building
blocks. More precisely, let $\\{X_{n}\\}_{n\in\mathbf{N}}$,
$\\{Y_{n}\\}_{n\in\mathbf{N}}$, and $\\{Z_{n}\\}_{n\in\mathbf{N}}$ be indexed
families of sets with operations $*_{n}:X_{n}\times Y_{n}\to Z_{n}$ for every
$n\in\mathbf{N}$. This naturally defines an operation
$*:\prod_{\mathcal{U}}X_{n}\times\prod_{\mathcal{U}}Y_{n}\to\prod_{\mathcal{U}}Z_{n}$
by

$\\{x_{n}\\}_{\mathcal{U}}*\\{y_{n}\\}_{\mathcal{U}}=\\{x_{n}*_{n}y_{n}\\}_{\mathcal{U}}$

We shall frequently encounter the following examples of ultraproducts:

  * •

The ultrapower ${}^{*}G$ of a group $G$ is itself a group. This can be seen by
noting that ${}^{*}G$ is the quotient of the direct product group
$\prod_{n\in\mathbf{N}}G$ by the normal subgroup comprising elements
$\\{g_{n}\\}_{n\in\mathbf{N}}$ with $\\{n:g_{n}=1\\}\in\mathcal{U}$.

  * •

The ultrapower ${}^{*}\mathbf{R}$ of $\mathbf{R}$ is a non-archimedean ordered
field called the _hyperreals_.

  * •

Let $\\{W_{n}\\}_{n\in\mathbf{N}}$ be a family of Banach spaces. Then
$\prod_{\mathcal{U}}W_{n}$ can be given the structure of a
${}^{*}\mathbf{R}$-vector space. In fact, it also comes equipped with a
${}^{*}\mathbf{R}$-valued norm.

One of the standard tools of non-standard analysis is the transfer principle
which relates the truth of statements concerning objects and their
counterparts in the ultraproduct universe. Intuitively, our standard universe
comprises all objects under normal consideration like $\mathbf{R}$,
$\mathbf{C}$, etc. but maybe formally modeled as follows: define

$V_{0}(\mathbf{R})=\mathbf{R}$
$V_{n+1}(\mathbf{R})=V_{n}(\mathbf{R})\cup\mathcal{P}\left(V_{n}(\mathbf{R})\right)$

where $\mathcal{P}\left(V_{n}(\mathbf{R})\right)$ denotes the power set of
$\left(V_{n}(\mathbf{R})\right)$. Then

$V(\mathbf{R})=\cup_{n\geq 0}V_{n}(\mathbf{R})$

is called the superstructure over $\mathbf{R}$, and can be interpreted as
comprising all the natural structures we study in mathematics. This shall
comprise our standard universe $Univ$.
We can construct a mapping ${}^{*}:V(\mathbf{R})\to V({}^{*}\mathbf{R})$ that
takes an object in the superstructure of $\mathbf{R}$ (our standard universe
$Univ$) to an object in the superstructure of ${}^{*}\mathbf{R}$ satisfying
the following:

  * •

(Extension Principle) The mapping $*$ maps $\mathbf{R}$ to ${}^{*}\mathbf{R}$.

  * •

(Transfer Principle) For any first-order formula $\phi$ involving $k$
variables, and $A_{1},\dots,A_{k}\in V(\mathbf{R})$, the statement
$\phi(A_{1},\dots,A_{k})$ is true in $V(\mathbf{R})$ iff
$\phi({}^{*}A_{1},\dots,{}^{*}A_{k})$ is true in $V({}^{*}\mathbf{R})$.

  * •

(Countable Saturation) Suppose $\\{X_{n}\\}_{n\in\mathbf{N}}$ is a collection
of sets in ${}^{*}\left(V(\mathbf{R})\right)$ such that the intersection of
any finite subcollection is non-empty. Then $\cap X_{n}$ is non-empty.

It is a basic result of non-standard analysis that such a mapping $*$ exists,
and can be constructed using a non-principal ultrafilter on $\mathbf{N}$. The
image of this mapping shall be referred to as our non-standard
universe${}^{*}Univ$, which is

${}^{*}Univ=\\{\\{X_{n}\\}_{\mathcal{U}}\lvert X_{n}\in Univ\\}$

comprising ultraproducts of elements of $Univ$. Note that
${}^{*}\mathbf{R},{}^{*}\mathbf{C},{}^{*}\Gamma$ are all contained in
${}^{*}Univ$.
Objects contained in $Univ$ are called standard, while objects contained in
${}^{*}Univ$ are called internal. In particular, a subset $S$ of an
ultraproduct $\prod_{\mathcal{U}}X_{n}$ is an _internal subset_ if there exist
subsets $S_{n}\subseteq X_{n}$ for every $n\in\mathbf{N}$ such that
$S=\prod_{\mathcal{U}}S_{n}$. A function
$f:\prod_{\mathcal{U}}X_{n}\to\prod_{\mathcal{U}}Y_{n}$ is said to be an
internal function if there exists a sequence $\\{f_{n}:X_{n}\to
Y_{n}\\}_{n\in\mathbf{N}}$ such that $f=\\{f_{n}\\}_{\mathcal{U}}$.
Objects that are not contained in ${}^{*}Univ$ are called _external_. Note
that standard objects like $\mathbf{R}$ and $\mathbf{C}$ too are external. We
can always consider objects that are neither in $Univ$ nor in ${}^{*}Univ$.
Two important examples of non-standard external subsets are

  * •

The set of bounded hyperreals, denoted ${}^{*}\mathbf{R}_{b}$, is the subset
comprising elements $\\{x_{n}\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}$ for which
there exists $C\in\mathbf{R}_{\geq 0}$ such that $\lvert x_{n}\rvert\leq C$
for every $n\in\mathbf{N}$.

  * •

The set of infinitesimal hyperreals, denoted ${}^{*}\mathbf{R}_{inf}$, is the
subset comprising elements $\\{x_{n}\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}$ such
that for _every_ real $\epsilon>0$, there exists a large set $S\in\mathcal{U}$
such that $\lvert x_{n}\rvert<\epsilon$ for every $n\in S$.

  * •

For $x,y\in{}^{*}\mathbf{R}$, denote by $x=O_{\mathcal{U}}(y)$ if
$x/y\in{}^{*}\mathbf{R}_{b}$, and by $x=o_{\mathcal{U}}(y)$ if
$x/y\in{}^{*}\mathbf{R}_{inf}$ (in particular,any bounded element
$x\in{}^{*}\mathbf{R}_{b}$ is $x=O_{\mathcal{U}}(1)$ while any infinitesimal
element $\epsilon\in{}^{*}\mathbf{R}_{inf}$ is $\epsilon=o_{\mathcal{U}}(1)$).

Note that the preimage of ${}^{*}\mathbf{R}_{b}$ under the map
$\prod_{n\in\mathbf{N}}\mathbf{R}\to{}^{*}\mathbf{R}$ includes the subset of
all bounded sequences, while the preimage of ${}^{*}\mathbf{R}_{inf}$ includes
the subset of infinitesimal sequences (that is, sequences that converge to
$0$).
The subset ${}^{*}\mathbf{R}_{b}$ forms a valuation ring with
${}^{*}\mathbf{R}_{inf}$ being the unique maximal ideal, with quotient
${}^{*}\mathbf{R}_{b}/{}^{*}\mathbf{R}_{inf}\cong\mathbf{R}$. The quotient map
$st:{}^{*}\mathbf{R}_{b}\to\mathbf{R}$ is known as the _standard part_ map or
_limit along the ultrafilter $\mathcal{U}$_.
The previous construction can also be replicated for metric spaces with
specified base points. Let $\\{(X_{n},d_{n},p_{n})\\}_{n\in\mathbf{N}}$ be an
indexed family of metric spaces (where the space $X_{n}$ comes with the metric
$d_{n}$ and the base point $p_{n}$). The ultraproduct
$\prod_{\mathcal{U}}X_{n}$ comes equipped with an ${}^{*}\mathbf{R}$-values
metric $\prod_{\mathcal{U}}d_{n}$ and base point $\prod_{\mathcal{U}}p_{n}$.
Consider the subset, denoted $\left(\prod_{\mathcal{U}}X_{n}\right)_{b}$, of
$\prod_{\mathcal{U}}X_{n}$ comprising $\\{x_{n}\\}_{\mathcal{U}}$ such that
$\\{d_{n}(x_{n},p_{n})\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{b}$ (such elements
are referred to as bounded or _admissible_), and a subset, denoted
$\left(\prod_{\mathcal{U}}X_{n}\right)_{inf}$, comprising
$\\{x_{n}\\}_{\mathcal{U}}$ such that
$\\{d_{n}(x_{n},p_{n})\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}$. Define an
equivalence relation $\sim$ on $\left(\prod_{\mathcal{U}}X_{n}\right)_{b}$ by
$\\{x_{n}\\}_{\mathcal{U}}\sim\\{y_{n}\\}_{\mathcal{U}}$ if
$\\{d_{n}(x_{n},y_{n})\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}$. The set of
equivalence classes $\left(\prod_{\mathcal{U}}X_{n}\right)_{b}/\sim$ is called
the _ultralimit_ of $\\{(X_{n},d_{n},p_{n})\\}_{n\in\mathbf{N}}$. We will
return to this notion in the context of Banach spaces in §4.

######  Remark 2.2.2.

For convenience, we shall henceforth denote by for $n\in\mathcal{U}$ to mean
for $n\in S$ for some $S\in\mathcal{U}$.

Returning to our setting, consider a sequence $\\{\phi_{n}:\Gamma\to
G_{n}\\}_{n\in\mathbf{N}}$ where $G_{n}\in\mathcal{G}$. This can be used to
construct an internal map
$\\{\phi_{n}\\}_{\mathcal{U}}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, and
allows us to redefine asymptotic homomorphisms and closeness to homomorphisms
in the ultraproduct.

  * •

Given two internal maps $\phi^{(1)}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$
and $\phi^{(2)}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, we denote by
$dist(\phi^{(1)},\phi^{(2)})\coloneq\\{dist_{\Gamma,G_{n}}(\phi^{(1)}_{n},\phi^{(2)}_{n})\\}_{\mathcal{U}}$.
The maps $\phi^{(1)}$ and $\phi^{(2)}$ are said to be (internally)
asymptotically close to each other if
$dist(\phi^{(1)},\phi^{(2)})\in{}^{*}\mathbf{R}_{inf}$.

  * •

An internal map $\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ is said to be
an internal homomorphism if there exists a sequence
$\\{\psi_{n}\\}_{n\in\mathbf{N}}$ of homomorphisms such that
$\psi=\\{\psi_{n}\\}_{\mathcal{U}}$.

  * •

An internal map
$\phi\coloneq\\{\phi_{n}\\}_{\mathcal{U}}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$
is said to be (internally) asymptotically close to an internal homomorphism if
there exists an internal homomorphism $\psi=\\{\psi_{n}:\Gamma\to
G_{n}\\}_{n\in\mathbf{N}}$ such that
$dist(\phi,\psi)\in{}^{*}\mathbf{R}_{inf}$ (in other words,
$\\{D_{\Gamma,G_{n}}\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}$).

  * •

An internal map
$\phi\coloneq\\{\phi_{n}\\}_{\mathcal{U}}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$
is called an internal asymptotic homomorphism if
$def(\phi)\coloneq\\{def(\phi_{n})\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}$.
For $\epsilon\in{}^{*}\mathbf{R}_{inf}$, we shall call an internal asymptotic
homomorphism $\phi$ an internal $\epsilon$-homomorphism if
$def(\phi)=\epsilon$ (we similarly define internal
$O_{\mathcal{U}}(\epsilon)$-homomorphisms and internal
$o_{\mathcal{U}}(\epsilon)$-homomorphisms).

