# Banach property (T) for $\rm SL_{n}(\mathbb{Z})$ and its applications

Izhar Oppenheim  Department of Mathematics, Ben-Gurion University of the
Negev, Be’er Sheva 84105, Israel

###### Abstract.

We prove that a large family of higher rank simple Lie groups (including $\rm
SL_{n}(\mathbb{R})$ for $n\geq 3$) and their lattices have Banach property (T)
with respect to all super-reflexive Banach spaces.

Two consequences of this result are: First, we deduce Banach fixed point
properties with respect to all super-reflexive Banach spaces for a large
family of higher rank simple Lie groups. For example, we show that for every
$n\geq 4$, the group $\rm SL_{n}(\mathbb{R})$ and all its lattices have the
Banach fixed point property with respect to all super-reflexive Banach spaces.
Second, we settle a long standing open problem and show that the Margulis
expanders (Cayley graphs of $\rm SL_{n}(\mathbb{Z}/m\mathbb{Z})$ for a fixed
$n\geq 3$ and $m$ tending to infinity) are super-expanders.

All of our results stem from proving Banach property (T) for $\rm
SL_{3}(\mathbb{Z})$. Our method of proof for $\rm SL_{3}(\mathbb{Z})$ relies
on a novel proof for relative Banach property (T) for the uni-triangular
subgroup of $\operatorname{SL}_{3}(\mathbb{Z})$. This proof of relative
property (T) is new even in the classical Hilbert setting and is interesting
in its own right.

###### 2010 Mathematics Subject Classification:

Primary 22D12, 22E40; Secondary 46B85, 20F65

The author was partially supported by ISF grant no. 293/18

##  1\. Introduction

Property (T) was introduced by Kazhdan in [Kaž67] as a tool to prove finite
generation. Since then it was found useful for a wide range of applications in
various different areas of mathematics (see [BdlHV08] and the introduction of
[BFGM07], and reference therein). We mention two such applications that are
relevant in the context of this paper: First, property (T) for a group $G$ is
equivalent (under some mild assumptions on $G$) to property (FH) which states
that every continuous isometric affine action of $G$ on a real Hilbert space
admits a fixed point. Second, Margulis gave the first explicit construction of
expanders using property (T).

More recently, Bader, Furman, Gelander and Monod [BFGM07] defined a Banach
version of property (T) (and its connection to Banach fixed point properties).
They conjectured that higher rank algebraic group should have this form of
Banach property (T) with respect to the class of all super-reflexive Banach
spaces. Roughly simultaneously to the work of [BFGM07], V. Lafforgue [Laf09]
proved that groups of the form $\operatorname{SL}_{3}(F)$ where $F$ is a non-
Archimedian local field have a strong form of Banach property (T) for large
classes of Banach spaces and that this strong form of Banach property (T)
implies the fix point property. In particular, his work corroborates the
conjecture of [BFGM07]: Namely, a consequence of Lafforgue’s work it that the
groups $\operatorname{SL}_{3}(F)$ where $F$ is a non-Archimedian local field
have Banach property (T) and the fixed point property with respect to all
super-reflexive Banach spaces. Later, Liao [Lia14] extended the work of
Lafforgue and proved the strong version of Banach property (T) holds for every
higher rank connected almost $F$-simple algebraic group, where $F$ is a non-
Archimedean local field. In his work [Laf09], Lafforgue also showed how to use
his result to construct super-expanders, i.e., families of graphs that are
expanders with respect to every super-reflexive Banach space (see exact
definition below).

Unlike in the non-Archimedian case, much less was known regarding Banach
property (T) for algebraic groups over $\mathbb{R}$ (and their lattices). In
the paper of Bader, Furman, Gelander and Monod [BFGM07] they showed that
higher rank algebraic groups have Banach property (T) (and fixed point
properties) for $L^{p}$ spaces where $1<p<\infty$ (the case $p=1$ was later
resolved in [BGM12]). For general super-reflexive spaces (that are not $L^{p}$
spaces), partial results were proven by de Laat, Mimura and de la Salle in
various collaborations [dLdlS15, Sal16, dLMdlS16, dLdlS18, dlS19]. However,
non of these works cover all super-reflexive spaces even for a single group of
the form $\operatorname{SL}_{n}(\mathbb{R})$ with some $n\geq 3$.

In this paper, we make a major breakthrough regarding Banach property (T): We
show that large family of connected simple (higher rank) Lie groups have
Banach property (T) with respect to all super-reflexive Banach spaces. In
particular, we show that for every $n\geq 3$, the groups
$\operatorname{SL}_{n}(\mathbb{Z})$ and $\operatorname{SL}_{n}(\mathbb{R})$
have Banach property (T) with respect to all super-reflexive Banach spaces.
This has several striking consequences: First, it allows us to prove that a
family of connected simple (higher rank) Lie groups and their lattices have
the fixed point property with respect to all super-reflexive Banach spaces. In
particular, we show that for every $n\geq 4$,
$\operatorname{SL}_{n}(\mathbb{R})$ and all its lattices have the fixed point
property with respect to all super-reflexive Banach spaces. Second, we settle
a long standing open problem and show that the Margulis expanders (i.e.,
Cayley graphs of $\operatorname{SL}_{n}(\mathbb{Z}/i\mathbb{Z})$ where $n\geq
3$ is a fixed integer) are super-expanders. Last, we show that for every
$n\geq 5$, the groups $\operatorname{SL}_{n}(\mathbb{Z})$ and
$\operatorname{SL}_{n}(\mathbb{R})$ have a strengthening of the fixed point
property (property $(FF)$ defined below) with respect to all super-reflexive
Banach spaces

Our method for proving Banach property (T) is also novel. The prior works of
de Laat, Mimura and de la Salle mentioned above were based on generalizing the
work of Lafforgue on strong (Hilbert) property (T) of
$\operatorname{SL}_{3}(\mathbb{R})$ to the Banach setting. Our approach is
very different: We first prove a relative version of Banach property (T) for
the uni-triangular matrices in $\operatorname{SL}_{3}(\mathbb{Z})$ with
respect to super-reflexive Banach spaces. We note that this proof is new even
in the Hilbert setting. After that, we use a bounded generation argument à la
Shalom to deduce Banach property (T) for $\operatorname{SL}_{3}(\mathbb{Z})$
for all super-reflexive spaces. Then using Howe-Moore we deduce Banach
property $(T)$ for simple Lie groups with whose Lie algebra contain
$\mathfrak{sl}_{3}(\mathbb{R})$ (see exact formulation below). We note that
most of this proof scheme was known to experts, but prior to this work, it was
not known how to prove the relative Banach property $(T)$ result.

###  1.1. Uniformly convex and super-reflexive Banach spaces

A Banach space $\operatorname{\mathbb{E}}$ is called uniformly convex if there
is a function $\delta:(0,2]\rightarrow(0,1]$ called the modulus of convexity
such that for every $0<\varepsilon\leq 2$ and every
$\xi,\eta\in\operatorname{\mathbb{E}}$ with $\|\xi\|=\|\eta\|=1$, if
$\|\xi-\eta\|\geq\varepsilon$, then
$\|\frac{\xi+\eta}{2}\|\leq(1-\delta(\varepsilon))$.

We will not recall the definition of super-reflexive Banach spaces, but only
note that by [BL00, Theorem A.6] a Banach space $\operatorname{\mathbb{E}}$ is
super-reflexive if and only if there is a equivalent uniformly convex norm on
$\operatorname{\mathbb{E}}$ (a reader who is not familiar with super-reflexive
Banach spaces can take this as a definition).

###  1.2. Banach property (T) for $\operatorname{SL}_{3}(\mathbb{Z})$ and
Simple Lie groups

Given a topological group $G$ and a Banach space $\operatorname{\mathbb{E}}$,
a linear isometric representation of $G$ on $\operatorname{\mathbb{E}}$ is a
continuous homomorphism $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$, where
$O(\operatorname{\mathbb{E}})$ denotes the group of all invertible linear
isometries of $\operatorname{\mathbb{E}}$ with the strong operator topology. A
linear isometric representation $\pi$ is said to have almost invariant vectors
if for every compact set $K\subseteq G$ and every $\varepsilon>0$, there is a
unit vector $\xi\in\operatorname{\mathbb{E}}$ such that

$\sup_{g\in K}\|\pi(g)\xi-\xi\|<\varepsilon.$

In [BFGM07], Bader, Furman, Gelander and Monod defined Banach property (T) of
a group $G$ as follows:

######  Definition 1.1.

Let $\operatorname{\mathbb{E}}$ be Banach space and $G$ be a topological
group. The group $G$ has property $(T_{\operatorname{\mathbb{E}}})$ if for
every continuous linear isometric representation $\pi:G\rightarrow
O(\operatorname{\mathbb{E}})$, the quotient representation
$\pi^{\prime}:G\rightarrow
O(\operatorname{\mathbb{E}}/\operatorname{\mathbb{E}}^{\pi(G)})$ does not have
almost invariant vectors.