The following lemmas are variants of Lemma 2.1.5 and Theorem 2.1.8 in the
ultraproduct setting:

######  Lemma 2.2.3.

The group $\Gamma$ is uniformly $\mathcal{G}$-stable iff every internal
asymptotic homomorphism $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ for
$G_{n}\in\mathcal{G}$ is asymptotically close to an internal homomorphism.

###### Proof.

We shall prove both directions by contradiction.
Let
$\phi=\\{\phi_{n}\\}_{\mathcal{U}}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ be
an internal asymptotic homomorphism that is _not_ asymptotically close to any
internal homomorphism. This means that
$\\{def(\phi_{n})\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}$, but there exists
some real $c>0$ and $S\in\mathcal{U}$ such that
$D_{\Gamma,G_{n}}(\phi_{n})\geq c$ for $n\in S$. In particular, for this $c$
and any $\epsilon>0$, there exists a map $\phi_{n}:\Gamma\to G_{n}$ with
defect $def(\phi_{n})\leq\epsilon$ and $D_{\Gamma,G_{n}}\geq c$, thus proving
that $\Gamma$ is _not_ uniformly $\mathcal{G}$-stable.
Conversely, suppose $\Gamma$ is _not_ uniformly $\mathcal{G}$-stable. This
means, by definition, that there exists $\delta>0$ such that for every
$\epsilon>0$, there exists $G\in\mathcal{G}$ and $\phi:\Gamma\to G$ such that
$def(\phi)\leq\epsilon$ and $dist_{\Gamma,G}(\phi)>\delta$. Let us fix this
$\delta$, and consider a sequence $\\{\epsilon_{n}\\}_{n\in\mathbf{N}}$ with
$\lim_{n\to\infty}\epsilon_{n}=0$. For each such $\epsilon_{n}$, there exists
$G_{n}\in\mathcal{G}$ and $\phi_{n}:\Gamma\to G_{n}$ with
$def(\phi_{n})\leq\epsilon_{n}$ and $dist_{\Gamma,G_{n}}(\phi_{n})>\delta$.
Consider the internal asymptotic homomorphism $\\{\phi_{n}\\}_{\mathcal{U}}$.
Clearly it is not asymptotically close to any internal homomorphism. ∎

######  Lemma 2.2.4.

The group $\Gamma$ is $\mathcal{G}$-uniformly stable with a linear estimate
iff for every internal asymptotic homomorphism
$\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, there exists an internal
homomorphism $\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ with
$dist(\phi,\psi)=O_{\mathcal{U}}\left(def(\phi)\right)$.

###### Proof.

(The proof is similar to Lemma 2.1.5) Suppose $\Gamma$ is not uniformly
$\mathcal{G}$-stable with a linear estimate. Consider a sequence
$\\{M_{n}\\}_{n\in\mathbf{N}}$ with $\lim_{n\to\infty}M_{n}=\infty$ and a
sequence $\\{\epsilon_{n}\\}_{n\in\mathbf{N}}$ with
$\lim_{n\to\infty}\epsilon_{n}=0$. For each $n\in\mathbf{N}$, let
$\phi_{n}:\Gamma\to G_{n}$ be the map with
$def_{\Gamma,G_{n}}(\phi_{n})\leq\epsilon_{n}$ but
$D_{\Gamma,G_{n}}(\phi_{n})>M_{n}\epsilon_{n}$. Then the ultraproduct
$\phi=\\{\phi_{n}\\}_{\mathcal{U}}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ is
an internal asymptotic homomorphism with
$def(\phi)=\epsilon\coloneq\\{\epsilon_{n}\\}_{\mathcal{U}}$ such that for any
internal homomorphism $\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$,
$dist(\phi,\psi)/def(\phi)$ is not in ${}^{*}\mathbf{R}_{b}$.
Conversely, suppose $\Gamma$ is $\mathcal{G}$-uniformly stable with a linear
estimate, and let $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ be an
internal asymptotic homomorphism. There exists $\epsilon_{0}>0$, $M>0$ and a
large subset $S_{\epsilon_{0}}\in\mathcal{U}$ such that for every $n\in
S_{\epsilon_{0}}$, $\phi_{n}:\Gamma\to G_{n}$ has defect
$def(\phi_{n})\leq\epsilon_{0}$, and $D_{\Gamma,G_{n}}(\phi_{n})\leq
Mdef(\phi_{n})$, allowing us to construct an internal homomorphism
$\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ that is asymptotically close to
$\phi$. ∎

We now reformulate the notion of defect diminishing in this ultraproduct
setting:

######  Definition 2.2.5.

The group $\Gamma$ is said to have the defect diminishing property with
respect to the family $\mathcal{G}$ if for any internal asymptotic
homomorphism $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, there exists an
asymptotic homomorphism $\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ such
that

  * •

The defect $def(\psi)=o_{\mathcal{U}}\left(def(\phi)\right)$.

  * •

The distance $dist(\phi,\psi)=O_{\mathcal{U}}\left(def(\phi)\right)$.

The following lemma reformulates Theorem 2.1.8 in the ultraproduct setting,
and the proof is on the same lines.

######  Theorem 2.2.6.

The group $\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate
iff $\Gamma$ has the defect diminishing property with respect to
$\mathcal{G}$.

###  2.3 Internal Liftings and Defect Diminishing

In this §, we shall reinterpret an internal asymptotic homomorphism as a
(true) homomorpism to a quotient group. This will allow us to describe defect
diminishing as a homomorphism lifting problem. In §3.1 we will restrict to a
family of metric groups of interest such that this homomorphism lifting
problem has an abelian kernel, allowing us to build a cohomological theory
capturing the obstruction to uniform stability.
Let $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ be an internal asymptotic
homomorphism. Consider the external subset
$\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$ defined as

$\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}\coloneq\Big{\\{}\\{g_{n}\\}_{\mathcal{U}}:\\{d_{n}(g_{n},1)\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}\Big{\\}}$

In other words, $\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$ comprises all
elements that are infinitesimally close to the identity in
$\prod_{\mathcal{U}}G_{n}$. It is easy to check that
$\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$ is not just a subset but a
normal subgroup of $\prod_{\mathcal{U}}G_{n}$. Since $\phi$ has defect
$def(\phi)\in{}^{*}\mathbf{R}_{inf}$, this means that for every
$x,y\in{}^{*}\Gamma$,
$\phi(xy)^{-1}\phi(x)\phi(y)\in\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$,
making
$\tilde{\phi}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
a homomorphism. Thus,

######  Lemma 2.3.1.

Given an internal asymptotic homomorphism
$\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, its composition, denoted
$\tilde{\phi}$, with the canonical quotient homomorphism
$\prod_{\mathcal{U}}G_{n}\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
is a homomorphism from ${}^{*}\Gamma$ to the group
$\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$.

Note that a homomorphism from ${}^{*}\Gamma$ to
$\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$ does
not necessarily arise as the composition of an internal map
${}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ and the quotient homomorphism
$\prod_{\mathcal{U}}G_{n}\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$.
However, we will be specifically interested in homomorphisms that arise this
way.

######  Definition 2.3.2.

Let
$F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
be a homomorphism. We say that $F$ has an internal lift
$\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ if $\hat{F}$ is internal,
and its composition with the canonical quotient homomorphism
$\prod_{\mathcal{U}}G_{n}\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
is $F$.
We say that $F$ has an internal lift homomorphism if there exists an internal
lift $\hat{F}$ of $F$ that is also a homomorphism from ${}^{*}\Gamma$ to
$\prod_{\mathcal{U}}G_{n}$.

Observe that for an internal asymptotic homomorphism
$\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, $\phi$ itself is an internal
lift of the homomorphism $\tilde{\phi}$. Conversely,

######  Lemma 2.3.3.

Suppose a homomorphism
$F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
has an internal lift $\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$. Then
$\hat{F}$ is an internal asymptotic homomorphism. Furthermore, suppose
$\hat{F}^{(1)}$ and $\hat{F}^{(2)}$ are two internal lifts of a homomorphism
$F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$,
then $\hat{F}^{(1)}$ and $\hat{F}^{(2)}$ are asymptotically close to each
other.

###### Proof.

The map $\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ , being internal, is
of the form $\hat{F}=\\{\hat{F}_{n}\\}_{\mathcal{U}}$ for
$\hat{F}_{n}:\Gamma\to G_{n}$ for every $n\in\mathbf{N}$. Since $F$ is a
homomorphism, for every
$x=\\{x_{n}\\}_{\mathcal{U}},y=\\{y_{n}\\}_{\mathcal{U}}\in{}^{*}\Gamma$,
$\\{\hat{F}_{n}(x_{n}y_{n})^{-1}\hat{F}_{n}(x_{n})\hat{F}_{n}(y_{n})\\}_{\mathcal{U}}\in\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
which means that
$\\{def(\hat{F}_{n})\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}$, making
$\hat{F}$ an internal asymptotic homomorphism.
Since both $\hat{F}^{(1)}$ and $\hat{F}^{(2)}$ are internal lifts of the
homomorphism
$F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$,
for every $x=\\{x_{n}\\}_{\mathcal{U}}\in{}^{*}\Gamma$,
$\\{d_{n}(\hat{F}^{(1)}_{n}(x_{n}),\hat{F}^{(2)}_{n}(x_{n}))\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{inf}$.
∎

This motivates the following equivalent condition for uniform
$\mathcal{G}$-stability in terms of internal lifts.

######  Lemma 2.3.4.

The group $\Gamma$ is uniformly $\mathcal{G}$-stable iff every homomorphism
$\tilde{\phi}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
that has an internal lift also has an internal lift homomorphism.

###### Proof.

Let $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ be an internal asymptotic
homomorphism, and
$\tilde{\phi}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
be the homomorphism obtained by composing $\phi$ with the quotient map
$\prod_{\mathcal{U}}G_{n}\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$.
Let $\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ be an internal lift
homomorphism of $\tilde{\phi}$. Then by the previous Lemma 2.3.3, $\phi$ is
asymptotically close to the internal homomorphism $\psi$. By Lemma 2.2.3, we
conclude that $\Gamma$ is uniformly $\mathcal{G}$-stable. The converse follows
by definition of uniform $\mathcal{G}$-stability. ∎

######  Remark 2.3.5.

The quotient group
$\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$ is
called the _metric ultraproduct_ of the sequence
$\\{G_{n}\\}_{n\in\mathbf{N}}$ of groups. For (pointwise) stability of groups,
it is shown in [AP15] that $\Gamma$ is (pointise) $\mathcal{G}$-stable if any
homomorphism
$\tilde{\phi}:\Gamma\to\prod_{\mathcal{U}}G_{n}/\left(\prod_{\mathcal{U}}G_{n}\right)_{inf}$
from $\Gamma$ to the metric ultraproduct, can be lifted to a homomorphism
$\psi:\Gamma\to\prod_{\mathcal{U}}G_{n}$. In our setting, the uniformity
requirement forces us to work with internal maps from the ultrapower
${}^{*}\Gamma$ as opposed to just maps from $\Gamma$.