The main result of this paper is proving Banach property
$(T_{\operatorname{\mathbb{E}}})$ with respect to every super-reflexive Banach
space $\operatorname{\mathbb{E}}$ for $\operatorname{SL}_{3}(\mathbb{Z})$:

######  Theorem 1.2.

Let $\operatorname{\mathbb{E}}$ be a super-reflexive Banach space. The group
$\operatorname{SL}_{3}(\mathbb{Z})$ has property
$(T_{\operatorname{\mathbb{E}}})$.

Using Howe-Moore Theorem, this allows us to deduce the following theorem:

######  Theorem 1.3.

Let $G$ be a connected simple real Lie group with a finite center and
$\mathfrak{g}$ the Lie algebra of $G$. If $\mathfrak{g}$ contains
$\mathfrak{sl}_{3}(\mathbb{R})$ as a Lie sub-algebra, then $G$ all its
lattices have property $(T_{\operatorname{\mathbb{E}}})$ for every super-
reflexive Banach space $\operatorname{\mathbb{E}}$.

This corroborates a conjecture stated in [BFGM07] in which it was conjectured
that all higher rank almost simple algebraic groups have property
$(T_{\operatorname{\mathbb{E}}})$ for every super-reflexive Banach space
$\operatorname{\mathbb{E}}$ (see [BFGM07, Remark 2.28]). An immediate
consequence is:

######  Theorem 1.4.

Let $n\geq 3$ and $\operatorname{\mathbb{E}}$ be a super-reflexive Banach
space. The group $\operatorname{SL}_{n}(\mathbb{R})$ and all its lattices have
property $(T_{\operatorname{\mathbb{E}}})$. In particular, for every $n\geq 3$
and every super-reflexive Banach space $\operatorname{\mathbb{E}}$, the group
$\operatorname{SL}_{n}(\mathbb{Z})$ has property
$(T_{\operatorname{\mathbb{E}}})$.

###  1.3. Applications

#### Banach fixed point property for simple Lie groups

Given a Banach space $\operatorname{\mathbb{E}}$, a topological group $G$ is
said to have property $(F_{\operatorname{\mathbb{E}}})$ is every affine
(continuous) isometric action of $G$ on $\operatorname{\mathbb{E}}$ admits a
fixed point.

As an application of Banach property $(T)$ for
$\operatorname{SL}_{3}(\mathbb{R})$ we prove:

######  Theorem 1.5.

Let $G$ be a connected simple real Lie group with a finite center and
$\mathfrak{g}$ be the Lie algebra of $G$. If $\mathfrak{g}$ contains
$\mathfrak{sl}_{4}(\mathbb{R})$ as a Lie sub-algebra, then $G$ and any lattice
$\Gamma<G$ have property $(F_{\operatorname{\mathbb{E}}})$ for every super-
reflexive Banach space $\operatorname{\mathbb{E}}$.

Again, this corroborates a conjecture stated in [BFGM07] in which it was
conjectured that all higher rank algebraic groups have property
$(F_{\operatorname{\mathbb{E}}})$ for every super-reflexive Banach space
$\operatorname{\mathbb{E}}$ (see [BFGM07, Conjecture 1.6]).

The method of proof of Theorem 1.5 is as follows: First, using ideas of
[BFGM07], we show that Banach property $(T_{\operatorname{\mathbb{E}}})$ for
$\operatorname{SL}_{3}(\mathbb{R})$ implies property
$(F_{\operatorname{\mathbb{E}}})$ for $\operatorname{SL}_{4}(\mathbb{R})$.
Second, via a (now standard) use of Maunter phenomenon, we deduce Theorem 1.5.

#### Super-expanders

A family of finite graphs with uniformly bounded degree is called a super-
expander family (or a super-expander) if it has a Poincaré inequality with
respect to every super-reflexive Banach space (see exact definition in section
9.2). The first examples of super-expanders were constructed by Lafforgue in
[Laf09] as a consequence of his work on strong Banach property (T) for
$\operatorname{SL}_{3}(F)$, where $F$ is a non-Archimedean local field. Since
Lafforgue’s work there have been several constructions of super-expanders
using two main techniques: Namely, the work of Mendel and Naor on non-linear
spectral calculus [MN14] which gave a zig-zag construction for super-
expanders. In a different direction, several works [Vig19, dLV19, FNvL19,
Saw20] gave constructions using warped cones of groups actions arising from
groups with Banach property (T).

It was an open problem to determine whether the Margulis expanders is a super-
expander family, i.e., if for a fixed $n\geq 3$ the Cayley graphs of
$\operatorname{SL}_{n}(\mathbb{Z}/i\mathbb{Z})$ form a super-expander family.
This open question appeared in the literature several times, including in
Assaf Naor’s Minerva lecture [Nao13] where it was attributed to Margulis and
in de la Salle’s 2022 ICM lecture [dlS22, Conjecture 4.4] (see also [Obe18,
Problem 5], [dLdlS18], [Mim13, Remark 5.3]). Partial results were achieved by
de Laat and de la Salle in [dLdlS18], but up until our work, the problem
remained open. As a consequence of our Theorem 1.4, we settle this problem to
the affirmative and prove the following:

######  Theorem 1.6.

Let $n\geq 3$ and $S$ be a finite generating set of
$\operatorname{SL}_{n}(\mathbb{Z})$ (e.g., $S=\\{e_{i,j}(\pm 1):1\leq i,j\leq
n,i\neq j\\}$). Also, let
$\phi_{i}:\operatorname{SL}_{n}(\mathbb{Z})\rightarrow\operatorname{SL}_{n}(\mathbb{Z}/i\mathbb{Z})$
be the natural surjective homomorphism for every $i\in\mathbb{N}$. Then the
family of Cayley graphs of
$\\{(\operatorname{SL}_{n}(\mathbb{Z}/i\mathbb{Z}),\phi_{i}(S))\\}_{i\in\mathbb{N}}$
is a super-expander family.

As noted above, one can construct super-expanders using warped cones arising
from an action of a Banach property (T) group on a compact Riemannian manifold
(see [Vig19, dLV19, FNvL19, Saw20]). Combining this machinery with our Theorem
1.4 also leads to a construction of super-expanders (see section 9.2 for more
details):

######  Theorem 1.7.

Let $n\geq 3$ and let $M$ be a compact Riemannian manifold such that
$\operatorname{SL}_{n}(\mathbb{Z})$ acts on $M$ by Lipschitz homeomorphisms.
For every increasing sequence
$\\{t_{i}\\}_{i\in\mathbb{N}}\subseteq\mathbb{R}_{>0}$ tending to infinity,
the family
$\\{(M,d^{t_{i}}_{\operatorname{SL}_{n}(\mathbb{Z})})\\}_{i\in\mathbb{N}}$ is
quasi-isometric to a super-expander.

#### Banach property $(FF_{\operatorname{\mathbb{E}}})$

In [Mim11], Mimura defined the notion of property
$(FF_{\operatorname{\mathbb{E}}})$ (see exact definition in section 9.3 below)
that is a Banach version of property (TT) defined by Monod in [Mon01]. A
result of de Laat, Mimura and de la Salle [dLMdlS16] allows one to deduce
property $(FF_{\operatorname{\mathbb{E}}})$ for the groups
$\operatorname{SL}_{n+2}(\mathbb{Z}),\operatorname{SL}_{n+2}(\mathbb{R})$ from
property $(T_{\operatorname{\mathbb{E}}})$ for the groups
$\operatorname{SL}_{n}(\mathbb{Z}),\operatorname{SL}_{n}(\mathbb{R})$. Thus,
we can deduce the following:

######  Corollary 1.8.

For every $n\geq 5$ and every super-reflexive Banach space
$\operatorname{\mathbb{E}}$, the groups
$\operatorname{SL}_{n}(\mathbb{Z}),\operatorname{SL}_{n}(\mathbb{R})$ have
property $(FF_{\operatorname{\mathbb{E}}})$.

###  1.4. Relative Banach property (T) for uni-triangular in
$\operatorname{SL}_{3}(\mathbb{Z})$

Here we will outline the proof of Theorem 1.2 from which all our other results
follow. The proof relies on a relative Banach property (T) argument that is
novel even in the Hilbert setting.

Generalizing the definition of relative property (T) given in [Jol05, Theorem
1.2 (b2)] to the Banach setting, we will define relative Banach property (T)
as follows:

######  Definition 1.9.

Let $G$ be a topological group and $H<G$ be a subgroup. For a Banach space
$\operatorname{\mathbb{E}}$, we will say that the pair $(G,H)$ has relative
Banach property $(T_{\operatorname{\mathbb{E}}})$ if for every continuous
linear isometric representation $\pi:G\rightarrow
O(\operatorname{\mathbb{E}})$ and every constant $\gamma>0$, there is a
compact set $K\subseteq G$ and a constant $\varepsilon>0$ such that for every
unit vector $\xi\in\operatorname{\mathbb{E}}$, if $\sup_{g\in
K}\|\pi(g)\xi-\xi\|<\varepsilon$, then there exits
$\eta\in\operatorname{\mathbb{E}}^{\pi(H)}$ such that $\|\xi-\eta\|<\gamma$.

######  Remark 1.10.

This definition is a-priori weaker than definition of weak relative Banach
property (T) given in [dLMdlS16] and is strictly weaker than the definition of
strong relative Banach property (T) given in [BFGM07].

Let $\operatorname{UT}_{3}(\mathbb{Z})$ and
$\operatorname{LT}_{3}(\mathbb{Z})$ denote the subgroups of uni-upper-
triangular and uni-lower-triangular matrices in
$\operatorname{SL}_{3}(\mathbb{Z})$, i.e.,

$\operatorname{UT}_{3}(\mathbb{Z})=\left\\{\left(\begin{matrix}1&a&c\\\
0&1&b\\\ 0&0&1\end{matrix}\right):a,b,c\in\mathbb{Z}\right\\},$

and

$\operatorname{LT}_{3}(\mathbb{Z})=\left\\{\left(\begin{matrix}1&0&0\\\
a&1&0\\\ c&b&0\end{matrix}\right):a,b,c\in\mathbb{Z}\right\\}.$

We prove the following (with respect to our definition of relative Banach
property $(T)$ stated above):

######  Theorem 1.11.

For any super-reflexive Banach space $\operatorname{\mathbb{E}}$ the pairs
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{UT}_{3}(\mathbb{Z}))$ and
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{LT}_{3}(\mathbb{Z}))$ both
have relative property $(T_{\operatorname{\mathbb{E}}})$.

The proof of this theorem is via defining a sequence of finitely supported
probability measures $f_{n}$ on $\operatorname{SL}_{3}(\mathbb{Z})$ and
showing that for every super-reflexive Banach space
$\operatorname{\mathbb{E}}$ and every continuous isometric representation
$\pi:\operatorname{SL}_{3}(\mathbb{Z})\rightarrow
O(\operatorname{\mathbb{E}})$ it holds that $\pi(f_{n})$ converges in the norm
topology and that the image of the limit operator is in
$\operatorname{\mathbb{E}}^{\pi(\operatorname{UT}_{3}(\mathbb{Z}))}$. We note
that this method is new even in the classical Hilbert setting. Moreover, the
proof is completely elementary in contrast with the more classical proofs of
relative property (T), e.g., the proof that
$(\operatorname{SL}_{2}(\mathbb{Z})\ltimes\mathbb{Z}^{2},\mathbb{Z}^{2})$ has
relative property (T) in [BdlHV08, Theorem 4.2.2] requires Fourier analysis
and projections valued measures while our proof requires neither.

Combining Theorem 1.11 with a bounded generation argument à la Shalom implies
Theorem 1.2.

#### Structure of this paper

This paper is organized as follows: In section 2, we cover some needed
preliminaries. In section 3, we gather some facts regarding Banach property
(T) (and relative versions of Banach property (T)). In section 4, we show how
bounded generation and relative Banach property (T) imply Banach property (T).
In section 5, we prove some bounds on the norms of averaging operations for
the Heisenberg group that are needed for our relative Banach property (T)
result. In section 6, we prove our relative Banach property (T) result
(Theorem 1.11). In section 7, we prove Banach property (T) for
$\operatorname{SL}_{3}(\mathbb{Z})$ (Theorem 1.2). In section 8, we prove
Banach property (T) for a large family of simple Lie groups and their lattices
(Theorem 1.3). Last, in section 9, we prove the applications stated above.

#### Acknowledgements

I thank Uri Bader and Mikael de la Salle for reading an early draft of this
paper and making several valuable suggestions that vastly improved it. I also
thank the anonymous referee of this paper for many corrections and suggestions
that improved the correctness and readability of this paper.

##  2\. Preliminaries

###  2.1. Uniformly convexity

Below, we will state some needed facts regarding uniformly convex spaces.

######  Proposition 2.1.

Let $\operatorname{\mathbb{E}}$ be a uniformly convex Banach space with a
modulus of convexity $\delta:(0,2]\rightarrow(0,1]$ and denote
$O(\operatorname{\mathbb{E}})$ to be the group of invertible linear isometries
of $\operatorname{\mathbb{E}}$. Then for every $0<\varepsilon\leq 2$, every
$S,T\in O(\operatorname{\mathbb{E}})$ such that $TS=ST$ and every
$\xi\in\operatorname{\mathbb{E}}$, if $\|(I-S)\xi\|\geq\varepsilon\|\xi\|$,
then

$\frac{1}{2}\left\|\frac{I+T}{2}\xi\right\|+\frac{1}{2}\left\|\frac{I+TS}{2}\xi\right\|\leq\max\left\\{1-\frac{1}{2}\delta(\delta(\varepsilon)),1-\frac{1}{4}\delta(\varepsilon)\right\\}\|\xi\|.$

This proposition is probably well-known and we give the proof for
completeness:

###### Proof.

Fix $\xi\in\operatorname{\mathbb{E}}$ and $0<\varepsilon\leq 2$, and assume
that $\|(I-S)\xi\|\geq\varepsilon\|\xi\|$. If
$\|(I-T)\xi\|\geq\delta(\varepsilon)$, then

$\displaystyle\frac{1}{2}\left\|\frac{I+T}{2}\xi\right\|+\frac{1}{2}\left\|\frac{I+TS}{2}\xi\right\|\leq\frac{1}{2}\left(1-\delta(\delta(\varepsilon))\right)\|\xi\|+\frac{1}{2}\|\xi\|=\left(1-\frac{1}{2}\delta(\delta(\varepsilon))\right)\|\xi\|,$

as needed.

Otherwise, $\|\frac{I-T}{2}\xi\|\leq\frac{\delta(\varepsilon)}{2}$ and

$\displaystyle\frac{1}{2}\left\|\frac{I+T}{2}\xi\right\|+\frac{1}{2}\left\|\frac{I+TS}{2}\xi\right\|\leq\frac{1}{2}\|\xi\|+\frac{1}{2}\left\|\frac{I+S}{2}\xi\right\|+\frac{1}{2}\left\|\frac{S(T-I)}{2}\xi\right\|\leq$
$\displaystyle\frac{1}{2}\|\xi\|+\frac{1}{2}(1-\delta(\varepsilon))\|\xi\|+\frac{1}{4}\delta(\varepsilon)\|\xi\|=\left(1-\frac{1}{4}\delta(\varepsilon)\right)\|\xi\|.$

∎

We will be interested in classes of uniformly convex Banach spaces defined as
follows: Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a monotone increasing
function. Denote $\mathcal{E}_{us}(\delta_{0})$ to be the class of all
uniformly convex Banach spaces $\operatorname{\mathbb{E}}$ with such that the
modulus of convexity of $\operatorname{\mathbb{E}}$ is bounded by
$\delta_{0}$, i.e., for a uniformly convex Banach space
$\operatorname{\mathbb{E}}$ with a modulus of convexity
$\delta:(0,2]\rightarrow(0,1]$ it holds that
$\operatorname{\mathbb{E}}\in\mathcal{E}_{uc}(\delta_{0})$ if and only if for
every $0<\varepsilon\leq 2$ it holds that
$\delta(\varepsilon)\geq\delta_{0}(\varepsilon)$. For these classes of Banach
space, we state the following immediate corollary of Proposition 2.1:

######  Corollary 2.2.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function and $0<\varepsilon\leq 2$
be a constant. There is $r_{0}=r_{0}(\delta_{0},\varepsilon)$, $0\leq r_{0}<1$
such that for every $\operatorname{\mathbb{E}}\in\mathcal{E}_{uc}(\delta_{0})$
and every two commuting operators $S,T\in O(\operatorname{\mathbb{E}})$ it
holds for every $\xi\in\operatorname{\mathbb{E}}$ that if
$\|(I-S)\xi\|\geq\varepsilon\|\xi\|$, then

$\frac{1}{2}\left\|\frac{I+T}{2}\xi\right\|+\frac{1}{2}\left\|\frac{I+TS}{2}\xi\right\|\leq
r_{0}\|\xi\|.$

We will need the following theorems:

######  Theorem 2.3.

[Day41, Theorem 3] Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function and
$\\{\operatorname{\mathbb{E}}_{n}\\}_{n\in\mathbb{N}}\subseteq\mathcal{E}_{uc}(\delta_{0})$
a sequence. Then the $\ell^{2}$-sum
$\bigoplus_{n}\operatorname{\mathbb{E}}_{n}$ is a uniformly convex Banach
space.