We shall further refine the internal lifting property parametrizing it by the
precise defect. Let
$\phi=\\{\phi_{n}\\}_{\mathcal{U}}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ be
an internal asymptotic homomorphism with defect
$def(\phi)=\epsilon\in{}^{*}\mathbf{R}_{inf}$. Consider the subset
$B(\epsilon)$ (elements _bounded_ by $\epsilon$) of $\prod_{\mathcal{U}}G_{n}$
defined as follows:

$B(\epsilon)\coloneqq\\{\\{g_{n}\\}_{\mathcal{U}}\in\prod_{\mathcal{U}}G_{n}:\\{d_{n}(g_{n},1_{n})\\}_{\mathcal{U}}=O_{\mathcal{U}}(\epsilon)\\}$

Note that $B(\epsilon)$ is an externally defined subset. Since the metric on
$G_{n}$ is bi-invariant, the subset $B(\epsilon)$ is a normal subgroup of
$\prod_{\mathcal{U}}G_{n}$. Let
$q_{B(\epsilon)}:\prod_{\mathcal{U}}G_{n}\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$
be the canonical quotient homomorphism, and denote by $\tilde{\phi}$ the
composition map
$q_{B(\epsilon)}\cdot\phi:{}^{*}\Gamma\to\left(\prod_{\mathcal{U}}G_{n}\right)/B(\epsilon)$.
The following lemma is a parametrized reformulation of Lemma 2.3.3, and the
proof is on the same lines, which we omit here:

######  Lemma 2.3.6.

The map
$\tilde{\phi}:{}^{*}\Gamma\to\left(\prod_{\mathcal{U}}G_{n}\right)/B(\epsilon)$
is a group homomorphism.

Conversely, let $F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$. An
internal map $\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ is said to be
an internal lift of $\tilde{\phi}$ if $q_{B(\epsilon)}\cdot\hat{F}=F$. We
again have an analogue of Lemma 2.3.3 here too, whose proof is similar:

######  Lemma 2.3.7.

Suppose a homomorphism $F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$
has an internal lift $\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$. Then
$\hat{F}$ is an internal $O_{\mathcal{U}}(\epsilon)$-homomorphism. Futhermore,
if $\hat{F}^{(1)}$ and $\hat{F}^{(2)}$ are two such internal lifts of $F$,
then $\hat{F}^{(1)}$ and $\hat{F}^{(2)}$ are internally
$O_{\mathcal{U}}(\epsilon)$-close to each other.

######  Lemma 2.3.8.

The group $\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate
iff for every $\epsilon\in{}^{*}\mathbf{R}_{inf}$, every homomorphism
$\tilde{\phi}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$ that has an
internal lift also has an internal lift homomorphism.

###### Proof.

Suppose $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ is an internal
asymptotic homomorphism with defect $\epsilon\in{}^{*}\mathbf{R}_{inf}$. Then
$\tilde{\phi}::{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$ is a
homomorphism which has internal lift $\phi$. Thus, it also has an internal
lift homomorphism $\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ which is
internally $O_{\mathcal{U}}(\epsilon)$-close to $\phi$. Hence, by Lemma 2.3.4,
$\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate. The
converse is immediate. ∎

Thus, for an infinitesimal $\epsilon\in{}^{*}\mathbf{R}_{inf}$, if a
homomorphism
$\tilde{\phi}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$ that has an
internal lift, can be internally lifted to an internal homomorphism
$\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, then $\Gamma$ is uniformly
$\mathcal{G}$-stable with a linear estimate. We shall now try to obtain such a
lift by a sequence of intermediate lifts.
For $\epsilon\in{}^{*}\mathbf{R}_{inf}$, denote by $I(\epsilon)$ the subset of
$\prod_{\mathcal{U}}G_{n}$ (elements infinitesimal with respect to $\epsilon$)
defined as

$I(\epsilon)\coloneq\\{\\{g_{n}\\}_{\mathcal{U}}\in\prod_{\mathcal{U}}G_{n}:\\{d_{n}(g_{n},1_{n})\\}_{\mathcal{U}}=o_{\mathcal{U}}(\epsilon)\\}$

Note that by the bi-invariance of the metric, $I(\epsilon)\subseteq
B(\epsilon)$ is a normal subgroup of $\prod_{\mathcal{U}}G_{n}$. Let
$q_{I(\epsilon)}:\prod_{\mathcal{U}}G_{n}\to\prod_{\mathcal{U}}G_{n}/I(\epsilon)$
be the canonical quotient homomorphism. The following lemma is similar to
Lemma 2.3.3, and the proof too is on the same lines:

######  Lemma 2.3.9.

Suppose $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ is an internal
$o_{\mathcal{U}}(\epsilon)$-homomorphism, then
$q_{I(\epsilon)}\cdot\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/I(\epsilon)$
is a homomorphism. Conversely, suppose a homomorphism
$F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/I(\epsilon)$ has an internal lift
$\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$, then $\hat{F}$ is an
internal $o_{\mathcal{U}}(\epsilon)$-homomorphism, and any two internal lifts
of $F$ are internally $o_{\mathcal{U}}(\epsilon)$-close to one another.

We can now reformulate the defect diminishing property in terms of internal
lifts.

######  Lemma 2.3.10.

The group $\Gamma$ has the defect diminishing property with respect to
$\mathcal{G}$ iff for every $\epsilon\in{}^{*}\mathbf{R}_{inf}$ and every
homomorphism $F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$ that has
an internal lift, $F$ has an internal lift
$\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ such that
$q_{I(\epsilon)}\cdot\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/I(\epsilon)$
is a homomorphism.

###### Proof.

Suppose $\Gamma$ has the defect diminishing property, then it is immediate
from Definition 2.2.5 that for every $\epsilon\in{}^{*}\mathbf{R}_{inf}$ and
every homomorphism $F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$
that has an internal lift, $F$ has an internal lift
$\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ such that
$q_{I(\epsilon)}\cdot F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/I(\epsilon)$
is a homomorphism.
Conversely, consider an internal asymptotic homomorphism
$\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ with defect
$def(\phi)=\epsilon$, and the induced homomorphism
$\tilde{\phi}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$. By the
hypothesis of the lemma (and Lemma 2.3.9), there exists an internal
$o_{\mathcal{U}}(\epsilon)$-homomorphism
$\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ which is
$O_{\mathcal{U}}(\epsilon)$-close to $\phi$. This shows that $\Gamma$ has the
defect diminishing property with respect to $\mathcal{G}$. ∎

We now recap the results obtained so far:

######  Theorem 2.3.11.

Let $\Gamma$ be a discrete group, and $\mathcal{G}$ be a family of metric
groups such that for every group $G\in\mathcal{G}$, its metric $d_{G}$ is
complete. Then the following are equivalent:

  1. 1.

The group $\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate.

  2. 2.

The group $\Gamma$ has the defect diminishing property with respect to
$\mathcal{G}$.

  3. 3.

For every $\epsilon\in{}^{*}\mathbf{R}_{inf}$ and every homomorphism
$F:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/B(\epsilon)$ that has an internal
lift, $F$ has an internal lift
$\hat{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ such that
$q_{I(\epsilon)}\cdot\tilde{F}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}/I(\epsilon)$
is a homomorphism.

##  3 A Cohomological Interpretation of Stability

We concluded the previous section by noting that in order to prove that
$\Gamma$ is uniformly $\mathcal{G}$-stable with a linear estimate, it is
sufficient to show that it has the defect diminishing property with respect to
$\mathcal{G}$. Furthermore, we interpreted this property in terms of internal
lifts of homomorphisms to quotient groups.
Recall that a matrix norm $\|\cdot\|$ on $M_{n}(\mathbf{C})$ is said to be
submultiplicative if for every $A,B\in M_{n}(\mathbf{C})$,
$\|AB\|\leq\|A\|\cdot\|B\|$. From now on, we shall work exclusively with the
family of unitary groups, each equipped with a unitarily bi-invariant
_submultiplicative_ matrix norm, and shall denote this family by
$\mathfrak{U}$.

$\mathfrak{U}\coloneq\Big{\\{}\left(U(n),\|\cdot\|\right):n\in\mathbf{N}\text{
and }\|\cdot\|\text{ is submultiplicative}\Big{\\}}$

In particular, these include the $p$-Schatten norms given by

$\|A\|_{p}=\begin{cases}(Tr|A|^{p})^{1/p}&1\leq p<\infty\\\
\sup_{\|\nu\|=1}\|A\nu\|&p=\infty\end{cases}$

Note that for $p=\infty$, this is the operator norm (as studied in [Kaz82] and
[BOT13]), while for $p=2$, this is the _Frobenius_ norm or the (unnormalized)
_Hilbert-Schmidt_ norm (as studied in [DCGLT20] and [AD22]).

Before we proceed further, we state and sketch the proof of the following
useful transference lemma for uniform $\mathfrak{U}$-stability with a linear
estimate, which we shall use in §6. The proof is on the lines of a relative
version of [BOT13, Theorem 3.2] (which reproves Kazhdan’s result on the Ulam
stability of amenable groups), which we can adapt here in the simpler setting
of finite index.

######  Lemma 3.0.1.

Let $\Lambda\leq\Gamma$ be a subgroup of finite index. Then $\Gamma$ is
uniformly $\mathfrak{U}$-stable with a linear estimate iff $\Lambda$ is
uniformly $\mathfrak{U}$-stable with a linear estimate.

###### Sketch.

Suppose $\Gamma$ is uniformly $\mathfrak{U}$-stable with a linear estimate.
Then the proof of [BOT13, Corollary 2.7] (further explained in [Gam11, Lemma
II.22]) implies that $\Lambda$ too is uniformly $\mathfrak{U}$-stable with a
linear estimate.
Conversely, suppose $\Lambda$ is uniformly $\mathfrak{U}$-stable with a linear
estimate, and let $\phi:\Gamma\to U(n)$ be an $\epsilon$-homomorphism. Since
the finite index subgroup $\Lambda$ is uniformly $\mathfrak{U}$-stable with a
linear estimate, we can assume that the restriction of $\phi$ to $\Lambda$ is
a homomorphism, and furthermore, $\phi(g\delta)=\phi(g)\phi(\delta)$ for every
$g\in\Gamma$, $\delta\in\Lambda$. Now define $\phi^{\prime}:\Gamma\to M(n)$ as

$\phi^{\prime}(g)\coloneq\frac{1}{|\Gamma:\Lambda|}\sum_{x\in\Gamma/\Lambda}\phi(gx)\phi(x)^{*}$

Note that $\phi^{\prime}(g)$ is just the average over coset representatives in
$\Gamma/\Lambda$, and $M(n)$ is the space of $n\times n$ matrices. Just as in
the proof of [BOT13, Theorem 3.2], this $\phi^{\prime}$ can be normalized to
obtain $\phi_{1}:\Gamma\to U(n)$ such that $\phi_{1}$ has defect
$C\epsilon^{2}$ (for a universal constant $C$ not depending on $n$), and we
repeat the process to obtain a (true) homomorphism as the limit. ∎

######  Remark 3.0.2.

In fact, it is further shown in [FFR23, Proposition 1.5] that for a subgroup
$\Lambda\leq\Gamma$ that is _co-amenable_ in $\Gamma$, if $\Lambda$ is
uniformly $\mathfrak{U}$-stable with a linear estimate, then $\Gamma$ too is
uniformly $\mathfrak{U}$-stable with a linear estimate. This generalizes one
direction of Lemma 3.0.1, though the converse is not true in this level of
generality.

######  Remark 3.0.3.

In particular, if $\Gamma_{1}$ and $\Gamma_{2}$ are commensurable, then
$\Gamma_{1}$ is uniformly $\mathfrak{U}$-stable with a linear estimate iff
$\Gamma_{2}$ is uniformly $\mathfrak{U}$-stable with a linear estimate.

Given a sequence of unitary groups $\\{U(k_{n})\\}_{n\in\mathbf{N}}$, we
denote its ultraproduct by $\prod_{\mathcal{U}}U(k_{n})$,and given an element
$u=\\{u_{n}\\}_{\mathcal{U}}\in\prod_{\mathcal{U}}U(k_{n})$ (where for each
$n\in\mathbf{N}$, $u_{n}\in U(k_{n})$), we denote its distance from the
identity $1\in\prod_{\mathcal{U}}U(k_{n})$ by
$\|u-1\|\coloneq\\{\|u_{n}-I\|\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{b}$, for
notational convenience.
Note that for $\epsilon\in{}^{*}\mathbf{R}_{inf}$ and an ultraproduct
$\prod_{\mathcal{U}}U(k_{n})$, the subsets
$B(\epsilon)\subseteq\prod_{\mathcal{U}}G_{n}$ and $I(\epsilon)\subseteq
B(\epsilon)$ can now be written as

$B(\epsilon)=\Big{\\{}u\in\prod_{\mathcal{U}}U(k_{n}):\|u-I\|=O_{\mathcal{U}}(\epsilon)\Big{\\}}$
$I(\epsilon)=\Big{\\{}u\in\prod_{\mathcal{U}}U(k_{n}):\|u-I\|=o_{\mathcal{U}}(\epsilon)\Big{\\}}$

Furthermore, the submultiplicativity of the norms implies that:

######  Lemma 3.0.4.

The group $B(\epsilon)/I(\epsilon)$ is abelian.

###### Proof.

Let $a,b\in B(\epsilon)$, and consider the commutator $aba^{*}b^{*}$. We shall
prove that $aba^{*}b^{*}\in I(\epsilon)$. Observe that
$\|aba^{*}b^{*}-I\|=\|ab-ba\|$ since the norm is unitarily invariant.

$\|ab-ba\|=\|(a-I)(b-I)-(b-I)(a-I)\|\leq 2\|a-I\|\|b-I\|$

This is because of the submultiplicativity of the norm. Since $a,b\in
B(\epsilon)$, we conclude that
$\|aba^{*}b^{*}-I\|=O_{\mathcal{U}}(\epsilon^{2})=o_{\mathcal{U}}(\epsilon)$.
∎

The fact that $B(\epsilon)/I(\epsilon)$ is an abelian group will allow us to
rephrase the lifting property discussed in §2.3 in terms of the vanishing of a
cohomology which we shall develop in detail in §3.2 and §4.
In §3.1, we explicitly work out the cocycles corresponding to possible lifts
of a homomorphism, which are then transferred to the linearized setting in
§3.2 where a cohomological theory begins to reveal itself. Finally, in §3.3,
we demonstrate the notions discussed in the case of discrete abelian groups,
showing that the vanishing of our second cohomology implies uniform stability.

###  3.1 Lifting with an Abelian Kernel

Let $\phi:{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$ be an internal
$\epsilon$-homomorphism that induces a homomorphism
$\tilde{\phi}:{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})/B(\epsilon)$. Let
$\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$ be an internal lift of
$\tilde{\phi}$.
To an internal lift $\psi:{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$ of
$\tilde{\phi}$, we associate an internal map

$\rho_{\psi}:{}^{*}\Gamma\times\prod_{\mathcal{U}}U(k_{n})\to\prod_{\mathcal{U}}U(k_{n})$
$\rho_{\psi}(g)(u)\coloneq\psi(g)\cdot u\cdot\psi(g)^{-1}$ (3.1)

For every $g\in{}^{*}\Gamma$ and $u\in B(\epsilon)$, $\rho_{\psi}(g)(u)\in
B(\epsilon)$, while for $u\in I(\epsilon)$, $\rho_{\psi}(g)(u)\in
I(\epsilon)$.

######  Lemma 3.1.1.

The internal map
$\rho_{\psi}:{}^{*}\Gamma\times\prod_{\mathcal{U}}U(k_{n})\to\prod_{\mathcal{U}}U(k_{n})$
induces an action, denoted $\tilde{\rho}_{\psi}$, of ${}^{*}\Gamma$ on the
abelian group $B(\epsilon)/I(\epsilon)$.

###### Proof.

For $g_{1},g_{2}\in{}^{*}\Gamma$ and $u\in B(\epsilon)$, we want to show that
$\rho_{\psi}(g_{1})(\rho_{\psi}(g_{2})(u))-\rho_{\psi}(g_{1}g_{2})(u)\in
I(\epsilon)$. Note that for every $g_{1},g_{2}\in{}^{*}\Gamma$,
$\psi(g_{1}g_{2})^{-1}\psi(g_{1})\psi(g_{2})\in B(\epsilon)$, so

$\|\psi(g_{1})\psi(g_{2})u\psi(g_{2})^{-1}\psi(g_{1})^{-1}-\psi(g_{1}g_{2})u\psi(g_{1}g_{2})^{-1}\|=\|\psi(g_{1}g_{2})^{-1}\psi(g_{1})\psi(g_{2})u-u\psi(g_{1}g_{2})^{-1}\psi(g_{1})\psi(g_{2})\|$

Since the elements $\psi(g_{1}g_{2})^{-1}\psi(g_{1})\psi(g_{2})$ and $u$ are
in $B(\epsilon)$, their commutator is in $I(\epsilon)$ (as in proof of Lemma
3.0.4). ∎

We shall call $\tilde{\rho}_{\psi}$ the _action induced from $\psi$_. Observe
that we have defined $\rho_{\psi}$ using a given internal lift $\psi$.
However, this induced action of ${}^{*}\Gamma$ on $B(\epsilon)/I(\epsilon)$ is
independent of the choice of internal lift of $\tilde{\phi}$.

######  Lemma 3.1.2.

For two internal lifts $\psi_{1}$ and $\psi_{2}$ of $\tilde{\psi}$,
$\tilde{\rho}_{\psi_{1}}=\tilde{\rho}_{\psi_{2}}$.

###### Proof.

Note that for $g\in{}^{*}\Gamma$, $\psi_{2}(g)^{-1}\psi_{1}(g)\in B(\epsilon)$
since they are both internal lifts of $\tilde{\phi}$. Hence for $u\in
B(\epsilon)$, again as in the proof of Lemma 3.0.4,
$\psi_{2}(g)^{-1}\psi_{1}(g)u-u\psi_{2}(g)^{-1}\psi_{1}(g)\in I(\epsilon)$,
which implies that
$\psi_{1}(g)u\psi_{1}(g)^{-1}-\psi_{2}(g)u\psi_{2}(g)^{-1}\in I(\epsilon)$. ∎

So while the internal map
$\rho_{\psi}:{}^{*}\Gamma\times\prod_{\mathcal{U}}U(k_{n})\to\prod_{\mathcal{U}}U(k_{n})$
depends on $\psi$, the induced action $\tilde{\rho}_{\psi}$ is independent of
the choice of internal lift of $\tilde{\phi}$. Hence we can denote this action
by $\tilde{\rho}_{\phi}$.
Define the internal map

$\alpha_{\psi}:{}^{*}\Gamma\times{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$
$\alpha_{\psi}(g_{1},g_{2})\coloneq\psi(g_{1})\psi(g_{2})\psi(g_{1}g_{2})^{-1}$

Note that by construction $\alpha_{\psi}$ takes values in $B(\epsilon)$ (and
our goal is to find some lift $\psi$ for which $\alpha_{\psi}$ takes values
only in $I(\epsilon)$). Observe that

######  Lemma 3.1.3.

For $g_{1},g_{2},g_{3}\in{}^{*}\Gamma$,

$\alpha_{\psi}(g_{1},g_{2})\cdot\alpha_{\psi}(g_{1}g_{2},g_{3})\cdot\alpha_{\psi}(g_{1},g_{2}g_{3})^{-1}=\psi(g_{1})\cdot\alpha_{\psi}(g_{2},g_{3})\psi(g_{1})^{-1}$

Let $q_{B/I}:B(\epsilon)\to B(\epsilon)/I(\epsilon)$ be the canonical quotient
homomorphism. Since $\alpha_{\psi}$ takes values in $B(\epsilon)$, let
$\tilde{\alpha}_{\psi}\coloneq
q_{B/I}\cdot\alpha_{\psi}:{}^{*}\Gamma\times{}^{*}\Gamma\to
B(\epsilon)/I(\epsilon)$. Since $B(\epsilon)/I(\epsilon)$ is abelian, Lemma
3.1.3 implies the following corollary:

######  Corollary 3.1.4.

For $g_{1},g_{2},g_{3}\in{}^{*}\Gamma$,

$\tilde{\rho}_{\phi}(g_{1})\cdot\tilde{\alpha}_{\psi}(g_{2},g_{3})-\tilde{\alpha}_{\psi}(g_{1}g_{2},g_{3})+\tilde{\alpha}_{\psi}(g_{1},g_{2}g_{3})-\tilde{\alpha}_{\psi}(g_{1},g_{2})=0$

While we noted that the action $\tilde{\rho}_{\psi}$ of ${}^{*}\Gamma$ on
$B(\epsilon)/I(\epsilon)$ does not depend on the choice of lift $\psi$, the
map $\tilde{\alpha}_{\psi}:{}^{*}\Gamma\times{}^{*}\Gamma\to
B(\epsilon)/I(\epsilon)$ does depend on the choice of lift $\psi$. Our hope is
to prove the existence of some choice of lift $\psi$ such that
$\tilde{\alpha}_{\psi}$ is trivial.
Consider $\tilde{\alpha}_{\psi_{1}}$ and $\tilde{\alpha}_{\psi_{2}}$ for two
different internal lifts $\psi_{1}$ and $\psi_{2}$ of $\tilde{\phi}$. Define
an internal map

$\beta_{\psi_{1},\psi_{2}}:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$
$\beta_{\psi_{1},\psi_{2}}(g)\coloneq\psi_{2}(g)\psi_{1}(g)^{-1}$

Since $\beta_{\psi_{1},\psi_{2}}$ takes values in $B(\epsilon)$, denote by
$\tilde{\beta}_{\psi_{1},\psi_{2}}:{}^{*}\Gamma\to B(\epsilon)/I(\epsilon)$
its composition with the quotient map $B(\epsilon)\to
B(\epsilon)/I(\epsilon)$. Then for $g_{1},g_{2}\in{}^{*}\Gamma$, a careful
computation shows that

$\tilde{\alpha}_{\psi_{2}}-\tilde{\alpha}_{\psi_{1}}=\tilde{\beta}_{\psi_{1},\psi_{2}}(g_{1})+\tilde{\rho}_{\phi}(g_{1})\cdot\tilde{\beta}_{\psi_{1},\psi_{2}}(g_{2})-\tilde{\beta}_{\psi_{1},\psi_{2}}(g_{1}g_{2})$
(3.2)

Suppose there exists an internal map
$\beta:{}^{*}\Gamma\to\prod_{\mathcal{U}}G_{n}$ that takes values in
$B(\epsilon)$ such that for the lift $\psi$, the following equation holds for
all $g_{1},g_{2}\in{}^{*}\Gamma$:

$\tilde{\alpha}_{\psi}(g_{1},g_{2})=\tilde{\beta}(g_{1})+\tilde{\rho}_{\phi}(g_{1})\cdot\tilde{\beta}(g_{2})-\tilde{\beta}(g_{1}g_{2})$

Then the internal map

$\psi^{\sim}:{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$
$\psi^{\sim}(g)\coloneq\psi(g)\beta(g)^{-1}$

is also an internal lift of $\tilde{\phi}$ such that for every
$g_{1},g_{2}\in{}^{*}\Gamma$,

$\tilde{\alpha}_{\psi^{\sim}}(g_{1},g_{2})=0$

In particular, this means that $\psi^{\sim}$ is the internal lift that we
want.
The above discussion hints at a cohomological theory that captures the
obstruction to such lifts. The idea is as follows: any candidate internal lift
$\psi$ of $\tilde{\phi}$, with defect $O_{\mathcal{U}}(\epsilon)$, gives us a
type of $2$-cocycle of ${}^{*}\Gamma$ with coefficients in
$B(\epsilon)/I(\epsilon)$, and if that cocycle happens to be a
$1$-couboundary, then the lift $\psi$ can be corrected to obtain another lift
that has defect $o_{\mathcal{U}}(\epsilon)$ and is still
$O_{\mathcal{U}}(\epsilon)$-close to $\psi$ (and $\phi$), thus implying the
defect diminishing property that we want.