######  Theorem 2.4.

[LT79, Theorem 1.e.9] Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function.
There is a function $\delta_{0}^{\prime}:(0,2]\rightarrow(0,1]$ that for every
finite measure space $(X,\mu)$ and every
$\operatorname{\mathbb{E}}\in\mathcal{E}_{uc}(\delta_{0})$ it holds that
$L^{2}(X,\mu;\operatorname{\mathbb{E}})\in\mathcal{E}_{uc}(\delta_{0}^{\prime})$.

######  Remark 2.5.

In the above theorem, we implicitly use the fact that
$L^{2}(X,\mu;\operatorname{\mathbb{E}}^{*})$ is isometrically isomorphic to
$(L^{2}(X,\mu;\operatorname{\mathbb{E}}))^{*}$. This follows from [DU77,
Section IV.1, Theorem 1] combined with the fact that reflexive Banach spaces
(and in particular uniformly convex Banach spaces) have the Radon-Nikodým
property (see [DU77, Section III.2, Corollary 13]).

###  2.2. Linear Representation of groups on Banach spaces

Given a topological group $G$ and a Banach space $\operatorname{\mathbb{E}}$,
a linear representation of $G$ on $\operatorname{\mathbb{E}}$ is a continuous
homomorphism $\pi:G\rightarrow\operatorname{GL}(\operatorname{\mathbb{E}})$,
where $\operatorname{GL}(\operatorname{\mathbb{E}})$ denotes the group of all
invertible linear transformations of $\operatorname{\mathbb{E}}$ with the
strong operator topology. Throughout this paper, $\pi$ will denote a
continuous representation.

For a given a linear representation of $G$ on $\operatorname{\mathbb{E}}$, the
contragredient representation is the map
$\pi^{*}:G\rightarrow\operatorname{GL}(\operatorname{\mathbb{E}}^{*})$ defined
as

$\forall g\in
G,\forall\xi\in\operatorname{\mathbb{E}},\eta\in\operatorname{\mathbb{E}}^{*},\langle\pi(g)\xi,\eta\rangle=\langle\xi,\pi^{*}(g^{-1})\eta\rangle.$

We note that if $\pi$ is an isometric representation, then $\pi^{*}$ is also
isometric, but in general $\pi^{*}$ need not be continuous. However, for every
reflexive (and hence super-reflexive) Banach space
$\operatorname{\mathbb{E}}$, if $\pi$ is continuous, then so is $\pi^{*}$.

Below, we will need the following result from [BFGM07]:

######  Proposition 2.6.

[BFGM07, Proposition 2.6] Let $\operatorname{\mathbb{E}}$ be a super-reflexive
Banach space, $G$ be a topological group and $\pi:G\rightarrow
O(\operatorname{\mathbb{E}})$ be a continuous linear isometric representation.
Denote $\operatorname{\mathbb{E}}^{\prime}(\pi)$ to be the annihilator of
$(\operatorname{\mathbb{E}}^{*})^{\pi^{*}(G)}$ in $\operatorname{\mathbb{E}}$,
i.e.,

$\operatorname{\mathbb{E}}^{\prime}(\pi)=\\{\xi\in\operatorname{\mathbb{E}}:\forall\eta\in(\operatorname{\mathbb{E}}^{*})^{\pi^{*}(G)},\langle\xi,\eta\rangle=0\\}.$

Then
$\operatorname{\mathbb{E}}=\operatorname{\mathbb{E}}^{\pi(G)}\oplus\operatorname{\mathbb{E}}^{\prime}(\pi)$.

###  2.3. Steinberg relations in $\operatorname{SL}_{3}(\mathbb{Z})$

For $1\leq i,j\leq 3,i\neq j$ and $m\in\mathbb{Z}$, denote $e_{i,j}(m)$ to be
the elementary matrix with $1$’s along the main diagonal, $m$ in the
$(i,j)$-entry and $0$ in all other entries. Using the convention
$[a,b]=a^{-1}b^{-1}ab$, the group $\operatorname{SL}_{3}(\mathbb{Z})$ has the
following relations that are called the Steinberg relations:

  1. (1)

For every $1\leq i,j\leq 3,i\neq j$ and every $m_{1},m_{2}\in\mathbb{Z}$,

$e_{i,j}(m_{1})e_{i,j}(m_{2})=e_{i,j}(m_{1}+m_{2}).$

  2. (2)

For every $1\leq i,j,k\leq 3,\\{i,j,k\\}=\\{1,2,3\\}$ and every
$m_{1},m_{2}\in\mathbb{Z}$,

$[e_{i,j}(m_{1}),e_{j,k}(m_{2})]=e_{i,k}(m_{1}m_{2}).$

  3. (3)

For every $1\leq i,j,k\leq 3,\\{i,j,k\\}=\\{1,2,3\\}$ and every
$m_{1},m_{2}\in\mathbb{Z}$,

$[e_{i,j}(m_{1}),e_{i,k}(m_{2})]=[e_{j,i}(m_{1}),e_{k,i}(m_{2})]=I.$

The group $\operatorname{SL}_{3}(\mathbb{Z})$ has other relations that do not
stem from the Steinberg relations. Forgetting the other relations of
$\operatorname{SL}_{3}(\mathbb{Z})$ yields the Steinberg group
$\operatorname{St}_{3}(\mathbb{Z})$. Explicitly, the Steinberg group
$\operatorname{St}_{3}(\mathbb{Z})$ is the group generated by the set
$S=\\{x_{i,j}:1\leq i,j\leq 3,i\neq j\\}$ with the following relations: For
every $m\in\mathbb{Z}$, denote $x_{i,j}(m)=x_{i,j}^{m}$. With this notation,
the relations defining $\operatorname{St}_{3}(\mathbb{Z})$ are:

  1. (1)

For every $1\leq i,j\leq 3,i\neq j$ and every $m_{1},m_{2}\in\mathbb{Z}$,

$x_{i,j}(m_{1})x_{i,j}(m_{2})=x_{i,j}(m_{1}+m_{2}).$

  2. (2)

For every $1\leq i,j,k\leq 3,\\{i,j,k\\}=\\{1,2,3\\}$ and every
$m_{1},m_{2}\in\mathbb{Z}$,

$[x_{i,j}(m_{1}),x_{j,k}(m_{2})]=x_{i,k}(m_{1}m_{2}).$

  3. (3)

For every $1\leq i,j,k\leq 3,\\{i,j,k\\}=\\{1,2,3\\}$ and every
$m_{1},m_{2}\in\mathbb{Z}$,

$[x_{i,j}(m_{1}),x_{i,k}(m_{2})]=[x_{j,i}(m_{1}),x_{k,i}(m_{2})]=I.$

###  2.4. The Heisenberg group $\rm H_{3}(\mathbb{Z})$

The Heisenberg group $\rm H_{3}(\mathbb{Z})$ is the group

$\rm
H_{3}(\mathbb{Z})=\langle\left.x,y,z\right|[x,y]=z,[x,z]=e,[y,z]=e\rangle.$

Below, we will use the following relations for the Heisenberg group that are
not hard verify: for every $k,m\in\mathbb{Z}$ it holds that
$y^{-k}x^{m}y^{k}=x^{m}z^{km}$ and $x^{-k}y^{m}x^{k}=y^{m}z^{-km}$.

In the sequel, we will use the fact that $\operatorname{SL}_{3}(\mathbb{Z})$
(and $\operatorname{St}_{3}(\mathbb{Z})$) contain several copies of $\rm
H_{3}(\mathbb{Z})$. Explicitly, for every $\\{i,j,k\\}=\\{1,2,3\\}$, if denote
$\tilde{x}=e_{i,j}(1),\tilde{y}=e_{j,k}(1),\tilde{z}=e_{i,k}(1)\in\operatorname{SL}_{3}(\mathbb{Z})$,
then
$\langle\tilde{x},\tilde{y},\tilde{z}\rangle<\operatorname{SL}_{3}(\mathbb{Z})$
is isomorphic to $\rm H_{3}(\mathbb{Z})$ (by the Steinberg relations) via the
isomorphism $x\mapsto\tilde{x},y\mapsto\tilde{y},z\mapsto\tilde{z}$.