######  Remark 3.1.5.

In [DCGLT20], it is shown (using the idea mentioned in Remark 2.3.5 and defect
diminishing) that $\Gamma$ is (pointwise) stable (with respect to unitary
matrices equipped with the Frobenius norm) if
$H^{2}\left(\Gamma,B(\epsilon)/I(\epsilon)\right)$ vanishes. From Lemma 3.1.3,
it might be tempting to simply consider $\tilde{\alpha}_{\psi}$ as a bounded
$2$-cocycle for the group ${}^{*}\Gamma$ with coefficients in the abelian
group $B(\epsilon)/I(\epsilon)$, and interpret Eq. 3.2 (and the ensuing
discussion) as insisting that $\tilde{\alpha}_{\psi}$ is the coboundary of a
bounded $1$-cochain of ${}^{*}\Gamma$ in $B(\epsilon)/I(\epsilon)$. But we
cannot simply work with
$\operatorname{H}_{b}^{2}\left({}^{*}\Gamma,B(\epsilon)/I(\epsilon)\right)$,
since we need to ensure that the bounded $2$-cocycle $\tilde{\alpha}_{\psi}$
(which was induced from an _internal_ map $\alpha_{\psi}$) is the coboundary
of a bounded $1$-cochain $\tilde{\beta}:{}^{*}\Gamma\to
B(\epsilon)/I(\epsilon)$ that is itself also induced from an _internal_ map
$\beta:{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$. This insistence on our
maps being induced from internal maps is essential in our setting of uniform
stability, and leads to the definition of an internal and asymptotic bounded
cohomology machinery that we construct in §4.

###  3.2 Linearization and the Lie Algebra

In the previous section we observed that the defect diminishing property could
be interpreted as a cohomological problem based on an action of ${}^{*}\Gamma$
on the abelian group $B(\epsilon)/I_{\epsilon}$. However, as pointed out in
Remark 3.1.5 the subtlety here involves the requirement that we deal only with
maps that are induced from some _internal_ mapping to
$\prod_{\mathcal{U}}U(k_{n})$. At that level we do not have the abelianness
that would allow us to properly formulate a cohomology theory. In this §, we
transfer to the Lie algebra allowing us to work with spaces of maps from the
group to Banach spaces.
For a matrix $A\in M_{n}(\mathbf{C})$, consider the matrix logarithm given by

$\log A\coloneq\sum\limits_{j=1}^{\infty}(-1)^{j-1}\frac{(A-I)^{j}}{j}$

The series above converges if $\|A-I\|<1$ (for some submultiplicative matrix
norm $\|\cdot\|$). By subadditivity and submultiplicativity of the norm
$\|\cdot\|$, if $\epsilon\leq 1/2$ and $\|u-I\|\leq\epsilon$,

$\|\log{u}\|\leq\sum\limits_{j=1}^{\infty}\|u-I\|^{j}\leq 2\epsilon$

######  Lemma 3.2.1.

For every $\epsilon<1/2$, $n\in\mathbf{N}$, $u\in U(n)$ and every
submultiplicative norm $\|\cdot\|$ on $M_{n}(\mathbf{C})$, if
$\|u-I\|\leq\epsilon$, then $\|\log{u}\|\leq 2\epsilon$.

It is a classical result that for a unitary matrix $u\in U(n)$, its logarithm
$\log{u}$ (whenever it is defined) is an _anti-Hermitian_ matrix. Denote by
$\mathfrak{u}(n)$ the (real) vector space of anti-Hermitian matrices in
$M_{n}(\mathbf{C})$.
In the other direction, we have the matrix exponential map defined as

$\exp(A)=\sum\limits_{j=0}^{\infty}\frac{A^{j}}{j!}$

which is well-defined for every $A\in M_{n}(\mathbf{C})$. For an anti-
hermitian matrix $W\in\mathfrak{u}(n)$ of the form $W=U\circ
diag(i\theta_{j})_{j=1}^{n}\circ U^{*}$ for $U\in U(n)$, its exponential
$exp(W)=U\circ diag(e^{i\theta_{j}})_{j=1}^{n}\circ U^{*}$. The following
trivial bound is sufficient for our purposes:

######  Lemma 3.2.2.

For every $\epsilon>0$, $n\in\mathbf{N}$ and a submultiplicative norm
$\|\cdot\|$ on $M_{n}(\mathbf{C})$, for a matrix $W\in\mathfrak{u}(n)$ with
$\|W\|<\epsilon$, $\|exp(W)-I\|\leq e^{\epsilon}-1$.

Note that since $\mathfrak{u}(n)$ is finite-dimensional, it is complete for
any norm, making it a finite-dimensional real Banach space. It comes with an
isometric (adjoint) action of $U(n)$ given as follows: for
$v\in\mathfrak{u}(n)$ and $U\in U(n)$, $Ad(U)(v)\coloneq
UvU^{*}\in\mathfrak{u}(n)$.
Consider the family of $\mathbf{R}$-vector spaces of anti-hermitian matrices
$\mathfrak{u}(n)$ each equipped with a submultiplicative norm.

$\Big{\\{}\left(\mathfrak{u}(n),\|\cdot\|\right):n\in\mathbf{N}\text{ and
}\|\cdot\|\text{ is submultiplicative}\Big{\\}}$

For an infinitesimal $\epsilon\in{}^{*}\mathbf{R}_{inf}$, define the internal
map

${}_{\epsilon}\log:\prod_{\mathcal{U}}U(k_{n})\to\prod_{\mathcal{U}}\mathfrak{u}(k_{n})$
${}_{\epsilon}\log{u}\coloneq\frac{1}{\epsilon}\\{\log{u_{k_{n}}}\\}_{\mathcal{U}}$

and similarly, the internal map
$\exp:\prod_{\mathcal{U}}\mathfrak{u}(k_{n})\to\prod_{\mathcal{U}}U(k_{n})$
given by

${}_{\epsilon}\exp{u}\coloneq\\{\exp{\epsilon_{k_{n}}u_{k_{n}}}\\}_{\mathcal{U}}$

Let us denote the ultraproduct $\prod_{\mathcal{U}}\mathfrak{u}(k_{n})$ by
$\mathcal{W}$ from now on. Note that $\mathcal{W}$ comes with a
${}^{*}\mathbf{R}$-valued norm, which we shall denote simply as $\|\cdot\|$,
obtained as the ultraproduct of the respective norms of each
$\mathfrak{u}(k_{n})$. The bounded elements of $\mathcal{W}$ shall be denoted
$\mathcal{W}_{b}$ while the infinitesimal elements of $\mathcal{W}$ are
denoted by $\mathcal{W}_{inf}$. That is,

$\mathcal{W}_{b}\coloneq\Big{\\{}w\in\mathcal{W}:\|w\|\in{}^{*}\mathbf{R}_{b}\Big{\\}}$
(3.3)
$\mathcal{W}_{inf}\coloneq\Big{\\{}w\in\mathcal{W}:\|w\|\in{}^{*}\mathbf{R}_{inf}\Big{\\}}$
(3.4)

The motivation behind scaling the definitions of $\log$ and $\exp$ by
$1/\epsilon$ and $\epsilon$ respectively is as follows:

######  Proposition 3.2.3.

The internal map ${}_{\epsilon}\log:\prod_{\mathcal{U}}U(k_{n})\to\mathcal{W}$
when restricted to $B(\epsilon)$, takes values in $\mathcal{W}_{b}$, and
elements in $I(\epsilon)$ are taken to $\mathcal{W}_{inf}$. This induces an
isomorphism of the abelian groups $\mathcal{W}_{b}/\mathcal{W}_{inf}$ and
$B(\epsilon)/I(\epsilon)$.

###### Proof.

It follows from Lemma 3.2.1 that for $u\in B(\epsilon)$,
${}_{\epsilon}\log{u}\in\mathcal{W}_{b}$, and for $u\in I(\epsilon)$,
${}_{\epsilon}\log{u}\in\mathcal{W}_{inf}$. The map is surjective on
$\mathcal{W}_{b}$ since the map ${}_{\epsilon}\log{(_{\epsilon}\exp{v})}=v$
for $v\in\mathcal{W}_{b}$ (and similarly for $\mathcal{W}_{inf}$ as well).
From properties of the logarithm map, it follows that for $u_{1},u_{2}\in
B(\epsilon)$,${}_{\epsilon}\log{u_{1}u_{2}}-(_{\epsilon}\log{u_{1}}+_{\epsilon}\log{u_{2}})\in\mathcal{W}_{inf}$.
Hence ${}_{\epsilon}\log$ induces a surjective group homomorphism from
$B(\epsilon)$ to $\mathcal{W}_{b}/\mathcal{W}_{inf}$ with kernel
$I(\epsilon)$. ∎

The ultralimit $\mathcal{W}_{b}/\mathcal{W}_{inf}$ shall be denoted
$\tilde{\mathcal{W}}$. The above lemma tells us that $\tilde{\mathcal{W}}\cong
B(\epsilon)/I(\epsilon)$. In fact, $\tilde{\mathcal{W}}\cong
B(\epsilon)/I(\epsilon)$ is not just an abelian group but has the structure of
a real Banach space. It is an example of a construction known as a _Banach
space ultralimit_. We shall not prove this result here, but refer to [Hei80]
and [ACH12] for more details:

######  Proposition 3.2.4 ([Hei80]).

The space $\tilde{W}=B(\epsilon)/I(\epsilon)$ is a real Banach space.

Recall that we had defined (Eq. 3.1) an internal map
$\rho_{\psi}:{}^{*}\Gamma\times\prod_{\mathcal{U}}U(k_{n})\to\prod_{\mathcal{U}}U(k_{n})$
defined as $\rho_{\psi}(g)v=\psi(g)v\psi(g)^{-1}$ which had induced an action
$\tilde{\rho}_{\phi}$ of ${}^{*}\Gamma$ on $B(\epsilon)/I(\epsilon)$. We can
similarly define an internal map

$\pi_{\psi}:{}^{*}\Gamma\times\mathcal{W}\to\mathcal{W}$

through the internal adjoint action of $\prod_{\mathcal{U}}U(k_{n})$ on
$\mathcal{W}$ (that is, conjugation),

$\pi_{\psi}v\coloneqq\psi(g)v\psi(g)^{-1}$ (3.5)

Again, by the submultiplicativity of the norms, for $v\in\mathcal{W}_{b}$ and
$g_{1},g_{2}\in{}^{*}\Gamma$,

$\pi_{\psi}(g_{1}g_{2})v-\pi_{\psi}(g_{1})\pi_{\psi}(g_{2})v\in\mathcal{W}_{inf}$

Thus, the internal map $\pi_{\psi}$ as defined above induces an action of
${}^{*}\Gamma$ on $\mathcal{W}_{b}/\mathcal{W}_{inf}$. Unless there is
ambiguity, we shall denote the induced action of $g\in{}^{*}\Gamma$ on
$\tilde{v}\in\mathcal{W}_{b}/\mathcal{W}_{inf}$ through $\pi_{\psi}$ by
$g\cdot\tilde{v}$.

######  Lemma 3.2.5.

The internal map ${}_{\epsilon}\log:\prod_{\mathcal{U}}U(k_{n})\to\mathcal{W}$
induces a ${}^{*}\Gamma$-equivariant (additive) group isomorphism between
$B(\epsilon)/I(\epsilon)$ (with the action induced from $\rho_{\psi}$) and
$\tilde{\mathcal{W}}$ (with the action induced from $\pi_{\psi}$).

###### Proof.

We already saw that ${}_{\epsilon}\log$ induces a group isomorphism between
$B(\epsilon)/I(\epsilon)$ and $\mathcal{W}_{b}/\mathcal{W}_{inf}$. Let
$g\in{}^{*}\Gamma$ and $u\in B(\epsilon)$. Then
$\rho_{\psi}(g)u=\psi(g)u\psi(g)^{-1}$, which means that
${}_{\epsilon}\log(\rho_{\psi}(g)u)=\psi(g)_{\epsilon}\log{u}\psi(g)^{-1}$
since the matrix logarithm is invariant with respect to conjugation by a
unitary matrix. ∎

In particular, $\tilde{\mathcal{W}}$ is a real Banach space with an isometric
action of ${}^{*}\Gamma$ (it is a real Banach ${}^{*}\Gamma$-module).

######  Remark 3.2.6.

The fact that $\mathcal{W}=B(\epsilon)/I(\epsilon)$ is a real Banach
${}^{*}\Gamma$-module is useful in reducing (pointwise) stability of $\Gamma$
with respect to the family $\mathfrak{U}$ to showing that
$H^{2}(\Gamma,\mathcal{W})=0$, and this line of study is pursued in [DCGLT20]
and [LO20]. However, in our setting of uniform stability, the Banach structure
of $\mathcal{W}$ is not as directly relevant.

Corresponding to the internal map
$\alpha_{\psi}:{}^{*}\Gamma\times{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$,
define the internal map

$\alpha:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$
$\alpha(g_{1},g_{2})\coloneqq_{\epsilon}\log{\alpha_{\psi}(g_{1},g_{2})}$
(3.6)

Since
$\alpha_{\psi}:{}^{*}\Gamma\times{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$
takes values only in $B(\epsilon)$, it is clear that
$\alpha:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$ takes values only in
$\mathcal{W}_{b}$. We shall denote by
$\tilde{\alpha}:{}^{*}\Gamma\times{}^{*}\Gamma\to\tilde{\mathcal{W}}$ the map
induced obtained by composing $\alpha$ with the canonical quotient map
$\mathcal{W}_{b}\to\tilde{\mathcal{W}}$.

######  Lemma 3.2.7.

The map $\alpha:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$ satisfies the
following condition: for any $g_{1},g_{2},g_{3}\in{}^{*}\Gamma$,

$\pi_{\psi}(g_{1})\alpha(g_{2},g_{3})-\alpha(g_{1}g_{2},g_{3})+\alpha(g_{1},g_{2}g_{3})-\alpha(g_{1},g_{2})\in\mathcal{W}_{inf}$

###### Proof.

Recall that $\alpha_{\psi}$ satisfies the following property: for
$g_{1},g_{2},g_{3}\in{}^{*}\Gamma$,

$\alpha_{\psi}(g_{1},g_{2})\alpha_{\psi}(g_{1}g_{2},g_{3})\alpha_{\psi}(g_{1},g_{2}g_{3})^{-1}=\rho_{\psi}(g_{1})\alpha_{\psi}(g_{2},g_{3})$

The conclusion then follows from the fact that the map
${}_{\epsilon}\log:\prod_{\mathcal{U}}U(k_{n})\to\mathcal{W}$ induces a
${}^{*}\Gamma$-equivariant group homomorphism between
$B(\epsilon)/I(\epsilon)$ and $\mathcal{W}_{b}/\mathcal{W}_{inf}$. ∎

Thus, the induced map
$\tilde{\alpha}:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$ satisifes the
$2$-cocycle condition given by: for $g_{1},g_{2},g_{3}\in{}^{*}\Gamma$,

$g_{1}\cdot\tilde{\alpha}(g_{2},g_{3})-\tilde{\alpha}(g_{1}g_{2},g_{3})+\tilde{\alpha}(g_{1},g_{2}g_{3})-\tilde{\alpha}(g_{1},g_{2})=0$
(3.7)

So transfering to the internal Lie algebra through the ${}_{\epsilon}\log$
map, we thus have a map $\alpha:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$
which takes values in $\mathcal{W}_{b}$ and satisfies the 2-cocycle condition
modulo $\mathcal{W}_{inf}$. Recall that the defect diminishing condition was
implied by the following statement: suppose
$\alpha_{\psi}:{}^{*}\Gamma\times{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$
is an internal function taking values in $B(\epsilon)$ and such that
$\tilde{\alpha}_{\phi}$ satisfies the $2$-cocycle condition. Then there exists
an internal $\beta:{}^{*}\Gamma\to\prod_{\mathcal{U}}U(k_{n})$ taking values
in $B(\epsilon)$ such that
$\tilde{\alpha}_{\phi}(g_{1},g_{2})=\tilde{\beta}(g_{1})+g_{1}\cdot\tilde{\beta}(g_{2})-\tilde{\beta}(g_{1}g_{2})$
Using the ${}_{\epsilon}\log$ map to transfer to $\mathcal{W}$, we conclude
with the following proposition summarizing our work so far:

######  Proposition 3.2.8.

Suppose for every internal
$\alpha:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$ with
$Im(\alpha)\subseteq\mathcal{W}_{b}$ that satisfies the $2$-cocycle condition
(Eq. 3.7), there exists an internal $\beta:{}^{*}\Gamma\to\mathcal{W}$ taking
values in $\mathcal{W}_{b}$ such that

$\tilde{\alpha}(g_{1},g_{2})=\tilde{\beta}(g_{1})+g_{1}\cdot\tilde{\beta}(g_{2})-\tilde{\beta}(g_{1}g_{2})$

then $\Gamma$ exhibits the defect diminishing property, and is therefore
uniformly $\mathfrak{U}$-stable with a linear estimate.

Since we shall work with internal maps from $({}^{*}\Gamma)^{2}$ (or
${}^{*}\Gamma$) to $\mathcal{W}$ that take values in $\mathcal{W}_{b}$, it is
helpful to describe such a map as an ultraproduct of bounded maps. Let
$\\{\alpha_{n}\in\ell^{\infty}(\Gamma^{2},\mathfrak{u}(k_{n}))\\}_{k=1}^{\infty}$
be a family of maps such that there exists a constant $C>0$ such that
$\|\alpha_{n}\|_{\infty}\leq C$ for every $n\in\mathbf{N}$. Then it is clear
that the ultraproduct $\alpha=\\{\alpha_{n}\\}_{\mathcal{U}}$ has image
$Im(\alpha)\subseteq\mathcal{W}_{b}$. Conversely,

######  Lemma 3.2.9.

Let $\alpha:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$ be an internal map
with $Im(\alpha)\subseteq\mathcal{W}_{b}$. Then there exists a family
$\\{\alpha_{n}\in\ell^{\infty}(\Gamma^{2},\mathfrak{u}(k_{n}))\\}_{k=1}^{\infty}$
such that $\alpha=\\{\alpha_{n}\\}_{\mathcal{U}}$. Conversely, if
$\\{\alpha_{n}\in\ell^{\infty}(\Gamma^{2},\mathfrak{u}(k_{n}))\\}_{k=1}^{\infty}$
is a family of maps such that
$\\{\|\alpha_{n}\|_{\infty}\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{b}$, then
$\alpha=\\{\alpha_{n}\\}_{\mathcal{U}}{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$
is an internal map with $Im(\alpha)\subseteq\mathcal{W}_{b}$.

###### Proof.

Since $\alpha$ is internal, it is of the form
$\alpha=\\{f_{n}\\}_{\mathcal{U}}$ for a family of maps
$f_{n}:\Gamma\times\Gamma\to\mathfrak{u}(k_{n})$. For a subset
$S\in\mathcal{U}$, suppose $f_{n}$ is unbounded for every $n\in S$. Then for
each $n\in S$, there exists $x_{n},y_{n}\in\Gamma$ such that
$f_{n}(x_{n},y_{n})$ has norm at least $n$. In particular, for
$x=\\{x_{n}\\}_{\mathcal{U}}$ and $y=\\{y_{n}\\}_{\mathcal{U}}$, we have
$\alpha(x,y)\notin\mathcal{W}_{b}$. This is a contradiction to the hypothesis
that $Im(\alpha)\subseteq\mathcal{W}_{b}$. The converse is immediate from the
definition of $\mathcal{W}_{b}$. ∎

Thus, an internal map $\alpha:{}^{*}\Gamma\times{}^{*}\Gamma\to\mathcal{W}$
with $Im(\alpha)\subseteq\mathcal{W}_{b}$ can be described as
$\\{\alpha_{n}\\}_{\mathcal{U}}$ where for every $k\in\mathbf{N}$,
$\alpha_{n}:\Gamma\times\Gamma\to\mathfrak{u}(k_{n})$ is a bounded map, and
such that
$\\{\|\alpha_{n}\|_{\infty}\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{b}$. In other
words, the internal map $\alpha$ is the ultraproduct of bounded maps, and is
also bounded as an ultraproduct.
From now on, we shall regard $\alpha$ as an element of
$\prod_{\mathcal{U}}\ell^{\infty}(({}^{*}\Gamma)^{2},\mathfrak{u}(k_{n})$,
which we shall henceforth denote
$\mathcal{L}^{\infty}(({}^{*}\Gamma)^{2},\mathcal{W})$ (understood as the
space of internal maps from $({}^{*}\Gamma)^{2}$ to $\mathcal{W}$. In fact,
$\alpha$ is actually an element of
$\mathcal{L}^{\infty}(({}^{*}\Gamma)^{2},\mathcal{W})_{b}$ since not only is
it internally bounded, but also
$\\{\|\alpha_{n}\|_{\infty}\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}_{b}$. In
general, we shall use the following notation:

  * •

For $m\in\mathbf{N}$, the internal space
$\mathcal{L}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})$ is defined as

$\mathcal{L}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})\coloneq\Big{\\{}\ell^{\infty}(\Gamma^{m},\mathfrak{u}(k_{n}))\Big{\\}}_{\mathcal{U}}$

  * •

The (external) subspace of
$\mathcal{L}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})$ comprising internal
functions with bounded (supremum) norm will be denoted
$\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{m},\mathcal{W})$ while the
(external) subspace of
$\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{m},\mathcal{W})$ comprising internal
functions with infinitesimal (supremum) norm will be denoted
$\mathcal{L}^{\infty}_{inf}(({}^{*}\Gamma)^{m},\mathcal{W})$.

  * •

The quotient
$\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{m},\mathcal{W})/\mathcal{L}^{\infty}_{inf}(({}^{*}\Gamma)^{m},\mathcal{W})$
shall be denoted
$\tilde{\mathcal{L}}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})$. This space
comprises bounded maps from $({}^{*}\Gamma)^{m}$ to $\tilde{\mathcal{W}}$ that
are induced from internal maps in
$\mathcal{L}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})$. As in Remark 3.2.6,
$\tilde{\mathcal{L}}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})$ is a real
Banach space.