##  3\. Banach property (T)

###  3.1. Banach property $(T)$ for super-reflexive Banach spaces

Bader, Furman, Gelander and Monod [BFGM07] gave an equivalent version to
Banach property (T) for super-reflexive spaces that is more convenient to work
with than their general definition. In [BFGM07], it is shown that if
$\operatorname{\mathbb{E}}$ is a super-reflexive Banach space and
$\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ is a linear isometric
representation, then one can pass to a compatible norm on
$\operatorname{\mathbb{E}}$ in which $\operatorname{\mathbb{E}}$ is uniformly
convex and $\pi$ remains a linear isometric representation with respect to
this new norm. It follows that for a given topological group $G$ the following
are equivalent:

  * •

The group $G$ has property $(T_{\operatorname{\mathbb{E}}})$ for every
uniformly convex Banach space $\operatorname{\mathbb{E}}$.

  * •

The group $G$ has property $(T_{\operatorname{\mathbb{E}}})$ for every super-
reflexive Banach space $\operatorname{\mathbb{E}}$.

Thus, below we will focus on property $(T_{\operatorname{\mathbb{E}}})$ for
uniformly convex Banach spaces $\operatorname{\mathbb{E}}$ and the general
result for super-reflexive Banach spaces will follow.

For uniformly convex Banach spaces, [BFGM07] gave the following equivalent
definition for property $(T_{\operatorname{\mathbb{E}}})$:

######  Definition 3.1.

[BFGM07, Remark 2.11] Let $\operatorname{\mathbb{E}}$ be uniformly convex
space and $G$ be a topological group. Denote
$\operatorname{\mathbb{E}}^{\prime}(\pi)$ to be the annihilator of
$(\operatorname{\mathbb{E}}^{*})^{\pi^{*}(G)}$ in $\operatorname{\mathbb{E}}$,
i.e.,

$\operatorname{\mathbb{E}}^{\prime}(\pi)=\\{\xi\in\operatorname{\mathbb{E}}:\forall\eta\in(\operatorname{\mathbb{E}}^{*})^{\pi^{*}(G)},\langle\xi,\eta\rangle=0\\}.$

The group $G$ has property $(T_{\operatorname{\mathbb{E}}})$ if for every
continuous linear isometric representation $\pi:G\rightarrow
O(\operatorname{\mathbb{E}})$, the restricted representation
$\pi^{\prime}:G\rightarrow\operatorname{\mathbb{E}}^{\prime}(\pi)$ does not
have almost invariant vectors, i.e., there is a Kazhdan pair $(K,\varepsilon)$
(that depends on $\pi$) where $K\subseteq G$ is compact and $\varepsilon>0$
such that for every vector $\xi\in\operatorname{\mathbb{E}}^{\prime}(\pi)$ it
holds that

$\sup_{g\in K}\|\pi^{\prime}(g)\xi-\xi\|\geq\varepsilon\|\xi\|.$

######  Observation 3.2.

Let $G$ be a topological group. By the above definition the following are
equivalent:

  1. (1)

The group $G$ has property $(T_{\operatorname{\mathbb{E}}})$ for every
uniformly convex Banach space $\operatorname{\mathbb{E}}$.

  2. (2)

For every uniformly convex Banach space $\operatorname{\mathbb{E}}$ and every
isometric representation $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ with
$\operatorname{\mathbb{E}}^{\pi(G)}=\\{0\\}$, there is a compact set
$K\subseteq G$ and a constant $\varepsilon>0$ such that for every unit vector
$\xi\in\operatorname{\mathbb{E}}$,

$\sup_{g\in K}\|\pi(g)\xi-\xi\|\geq\varepsilon\|\xi\|.$

###  3.2. Relative Banach property (T)

Here we introduce a variation of relative Banach property (T) called relative
Banach property $(T^{\operatorname{proj}})$ that can be seen as a
generalization of the definition of Banach property
$(T^{\operatorname{proj}})$ given in [dLdlS18]. We will show that relative
Banach property $(T^{\operatorname{proj}})$ is a-priori stronger than the
definition of relative Banach property (T) given in the introduction (we do
not know if the two definitions do in fact coincide - see Remark 3.5 below).
In order to define relative Banach property $(T^{\operatorname{proj}})$, we
will need to first introduce some notation and terminology.

Let $G$ be a locally compact group with Haar measure $\mu$. We denote
$C_{c}(G)$ to be the compactly supported continuous functions
$f:G\rightarrow\mathbb{C}$ with the convolution product. We further denote
$\operatorname{Prob}_{c}(G)\subseteq C_{c}(G)$ to be functions
$f:G\rightarrow[0,\infty)$ such that $\int_{G}f(g)d\mu(g)=1$. Given a
continuous representation
$\pi:G\rightarrow\operatorname{GL}(\operatorname{\mathbb{E}})$ where
$\operatorname{\mathbb{E}}$ is a Banach space, we define for every $f\in
C_{c}(G)$ an operator $\pi(f)$ via the Bochner integral

$\pi(f)\xi=\int_{G}f(g)\pi(g)\xi
d\mu(g),\forall\xi\in\operatorname{\mathbb{E}}.$

For a class of Banach spaces $\mathcal{E}$, we denote
$\mathcal{U}(G,\mathcal{E})$ to be the class of all continuous isometric
linear representations $(\pi,\operatorname{\mathbb{E}})$ where
$\operatorname{\mathbb{E}}\in\mathcal{E}$. When $G$ is obvious from the
context, we will denote $\mathcal{U}(\mathcal{E})=\mathcal{U}(G,\mathcal{E})$.
We define a norm $\|.\|_{\mathcal{U}(G,\mathcal{E})}$ on $C_{c}(G)$ by

$\|f\|_{\mathcal{U}(G,\mathcal{E})}=\sup_{\pi\in\mathcal{U}(G,\mathcal{E})}\|\pi(f)\|,$

and denote $C_{\mathcal{U}(G,\mathcal{E})}(G)$ to be the completion of
$C_{c}(G)$ with respect to this norm. We note that for every $f\in
C_{\mathcal{U}(G,\mathcal{E})}(G)$ and every
$\pi\in\mathcal{U}(G,\mathcal{E})$, the operator $\pi(f)\in
B(\operatorname{\mathbb{E}})$ is well-defined as a limit of operators
$\pi(f_{n})$ with $f_{n}\in C_{c}(G)$.

######  Definition 3.3.

Let $G$ be a locally compact group with a subgroup $H<G$. We will say that
$(G,H)$ has relative property $(T_{\mathcal{E}}^{\operatorname{proj}})$ if
there is a sequence $h_{n}\in\operatorname{Prob}_{c}(G)$ that converges to
$f\in C_{\mathcal{U}(G,\mathcal{E})}(G)$ (with respect to the norm
$\|.\|_{\mathcal{U}(G,\mathcal{E})}$) such that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(G,\mathcal{E})$,
$\operatorname{Im}(\pi(f))\subseteq\operatorname{\mathbb{E}}^{\pi(H)}$.

We show that relative property $(T^{\operatorname{proj}}_{\mathcal{E}})$ imply
relative property $(T_{\operatorname{\mathbb{E}}})$ for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$ as defined above (see Definition
1.9):

######  Proposition 3.4.

Let $G$ be a locally compact group, $H<G$ a subgroup and $\mathcal{E}$ a class
of Banach spaces. Assume that $(G,H)$ has relative property
$(T^{\operatorname{proj}}_{\mathcal{E}})$. Then for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$, the pair $(G,H)$ has relative
property $(T_{\operatorname{\mathbb{E}}})$.

###### Proof.

We need to show that for every $\gamma>0$, there are $K\subseteq G$ compact
and $\varepsilon>0$, such that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E})$ and every unit
vector $\xi\in\operatorname{\mathbb{E}}$, if $\sup_{g\in
K}\|\pi(g)\xi-\xi\|<\varepsilon$, then there is
$\eta\in\operatorname{\mathbb{E}}^{\pi(H)}$ such that $\|\xi-\eta\|<\gamma$.

Let $\gamma>0$ arbitrary. By the assumption that $(G,H)$ has relative property
$(T^{\operatorname{proj}}_{\mathcal{E}})$ it follows that there is a sequence
$h_{n}\in\operatorname{Prob}_{c}(G)$ that converges to $f\in
C_{\mathcal{U}(\mathcal{E})}(G)$ such that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E})$,
$\operatorname{Im}(\pi(f))\subseteq\operatorname{\mathbb{E}}^{\pi(H)}$.

For $f\in C_{\mathcal{U}(\mathcal{E})}(G)$ as above, there is
$h\in\operatorname{Prob}_{c}(G)$ such that
$\|f-h\|_{\mathcal{U}(\mathcal{E})}<\frac{\gamma}{2}$. We take $K$ to be a
compact set such that $\operatorname{supp}(h)\subseteq K$ and
$\varepsilon=\frac{\gamma}{2}$ and show that this choice of $K,\varepsilon$
fulfils the needed condition.