  * •

For a map $f\in\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{m},\mathcal{W})$, we
shall denote by
$\tilde{f}\in\tilde{\mathcal{L}}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})$ the
composition of $f$ with the canonical quotient map
$\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{m},\mathcal{W})\to\tilde{\mathcal{L}}^{\infty}(({}^{*}\Gamma)^{m},\mathcal{W})$.

For convenience, we restate Proposition 3.2.8 in this notation:

######  Proposition 3.2.10.

Suppose for every
$\alpha\in\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{2},\mathcal{W})$ that
satisfies the $2$-cocycle condition (Eq. 3.7), there exists a
$\beta\in\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{1},\mathcal{W})$ such that

$\tilde{\alpha}(g_{1},g_{2})=\tilde{\beta}(g_{1})+g_{1}\cdot\tilde{\beta}(g_{2})-\tilde{\beta}(g_{1}g_{2})$

then $\Gamma$ exhibits the defect diminishing property, and is therefore
uniformly $\mathfrak{U}$-stable with a linear estimate.

###  3.3 Uniform Stability of Amenable groups

In this section, we shall demonstrate an application of Proposition 3.2.10 to
amenable groups. The first step is to recognize the duality of $\mathcal{W}$
in an internal way.
Consider the space $\mathcal{W}=\prod_{\mathcal{U}}\mathfrak{u}(k_{n})$, and
let $\mathcal{W}^{\sharp}\coloneqq\prod_{\mathcal{U}}(\mathfrak{u}(n))^{*}$.
For the Banach space $\mathfrak{u}(n)$, consider its dual space
$(\mathfrak{u}(n))^{*}$ and let $\langle\cdot|\cdot\rangle$ the canonical
duality (for instance, if $\mathfrak{u}(n)$ is equipped with the Schatten
$p$-norm for $p>1$, then $(\mathfrak{u}(n))^{*}$ comes equipped with the
Schatten $q$-norm, where $1/p+1/q=1$). Denote by $\mathcal{W}^{\sharp}$ the
ultraproduct $\prod_{\mathcal{U}}(\mathfrak{u}(k_{n}))^{*}$, and its external
subsets $\mathcal{W}^{\sharp}_{b}$ and $\mathcal{W}^{\sharp}_{inf}$ as in (Eq.
3.3) and (Eq. 3.4). Note that the internal pairing

$\langle\cdot|\cdot\rangle_{\mathcal{U}}:\mathcal{W}^{\sharp}\times\mathcal{W}\to{}^{*}\mathbf{R}$

induces a pairing

$\langle\cdot|\cdot\rangle:\tilde{\mathcal{W}}^{\sharp}\times\tilde{\mathcal{W}}\to\mathbf{R}$

Equivalently, $\mathcal{W}_{b}^{\sharp}$ comprises
$\lambda\in\mathcal{W}^{\sharp}$ such that $\lambda(v)\in{}^{*}\mathbf{R}_{b}$
for every $v\in\mathcal{W}_{b}$, while $\mathcal{W}_{inf}^{\sharp}$ comprises
$\lambda\in\mathcal{W}^{\sharp}$ such that
$\lambda(v)\in{}^{*}\mathbf{R}_{inf}$ for every $v\in\mathcal{W}_{b}$.
We can use the internal map
$\pi_{\psi}:{}^{*}\Gamma\times\mathcal{W}\to\mathcal{W}$ to define the
internal map

$\pi_{\psi}^{\sharp}:{}^{*}\Gamma\times\mathcal{W}^{\sharp}\to\mathcal{W}^{\sharp}$
(3.8)

which on $\lambda\in\mathcal{W}^{\sharp}$ and $v\in\mathcal{W}$ is defined to
be

$\pi_{\psi}^{\sharp}(g)(\lambda)(v)\coloneqq\lambda(\pi_{\psi}(g^{-1})v)$

######  Lemma 3.3.1.

The internal map $\pi_{\psi}^{\sharp}$ restricts to a map on
$\mathcal{W}^{\sharp}_{b}$ that induces an action of ${}^{*}\Gamma$ on
$\mathcal{W}^{\sharp}_{b}/\mathcal{W}^{\sharp}_{inf}$.

###### Proof.

Let $\lambda\in\mathcal{W}_{b}^{\sharp}$. For $v\in\mathcal{W}$, since
$\pi_{\psi}^{\sharp}(g)(\lambda)(v)=\lambda(\pi_{\psi}(g^{-1})v)$, note that
$\pi_{\psi}(g^{-1})v\in\mathcal{W}_{b}$ for $v\in\mathcal{W}_{b}$. Hence
$\pi_{\psi}^{\sharp}(g)(\lambda)\in\mathcal{W}_{b}^{\sharp}$ for every
$g\in{}^{*}\Gamma$. Similarly, for $\lambda\in\mathcal{W}_{inf}^{\sharp}$,
$\pi_{\psi}^{\sharp}(\lambda)\in\mathcal{W}_{inf}^{\sharp}$. That this induces
an action of ${}^{*}\Gamma$ on
$\mathcal{W}_{b}^{\sharp}/\mathcal{W}_{inf}^{\sharp}$ follows easily from the
fact that $\pi_{\psi}$ induces an action of ${}^{*}\Gamma$ on
$\mathcal{W}_{b}/\mathcal{W}_{inf}$. ∎

Going one step further, we can obtain a canonical identification of
$\mathcal{W}$ with $(\mathcal{W}^{\sharp})^{\sharp}$ (which has the same norm
as $\mathcal{W}$) so that we can regard $\mathcal{W}$ as
$(\mathcal{W}^{\sharp})^{\sharp}$ with
$(\pi_{\psi}^{\sharp})^{\sharp}=\pi_{\psi}$. This is true for $\mathcal{W}$
since $\mathcal{W}=\prod_{\mathcal{U}}\mathfrak{u}(n)$, and $\mathfrak{u}(n)$
is finite-dimensional for each $n\in\mathbf{N}$.

######  Remark 3.3.2.

We shall often use this reflexivity property of $\mathcal{W}$ to regard its
dual $\mathcal{W}^{\sharp}$ as its predual, so that $v\in\mathcal{W}$ acts on
$\lambda\in\mathcal{W}^{\sharp}$ by $v\cdot\lambda=\lambda(v)$. However, note
that for the following discussion of amenability, what we actually need is not
reflexivity but merely the property that $\mathcal{W}$ is dual.

Let us now recall the definition of amenability for discrete groups. While
there are innumerable equivalent definitions of amenability, here we shall see
the definition that is most relevant to us (later on in §4.3, we shall study
amenability and amenable actions in the locally compact case). Consider the
Banach space $\ell^{\infty}(\Gamma)$ with the following action of $\Gamma$:
for $g,x\in\Gamma$ and $f\in\ell^{\infty}(\Gamma)$, $(g\cdot
f)(x)=f(g^{-1}x)$. A _mean_ on $\ell^{\infty}(\Gamma)$ is a bounded linear
functional $m:\ell^{\infty}(\Gamma)\to\mathbf{R}$ such that $\|m\|\leq 1$,
$m(1)=1$ and $m(f)\geq 0$ whenever $f\geq 0$. The mean $m$ is said to be
$\Gamma$-_invariant_ if for every $g\in\Gamma$ and
$f\in\ell^{\infty}(\Gamma)$, $m(g\cdot f)=m(f)$.

######  Definition 3.3.3.

The discrete group $\Gamma$ is said to be amenable if there exists a
$\Gamma$-invariant mean on $\ell^{\infty}(\Gamma)$.

While the definition of amenability asks for a $\Gamma$-invariant mean on
$\ell^{\infty}(\Gamma)$, this can be easily extended to obtain a
$\Gamma$-_equivariant_ mean on $\ell^{\infty}(\Gamma,W)$ for a dual normed
$W$-module (where the action of $\Gamma$ on $\ell^{\infty}(\Gamma,W)$ is given
by $(g\cdot f)(x)=g\cdot f(g^{-1}x)$. The following lemma builds on this idea
to construct an internal mean on
$\mathcal{L}^{\infty}({}^{*}\Gamma,\mathcal{W})$.

######  Lemma 3.3.4.

Suppose $\Gamma$ is amenable. Then there exists an internal map
$m_{in}:\mathcal{L}^{\infty}({}^{*}\Gamma,\mathcal{W})\to\mathcal{W}$ such
that $m_{in}$ induces a linear map
$\tilde{m}:\tilde{\mathcal{L}}^{\infty}({}^{*}\Gamma,\mathcal{W})\to\tilde{\mathcal{W}}$
satisfying the following two conditions:

  * •

Suppose $\tilde{f}\in\tilde{\mathcal{L}}^{\infty}({}^{*}\Gamma,\mathcal{W})$
is the constant function $\tilde{f}(g)=\tilde{v}$ for every
$g\in{}^{*}\Gamma$, then $\tilde{m}(\tilde{f})=\tilde{v}$.

  * •

For $\tilde{f}\in\tilde{\mathcal{L}}^{\infty}({}^{*}\Gamma,\mathcal{W})$,
$\|\tilde{m}(\tilde{f})\|\leq\|\tilde{f}\|$.

  * •

For $g\in{}^{*}\Gamma$ and
$\tilde{f}\in\tilde{\mathcal{L}}^{\infty}({}^{*}\Gamma,\mathcal{W})$,
$\tilde{m}\left(g\cdot\tilde{f}\right)=g\cdot\tilde{m}(\tilde{f})$.

###### Proof.

Consider
$f=\\{f_{n}\\}_{\mathcal{U}}\in\mathcal{L}^{\infty}({}^{*}\Gamma,\mathcal{W})$.
Since $\mathcal{W}=(\mathcal{W}^{\sharp})^{\sharp}$, for each
$\lambda\in\mathcal{W}^{\sharp}$, we get an internal map

$f_{\lambda}:{}^{*}\Gamma\to{}^{*}\mathbf{R}$ $f_{\lambda}(x)\coloneqq
f(x)(\lambda)$

Note that $f_{\lambda}$ being internal, is of the form
$\\{(f_{\lambda})_{n}\\}_{\mathcal{U}}$ where
$(f_{\lambda})_{n}\in\ell^{\infty}(\Gamma)$. This allows us to construct the
internal map
$m_{in}^{\lambda}:\mathcal{L}^{\infty}({}^{*}\Gamma,\mathcal{W})\to{}^{*}\mathbf{R}$
as

$m_{in}^{\lambda}(f)=\\{m\left((f_{\lambda})_{n}\right)\\}_{\mathcal{U}}$

and finally
$m_{in}:\mathcal{L}^{\infty}({}^{*}\Gamma,\mathcal{W})\to(\mathcal{W}^{\sharp})^{\sharp}$
as

$m_{in}(f)(\lambda)\coloneqq m_{in}^{\lambda}(f)$

It is straightforward to check that $m_{in}$ as defined induces a linear map
$\tilde{m}:\tilde{\mathcal{L}}({}^{*}\Gamma,\mathcal{W})\to\mathcal{W}_{b}/\mathcal{W}_{inf}$.
As for ${}^{*}\Gamma$-equivariance, this follows from the observation that
$(g\cdot f)_{\lambda}(x)=\pi(g)f(g^{-1}x)(\lambda)$ while $(g\cdot
f_{\lambda})(x)=f(g^{-1}x)(\lambda)$. The conditions on $\tilde{m}$ follow
from the definition and properties of the $\Gamma$-invariant mean $m$ on
$\ell^{\infty}(\Gamma)$. ∎

We shall denote the internal mean above by $m_{in}^{x}$ (or $\tilde{m}^{x}$)
when the mean is understood to be taken over $x\in\Gamma$. This would be
particularly useful when working with multivariate maps where we fix certain
coordinates to obtain a univariate map which we can take a mean over (as in
the following Proposition 3.3.5). Note that in this notation, it is easy to
see that the ${}^{*}\Gamma$-equivariance of the mean constructed above
translates to the simpler (invariant) form: for
$\tilde{f}\in\tilde{\mathcal{L}}^{\infty}({}^{*}\Gamma,\mathcal{W})$ and
$g\in{}^{*}\Gamma$,

$\tilde{m}^{x}\left(\tilde{f}(gx)\right)=\tilde{m}^{x}\left(\tilde{f}(x)\right)$

We shall now use this internal map $m_{in}$ and the ${}^{*}\Gamma$-equivariant
map $\tilde{m}$ to show the following:.

######  Proposition 3.3.5.

Suppose $\Gamma$ is amenable. Then for every
$\alpha\in\mathcal{L}^{\infty}_{b}(({}^{*}\Gamma)^{2},\mathcal{W})$ that
satisfies the $2$-cocycle condition (Eq. 3.7), there exists a
$\beta\in\mathcal{L}^{\infty}_{b}({}^{*}\Gamma,\mathcal{W})$ such that

$\tilde{\alpha}(g_{1},g_{2})=\tilde{\beta}(g_{1})+g_{1}\cdot\tilde{\beta}(g_{2})-\tilde{\beta}(g_{1}g_{2})$

###### Proof.

Suppose $\alpha\in\mathcal{L}^{\infty}L(({}^{*}\Gamma)^{2},\mathcal{W})$
satisfies the $2$-cocyle condition: for every $g_{1},g_{2},x\in{}^{*}\Gamma$,

$\tilde{\alpha}(g_{1},g_{2})=g_{1}\cdot\tilde{\alpha}(g_{2},x)-\tilde{\alpha}(g_{1}g_{2},x)+\tilde{\alpha}(g_{1},g_{2}x)$

For a fixed $g$, the map $\alpha_{g}:{}^{*}\Gamma\to\mathcal{W}$ defined as
$\alpha_{g}(x)\coloneq\alpha(g,x)$ is clearly contained in
$\mathcal{L}^{\infty}({}^{*}\Gamma,\mathcal{W})$. Define
$\beta\in\mathcal{L}^{\infty}({}^{*}\Gamma,\mathcal{W})$ as

$\beta(g)\coloneq m_{in}\left(\alpha_{g}\right)$

In other words, $\beta(g)=m_{in}^{x}\left(\alpha(g,x)\right)$. Then the
$2$-cocycle condition satisfied by $\alpha$ immediately implies that

$\tilde{\alpha}(g_{1},g_{2})=g_{1}\cdot\tilde{\beta}(g_{2})-\tilde{\beta}(g_{1}g_{2})+\tilde{\beta}(g_{1})$

∎

Observe that the proof of Proposition 3.3.5 is almost exactly on the lines of
the proof that $\operatorname{H}_{b}^{2}(\Gamma,V)=0$ for amenable $\Gamma$
and dual normed $\Gamma$-module $W$ (refer to Theorem $3.6$ in [Fri17] for
more details).
In light of Proposition 3.3.5 and Proposition 3.2.10, we conclude that:

######  Corollary 3.3.6.

If $\Gamma$ is a discrete amenable group, then $\Gamma$ is uniformly
$\mathfrak{U}$-stable with a linear estimate.

Note that while on the one hand Corollary 3.3.6 generalizes Kazhdan’s result
[Kaz82] to a larger family (where we allow any submultiplicative matrix norm
as opposed to just the operator norm as in [Kaz82] and [BOT13]), we _do not_
prove here the analogous result of strong Ulam stability, where the family
comprises groups of unitary operators on (possibly infinite-dimensional)
Hilbert spaces.

##  4 Asymptotic Cohomology of Groups

In this section, we shall formally define the asymptotic cohomology theory of
(topological) groups, and study some basic properties along the lines of the
theory of bounded cohomology. Recall that our goal is to prove uniform
$\mathfrak{U}$-stability for lattices in higher rank Lie groups, so this
forces us to develop the cohomology theory for locally compact groups (as
opposed to just discrete groups as we briefly saw in §3.2 and §3.3).
The basic objects we shall deal with are defined in §4.1, where we describe
the category of asymptotic cohomology abstractly using tools from
cohomological algebra. In §4.3, we define the asymptotic cohomology of groups
and relate it to the way it was motivated in §3.3. Finally, in §4.3, we use
Zimmer amenability and the functorial relations of §4.1 to obtain other
complexes that compute the same cohomology, which we shall use in §5 and §6.

###  4.1 Basic Definitions and Some Cohomological Algebra

Recall, from §2.2, that we fix a non-principal ultrafilter $\mathcal{U}$ on
$\mathbf{N}$ to define ultraproducts and internal objects. For convenience, we
set some notation and conventions now. Let $C$ be a category, with $C$-objects
and $C$-morphisms. We shall define a category ${}^{*}C$ in ${}^{*}Univ$ as
follows:

  * •

The objects of ${}^{*}C$, referred to as _internal_ $C$-objects, shall be
ultraproducts $\prod_{\mathcal{U}}X_{n}$ where $\\{X_{n}\\}_{n\in\mathbf{N}}$
is an indexed collection of $C$-objects.

  * •

The morphisms of ${}^{*}C$, referred to as _internal morphism_ , are of the
form $\phi=\\{\phi_{n}\\}_{\mathcal{U}}$ (also denoted
$\prod_{\mathcal{U}}\phi_{n}$), where $\\{\phi_{n}\\}_{n\in\mathbf{N}}$ is an
indexed collection of $C$-morphisms.

Given a category $C$, the internal $C$-objects and internal $C$-morphisms form
a category ${}^{*}C$ in ${}^{*}Univ$, and these two categories have the same
first-order theories, allowing us to use the transfer principle, as remarked
in §2.2.

######  Definition 4.1.1.

Let $A$ be a property of $C$-objects (resp, $C$-morphisms). Then
$\prod_{\mathcal{U}}X_{n}$ (resp,
$\phi:\prod_{\mathcal{U}}X_{n}\to\prod_{\mathcal{U}}Y_{n}$) has _internal $A$_
if $X_{n}$ (resp, $\phi_{n}$) has $A$ for every $n\in S$ with
$S\in\mathcal{U}$.

For example, let $G$ be a locally compact, second countable topological group,
and consider the ultrapower group ${}^{*}G$ and let
$\mathcal{E}=\\{E_{n}\\}_{\mathcal{U}}$ be an internal Banach space. An
internal map $\pi:{}^{*}G\times\mathcal{E}\to\mathcal{E}$, where
$\pi=\\{\pi_{n}:G\times E_{n}\to E_{n}\\}$, is an _internal action_ of
${}^{*}G$ on $\mathcal{E}$ (or $\mathcal{E}$ is an _internal
${}^{*}G$-representation_) if the map $\pi_{n}:G\times E_{n}\to E_{n}$ is an
isometric $G$-representation for every $n\in\mathbf{N}$, and the internal map
$\pi$ is _internally continuous_ if $\pi_{n}:G\times E_{n}\to E_{n}$ is
continuous for every $n\in\mathbf{N}$ (here $G\times E_{n}$ is endowed with
the product topology). All of these notions simply involve passing from
standard categories to their internal counterparts in ${}^{*}Univ$.
We shall now work with the category $Ban$ whose objects are (real) Banach
spaces and whose morphisms are bounded linear maps, and study ${}^{*}Ban$.
Consider the ultraproduct $\mathcal{E}=\prod_{\mathcal{U}}E_{n}$ of the real
Banach spaces $\\{E_{n}\\}_{n\in\mathbf{N}}$, which is an internal Banach
space. For an element $v\in\mathcal{E}$ where $v=\\{v_{n}\\}_{\mathcal{U}}$,
we denote by $\|v\|\coloneq\\{\|v_{n}\|\\}_{\mathcal{U}}\in{}^{*}\mathbf{R}$.
Given two internal Banach spaces $\mathcal{E}=\\{E_{n}\\}_{\mathcal{U}}$ and
$\mathcal{F}=\\{F_{n}\\}_{\mathcal{U}}$, we shall denote by
$\mathcal{Hom}(\mathcal{E},\mathcal{E})$ the set of internal morphisms between
$\mathcal{E}$ and $\mathcal{F}$. These are exactly of the form
$\phi\coloneq\\{\phi_{n}:E_{n}\to F_{n}\\}_{\mathcal{U}}$ where
$\phi_{n}:E_{n}\to F_{n}$ is a bounded linear map for every $n\in\mathbf{N}$
(such maps are internal morphisms). Note that
$\mathcal{Hom}(\mathcal{E},\mathcal{E})$ itself is an internal Banach space
when endowed with the internal operator norm (that is,
$\|\phi\|\coloneq\\{\|\phi_{n}\|_{op}\\}_{\mathcal{U}}$).
Consider the set $\mathcal{Hom}(\mathcal{E},{}^{*}\mathbf{R})$, which is the
_internal dual_ of $\mathcal{E}$, denoted $\mathcal{E}^{\sharp}$. Explicitly,
for each Banach space $E_{n}$ as above, let $E_{n}^{\sharp}$ denote its
(constinuous) dual Banach space, and let $\langle\cdot|\cdot\rangle$ the
canonical duality, and $\mathcal{E}^{\sharp}$ denote the ultraproduct
$\prod_{\mathcal{U}}E_{n}^{\sharp}$. Note that we have an internal pairing

$\langle\cdot|\cdot\rangle_{\mathcal{U}}:\mathcal{E}^{\sharp}\times\mathcal{E}\to{}^{*}\mathbf{R}$

We shall call $\mathcal{E}$ an internal dual Banach space if $\mathcal{E}$ is
the internal dual of some internal Banach space (which we shall denote
$\mathcal{E}^{\flat}=\prod_{\mathcal{U}}E_{n}^{\flat}$). For
$\phi\in\mathcal{Hom}(\mathcal{E},\mathcal{E})$, we shall denote by
$\phi^{\sharp}\in\mathcal{Hom}(\mathcal{E}^{\sharp},\mathcal{E}^{\sharp})$ the
internal adjoint map with respect to the internal pairing above: that is
$\phi^{\sharp}$ is such that for every $v\in\mathcal{E}$ and
$w\in\mathcal{E}^{\sharp}$,
$\langle\phi^{\sharp}w,v\rangle_{\mathcal{U}}=\langle w,\phi
v\rangle_{\mathcal{U}}$.
We shall now see some general functorial aspects of (standard) real Banach
spaces and extend them naturally to internal Banach spaces. Let $X$, $Y$, $E$
and $F$ be (standard) real Banach spaces. Let $B(X\times E)$ denote the Banach
space of bounded bilinear forms $X\times E\to\mathbf{R}$, and $L(E,F)$ denote
the Banach space of bounded linear functions from $E$ to $F$. Through the
canonical pairing, note that $B(X\times E)$ is naturally isometrically
isomorphic to the Banach space of bounded linear maps $E\to X^{\sharp}$ (and
also to the Banach space of bounded linear maps $X\to E^{\sharp}$). That is,

$B(X\times E)\cong L(E,X^{\sharp})\cong L(X,E^{\sharp})$ (4.1)