Indeed, for every $(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E})$
and every unit vector $\xi\in\operatorname{\mathbb{E}}$, if $\sup_{g\in
K}\|\pi(g)\xi-\xi\|<\varepsilon=\frac{\gamma}{2}$, then for
$\eta=\pi(f)\xi\in\operatorname{\mathbb{E}}^{\pi(H)}$ it holds that

$\|\xi-\eta\|\leq\|\xi-\pi(h)\xi\|+\|\pi(h)\xi-\eta\|<\left\|\int_{G}h(g)(\xi-\pi(g)\xi)d\mu(g)\right\|+\frac{\gamma}{2}\leq\\\
{\int_{G}h(g)\left\|\xi-\pi(g)\xi\right\|d\mu(g)}+\frac{\gamma}{2}\leq\max_{g\in\operatorname{supp}(h)}\left\|\xi-\pi(g)\xi\right\|+\frac{\gamma}{2}\leq\gamma$

as needed. ∎

######  Remark 3.5.

We do not know if the opposite direction of the above proposition is also
true, i.e., if relative property $(T_{\operatorname{\mathbb{E}}})$ for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$ implies property
$(T^{\operatorname{proj}}_{\mathcal{E}})$. The problem is that even in the
classical setting of Hilbert spaces there is not natural candidate for the
sequence $h_{n}\in\operatorname{Prob}_{c}(G)$. To illustrate this, we consider
what should be a simple situation: Let $G$ be a finitely generating group with
a finite generating set $S$ and $N\triangleleft G$ a normal subgroup and
$\mathcal{H}$ be the class of all Hilbert spaces.

We recall that from that fact that $N$ is a normal subgroup it follows for
every unitary $(\pi,\mathbb{H})$ of $G$ on a Hilbert space $\mathbb{H}$ the
subspaces $\mathbb{H}^{\pi(N)},(\mathbb{H}^{\pi(N)})^{\perp}$ are
$G$-invariant subspaces with respect to the $G$ action. In this setting,
relative property $(T)$ for $(G,N)$ can be described by the following
formulation in [Jol05, Theorem 1.2 (b2)]: There is $\varepsilon_{0}>0$ such
that for every $\alpha>0$, every unitary representation $(\pi,\mathbb{H})$ of
$G$ on a Hilbert space $\mathbb{H}$ and every unit vector $\xi\in\mathbb{H}$,
if

$\max_{s\in S}\|\pi(s)\xi-\xi\|\leq\alpha\varepsilon_{0}$

then $\|\xi-P_{\mathbb{H}^{\pi(N)}}\xi\|\leq\alpha$ where
$P_{\mathbb{H}^{\pi(N)}}$ is the orthogonal projection on
$\mathbb{H}^{\pi(N)}$.

Our naive guess for $h_{n}\in\operatorname{Prob}_{c}(G)$ is the sequence

$h_{n}=\left(\frac{1}{2}I+\frac{1}{2|S|}\sum_{s\in S}s\right)^{n},$

(which is the sequence that converges to a Kazhdan projection when $N=G$). For
every unitary representation $(\pi,\mathbb{H})$,
$\left.\pi\right|_{(\mathbb{H}^{\pi(N)})^{\perp}}(h_{n})$ indeed converges to
$0$ and the rate of convergence can be bounded independently of $\pi$.
However, we see no reason that $\left.\pi\right|_{\mathbb{H}^{\pi(N)}}(h_{n})$
will converge when $N\neq G$ and thus (as far as we can tell) this naive
attempt fails.

###  3.3. Hereditary properties of property $(T_{\operatorname{\mathbb{E}}})$

Lafforgue showed that property $(T_{\operatorname{\mathbb{E}}})$ is inherited
by lattices via an induction of representation:

######  Proposition 3.6.

[Laf08, Proposition 4.5], [Laf09, Proposition 5.3] Let $G$ be a locally
compact group, $\Gamma<G$ a lattice and $\mathcal{E}$ a class of Banach
spaces. Also let $\mathcal{E}^{\prime}$ be a class of Banach spaces such that
for every $\operatorname{\mathbb{E}}\in\mathcal{E}$ and every finite measure
space $(X,\mu)$ it holds that
$L^{2}(X,\mu;\operatorname{\mathbb{E}})\in\mathcal{E}^{\prime}$. If $G$ has
property $(T_{\operatorname{\mathbb{E}}^{\prime}})$ for every
$\operatorname{\mathbb{E}}^{\prime}\in\mathcal{E}^{\prime}$ , then $\Gamma$
has property $(T_{\operatorname{\mathbb{E}}})$ for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$.

######  Remark 3.7.

The above formulation differs from the formulation in [Laf08, Laf09] since we
do not assume that $\mathcal{E}$ is closed under passing to $L^{2}$-sums.

######  Corollary 3.8.

Let $G$ be a locally compact group and $\Gamma<G$ a lattice. If $G$ has
property $(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach
space $\operatorname{\mathbb{E}}$, then $\Gamma$ has property
$(T_{\operatorname{\mathbb{E}}})$ for every super-reflexive Banach space
$\operatorname{\mathbb{E}}$.

###### Proof.

This follows immediately from Proposition 3.6 and Theorem 2.4. ∎

##  4\. Bounded generation and Banach property (T)

In this section, we adapt a bounded generation argument of Shalom [Sha99] to
our setting and show that, in our setting, relative Banach property (T) and
bounded generation imply Banach property (T).

######  Definition 4.1.

Let $G$ be a group with subgroups $H_{1},...,H_{k}$. We say that
$H_{1},...,H_{k}$ boundedly generate $G$ if there is a number
$\nu=\nu(H_{1},...,H_{k})\in\mathbb{N}$ such that every element $g\in G$ can
be written by at most $\nu$ elements of $H_{1}\cup...\cup H_{k}$.

######  Lemma 4.2.

Let $G$ be a group with subgroups $H_{1},...,H_{k}$ that boundedly generate
$G$ and denote $\nu=\nu(H_{1},...,H_{k})$ as above. Also, let
$\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ be a continuous linear
isometric representation. Assume that there are
$\eta_{1},...,\eta_{k}\in\operatorname{\mathbb{E}}$ such that for every $1\leq
i\leq k$, $\eta_{i}\in\operatorname{\mathbb{E}}^{\pi(H_{i})}$. Then for every
$\xi\in\operatorname{\mathbb{E}}$ and every $g\in G$,

$\|\pi(g)\xi-\xi\|\leq 2\nu\max_{1\leq i\leq k}\|\xi-\eta_{i}\|.$

###### Proof.

Let $g\in G$ such that $g=g_{1}...g_{j}$ with
$g_{1},...,g_{j}\in\bigcup_{i=1}^{k}H_{i}$. We will prove by induction that
for every $\xi\in\operatorname{\mathbb{E}}$,

(1) $\|\pi(g)\xi-\xi\|\leq 2j\max_{1\leq i\leq k}\|\xi-\eta_{i}\|.$

For $j=1$, there is $1\leq i_{0}\leq k$ such that $g\in H_{i_{0}}$. Then

$\displaystyle\|\pi(g)\xi-\xi\|=\|\pi(g)\xi-\pi(g)\eta_{i_{0}}+\eta_{i_{0}}-\xi\|\leq\|\pi(g)(\xi-\eta_{i_{0}})\|+\|\eta_{i_{0}}-\xi\|=$
$\displaystyle 2\|\xi-\eta_{i_{0}}\|\leq 2\max_{1\leq i\leq
k}\|\xi-\eta_{i}\|.$

Assume (1) holds for $j$ and let $g=g_{1}...g_{j+1}$ with
$g_{1},...,g_{j+1}\in\bigcup_{i=1}^{k}H_{i}$. Then for every
$\xi\in\operatorname{\mathbb{E}}$,

$\displaystyle\|\pi(g)\xi-\xi\|=\|\pi(g_{1}...g_{j}g_{j+1})\xi-\pi(g_{1}...g_{j})\xi+\pi(g_{1}...g_{j})\xi-\xi\|\leq$
$\displaystyle\|\pi(g_{1}...g_{j})(\pi(g_{j+1})\xi-\xi)\|+\|\pi(g_{1}...g_{j})\xi-\xi\|=$
$\displaystyle\|(\pi(g_{j+1})\xi-\xi)\|+\|\pi(g_{1}...g_{j})\xi-\xi\|\leq^{\text{The
induction assumption}}$ $\displaystyle 2\max_{1\leq i\leq
k}\|\xi-\eta_{i}\|+2j\max_{1\leq i\leq k}\|\xi-\eta_{i}\|=2(j+1)\max_{1\leq
i\leq k}\|\xi-\eta_{i}\|.$

By the assumption of bounded generation, every $g\in G$ can be written as
$g=g_{1}...g_{\nu}$ with $g_{1},...,g_{\nu}\in\bigcup_{i=1}^{k}H_{i}$ and thus
it follows that for every $g\in G$ and every
$\xi\in\operatorname{\mathbb{E}}$,

$\|\pi(g)\xi-\xi\|\leq 2\nu\max_{1\leq i\leq k}\|\xi-\eta_{i}\|,$

as needed. ∎

######  Theorem 4.3.

Let $G$ be a locally compact group and $H_{1},...,H_{k}<G$ subgroups that
boundedly generate $G$. If the pairs $(G,H_{1}),...,(G,H_{k})$ has relative
property $(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach
space $\operatorname{\mathbb{E}}$, then $G$ has property
$(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach space
$\operatorname{\mathbb{E}}$.

###### Proof.

By Observation 3.2, we need to show that for every uniformly convex Banach
space $\operatorname{\mathbb{E}}$ and every $\pi:G\rightarrow
O(\operatorname{\mathbb{E}})$ with
$\operatorname{\mathbb{E}}^{\pi(G)}=\\{0\\}$ there is a compact set
$K\subseteq G$ and $\varepsilon>0$ such that for every unit vector
$\xi\in\operatorname{\mathbb{E}}$,

$\sup_{g\in K}\|\pi(g)\xi-\xi\|\geq\varepsilon.$

Denote $\nu=\nu(H_{1},...,H_{k})\in\mathbb{N}$ as in the definition above. By
assumption, there are compact sets $K_{1},...,K_{k}$ and constants
$\varepsilon_{1},...,\varepsilon_{k}>0$ such that for every $i=1,...,k$ and
every unit vector $\xi\in\operatorname{\mathbb{E}}$ if

$\sup_{g\in K_{i}}\|\pi(g)\xi-\xi\|<\varepsilon_{i},$

then there is $\eta_{i}\in\operatorname{\mathbb{E}}^{\pi(H_{i})}$ such that

$\|\xi-\eta_{i}\|\leq\frac{1}{4(\nu+1)}.$

Denote $K=\bigcup_{i=1}^{k}K_{i}$ and
$\varepsilon=\min\\{\varepsilon_{1},...,\varepsilon_{k}\\}$. We will show that
for this choice of $K,\varepsilon$ it holds for every unit vector
$\xi\in\operatorname{\mathbb{E}}$ that

$\sup_{g\in K}\|\pi(g)\xi-\xi\|\geq\varepsilon.$

Assume towards contradiction that there is a unit vector
$\xi\in\operatorname{\mathbb{E}}$ such that

$\sup_{g\in K}\|\pi(g)\xi-\xi\|<\varepsilon.$

Thus, for every $i=1,...,k$ there is
$\eta_{i}\in\operatorname{\mathbb{E}}^{\pi(H_{i})}$ such that
$\|\xi-\eta_{i}\|<\frac{1}{4(\nu+1)}$. Applying Lemma 4.2, it follows that for
every $g\in G$,

$\|\pi(g)\xi-\xi\|\leq\frac{\nu}{2(\nu+1)}\leq\frac{1}{2}.$

Thus the orbit of $\xi$ in $\operatorname{\mathbb{E}}$ under the action of $G$
is contained in a closed ball of radius $\frac{1}{2}$ around $\xi$. Denote $C$
to be the closure of the convex hull of the orbit of $\xi$. Recall that $\xi$
is a unit vector and thus $0\notin C$. By uniform convexity there is a unique
vector with a minimal norm in $C$ and thus this vector is fixed by the $\pi$
action of $G$. It follows that
$C\cap\operatorname{\mathbb{E}}^{\pi(G)}\neq\emptyset$ which contradicts the
assumption that $\operatorname{\mathbb{E}}^{\pi(G)}\neq\\{0\\}$. ∎

##  5\. Averaging operations on $\rm H_{3}(\mathbb{Z})$

In this section, we will prove norm bounds on averaging operations on the
Heisenberg group that are needed in our proof of relative Banach property (T)
stated in the introduction.

For every $k\in\mathbb{N}\cup\\{0\\}$, we define
$X_{k},Y_{k},Z_{k}\in\operatorname{Prob}_{c}(\rm H_{3}(\mathbb{Z}))$ by

$X_{k}=\frac{e+x^{2^{k}}}{2},Y_{k}=\frac{e+y^{2^{k}}}{2},Z_{k}=\frac{e+z^{2^{k}}}{2}.$

Also, for $d\in\mathbb{N}$, we define

$X^{d}=\frac{1}{2^{d}}\sum_{a=0}^{2^{d}-1}x^{a},Y^{d}=\frac{1}{2^{d}}\sum_{b=0}^{2^{d}-1}y^{b},Z^{d}=\frac{1}{2^{d}}\sum_{c=0}^{2^{d}-1}z^{c}.$

######  Observation 5.1.

For every $d\in\mathbb{N}$ it holds that

$X^{d}=\prod_{a=0}^{d-1}X_{a},Y^{d}=\prod_{b=0}^{d-1}Y_{b},Z^{d}=\prod_{c=0}^{d-1}Z_{c}.$

######  Theorem 5.2.

Let $d\in\mathbb{N}$ be a constant and $A,B\subseteq\\{0,...,d-1\\}$ be sets
such that $(\max A)(\max B)\leq d-2$, then for every Banach space
$\operatorname{\mathbb{E}}$ and every isometric linear representation
$\pi:H_{3}(\mathbb{Z})\rightarrow O(\operatorname{\mathbb{E}})$ it holds that

$\left\|\pi\left(\left(\left(\prod_{a\in A}X_{a}\right)\left(\prod_{b\in
B}Y_{b}\right)-\left(\prod_{b\in B}Y_{b}\right)\left(\prod_{a\in
A}X_{a}\right)\right)Z^{d}\right)\right\|\leq
8\left(\frac{1}{2}\right)^{d-\max A-\max B}.$

In particular, for $d_{1},d_{2},d_{3}\in\mathbb{N}\cup\\{0\\}$, if
$d_{1}+d_{2}\leq d_{3}-2$, then for any class of Banach spaces $\mathcal{E}$
it holds that

$\left\|\left(X^{d_{1}}Y^{d_{2}}-Y^{d_{2}}X^{d_{1}}\right)Z^{d}\right\|_{\mathcal{U}(\mathcal{E})}\leq
8\left(\frac{1}{2}\right)^{d_{3}-(d_{1}+d_{2})}.$

###### Proof.

We note that

$\displaystyle\left(\prod_{a\in A}X_{a}\right)\left(\prod_{b\in
B}Y_{b}\right)-\left(\prod_{b\in B}Y_{b}\right)\left(\prod_{a\in
A}X_{a}\right)=$
$\displaystyle\frac{1}{2^{|A|+|B|}}\sum_{f\in\\{0,1\\}^{A},h\in\\{0,1\\}^{B}}x^{\sum_{k\in
A}f(k)2^{k}}y^{\sum_{l\in B}h(l)2^{l}}-y^{\sum_{l\in B}h(l)2^{l}}x^{\sum_{k\in
A}f(k)2^{k}}=$
$\displaystyle\frac{1}{2^{|A|+|B|}}\sum_{f\in\\{0,1\\}^{A},h\in\\{0,1\\}^{B}}x^{\sum_{k\in
A}f(k)2^{k}}y^{\sum_{l\in B}h(l)2^{l}}(e-z^{(\sum_{k\in
A}f(k)2^{k})(\sum_{l\in B}h(l)2^{l})}).$

Note that for every $f\in\\{0,1\\}^{A},h\in\\{0,1\\}^{B}$ it holds that

$(\sum_{k\in A}f(k)2^{k})(\sum_{l\in B}h(l)2^{l})\leq 2^{\max A+\max B+2}.$

Thus, it is enough to show that for every $1\leq m\leq 2^{\max A+\max B+2}$ it
holds that

$\|\pi((e-z^{m})Z^{d})\|\leq\frac{2m}{2^{d}}\leq\left(\frac{1}{2}\right)^{d-\max
A-\max B-3},$

but this follows immediately from the fact that

$(e-z^{m})Z^{d}=\frac{1}{2^{d}}\left(\sum_{k=0}^{m-1}z^{k}-\sum_{k=2^{d}}^{2^{d}+m-1}z^{k}\right).$

∎

######  Lemma 5.3.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function. There is a constant
$0\leq r_{0}<1$ such that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
every $k,m\in\mathbb{N}\cup\\{0\\}$ and every
$\zeta\in\operatorname{\mathbb{E}}$, if
$\|\pi(e-z^{m})\zeta\|\geq\frac{1}{2}\|\zeta\|$, then

$\left\|\pi\left(\frac{e+xz^{k}}{2}\frac{e+y^{m}}{2}\right)\zeta\right\|\leq
r_{0}\|\zeta\|.$

###### Proof.

We will show that the needed inequality holds for
$r_{0}=r_{0}(\delta_{0},\frac{1}{2})$ where this is the constant of Corollary
2.2.

Fix
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
$k,m\in\mathbb{N}\cup\\{0\\}$ and $\zeta\in\operatorname{\mathbb{E}}$ such
that $\|\pi(e-z^{m})\zeta\|\geq\frac{1}{2}\|\zeta\|$.

We note that

$\displaystyle\frac{e+xz^{k}}{2}\frac{e+y^{m}}{2}=\frac{1}{2}\left(\frac{e+xz^{k}}{2}\right)+\frac{1}{2}\left(\frac{y^{m}+xy^{m}z^{k}}{2}\right)=$
$\displaystyle\frac{1}{2}\left(\frac{e+xz^{k}}{2}\right)+\frac{y^{m}}{2}\left(\frac{e+y^{-m}xy^{m}z^{k}}{2}\right)=\frac{1}{2}\left(\frac{e+xz^{k}}{2}\right)+\frac{y^{m}}{2}\left(\frac{e+xz^{k}z^{m}}{2}\right).$

Thus

$\displaystyle\left\|\pi\left(\frac{e+xz^{k}}{2}\frac{e+y^{m}}{2}\right)\zeta\right\|\leq\frac{1}{2}\left\|\pi\left(\frac{e+xz^{k}}{2}\right)\zeta\right\|+\frac{1}{2}\left\|\pi(y^{m})\pi\left(\frac{e+xz^{k}z^{m}}{2}\right)\zeta\right\|\leq^{\|\pi(y^{m})\|=1}$
$\displaystyle\frac{1}{2}\left\|\frac{I+\pi(xz^{k})}{2}\zeta\right\|+\frac{1}{2}\left\|\frac{I+\pi(xz^{k})\pi(z^{m})}{2}\zeta\right\|.$

Denote $T=\pi(xz^{k})$ and $S=\pi(z^{m})$. Note that $T,S\in
O(\operatorname{\mathbb{E}})$ are commuting operators and that
$\|(I-S)\zeta\|\geq\frac{1}{2}\|\zeta\|$. Thus the conditions of Corollary 2.2
and it follows that

$\frac{1}{2}\left\|\frac{I+\pi(xz^{k})}{2}\zeta\right\|+\frac{1}{2}\left\|\frac{I+\pi(xz^{k})\pi(z^{m})}{2}\zeta\right\|=\frac{1}{2}\left\|\frac{I+T}{2}\zeta\right\|+\frac{1}{2}\left\|\frac{I+TS}{2}\zeta\right\|\leq
r_{0}\|\zeta\|,$

as needed. ∎

######  Lemma 5.4.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function. There is a constant
$0\leq r_{1}<1$ such that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
every $n,m\in\mathbb{N}$ such that $1\leq m\leq 2^{n-1}$ and every
$\zeta\in\operatorname{\mathbb{E}}$, if
$\|\pi(e-z^{m})\zeta\|\geq\frac{1}{2}\|\zeta\|$, then

$\left\|\pi\left(X_{0}Y^{n}\right)\zeta\right\|\leq r_{1}\|\zeta\|.$

###### Proof.

Fix
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$
and $n,m\in\mathbb{N}$ as above. Let $0\leq r_{0}<1$ be the constant of Lemma
5.3. We will show that for $r_{1}=\frac{r_{0}+1}{2}$ the needed inequality
holds.

Fix $\zeta\in\operatorname{\mathbb{E}}$ such that
$\|\pi(e-z^{m})\zeta\|\geq\frac{1}{2}\|\zeta\|$.

Denote $t=\lfloor\frac{2^{n}}{2m}\rfloor$ and $k=2^{n}-2mt$ and note that
$t\geq 1,0\leq k\leq\min\\{2^{n-1},2m-1\\}$ and $2^{n}=2mt+k$. It follows that

$\displaystyle
Y^{n}=\frac{1}{2^{n}}\sum_{b=0}^{2^{n}-1}y^{b}=\frac{2mt}{2^{n}}\left(\frac{1}{2mt}\sum_{b=0}^{2mt-1}y^{b}\right)+\frac{k}{2^{n}}\left(\frac{1}{k}\sum_{b=2mt}^{2^{n}-1}y^{b}\right).$

We claim it is sufficient to prove that

(2)
$\left\|\pi\left(X_{0}\left(\frac{1}{2mt}\sum_{b=0}^{2mt-1}y^{b}\right)\right)\zeta\right\|\leq
r_{0}\|\zeta\|.$

Indeed, note that

$\left\|\pi\left(X_{0}\left(\frac{1}{k}\sum_{b=2mt}^{2^{n}-1}y^{b}\right)\right)\right\|\leq
1$

and thus if (2) holds, then

$\displaystyle\left\|\pi\left(X_{0}Y^{n}\right)\zeta\right\|\leq\frac{2mt}{2^{n}}\left\|\pi\left(X_{0}\left(\frac{1}{2mt}\sum_{b=0}^{2mt-1}y^{b}\right)\right)\zeta\right\|+\frac{k}{2^{n}}\left\|\pi\left(X_{0}\left(\frac{1}{k}\sum_{b=2mt}^{2^{n}-1}y^{b}\right)\right)\zeta\right\|\leq$
$\displaystyle\left(\frac{2mt}{2^{n}}r_{0}+\frac{k}{2^{n}}\right)\|\zeta\|\leq^{k\leq
2^{n-1}}\frac{r_{0}+1}{2}\|\zeta\|,$

as needed.

We will finish the proof by proving (2). We note that

$\displaystyle\frac{1}{2mt}\sum_{b=0}^{2mt-1}y^{b}=\left(\frac{1}{m}\sum_{b_{1}=0}^{m-1}y^{b_{1}}\right)\left(\frac{1}{t}\sum_{b_{2}=0}^{t-1}y^{2mb_{2}}\right)\left(\frac{e+y^{m}}{2}\right).$

Thus,

$\displaystyle
X_{0}\left(\frac{1}{2mt}\sum_{b=0}^{2mt-1}y^{b}\right)=\frac{e+x}{2}\left(\frac{1}{m}\sum_{b_{1}=0}^{m-1}y^{b_{1}}\right)\left(\frac{1}{t}\sum_{b_{2}=0}^{t-1}y^{2mb_{2}}\right)\left(\frac{e+y^{m}}{2}\right)=$
$\displaystyle\frac{1}{mt}\sum_{b_{1}=0}^{m-1}\sum_{b_{2}=0}^{t-1}y^{b_{1}+2mb_{2}}\frac{e+y^{-(b_{1}+2mb_{2})}xy^{b_{1}+2mb_{2}}}{2}\left(\frac{e+y^{m}}{2}\right)=$
$\displaystyle\frac{1}{mt}\sum_{b_{1}=0}^{m-1}\sum_{b_{2}=0}^{t-1}y^{b_{1}+2mb_{2}}\frac{e+xz^{b_{1}+2mb_{2}}}{2}\left(\frac{e+y^{m}}{2}\right).$

Using the fact that $\|\pi(y^{b_{1}+2mb_{2}})\|=1$ for every
$b_{1},b_{2}\in\mathbb{N}$, it follows that

$\displaystyle\left\|\pi\left(X_{0}\left(\frac{1}{2mt}\sum_{b=0}^{2mt-1}y^{b}\right)\right)\zeta\right\|\leq$
$\displaystyle\frac{1}{mt}\sum_{b_{1}=0}^{m-1}\sum_{b_{2}=0}^{t-1}\left\|\pi\left(\frac{e+xz^{b_{1}+2mb_{2}}}{2}\right)\pi\left(\frac{e+y^{m}}{2}\right)\zeta\right\|\leq^{\text{Lemma
}\ref{product inequality
lemma}}\frac{1}{mt}\sum_{b_{1}=0}^{m-1}\sum_{b_{2}=0}^{t-1}r_{0}\|\zeta\|=r_{0}\|\zeta\|,$

and the proof of (2) is concluded. ∎

######  Theorem 5.5.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function. Let $0\leq r_{1}<1$ be
the constant given in Lemma 5.4 above. For every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
every $n\in\mathbb{N},n\geq 2$ and every $\xi\in\operatorname{\mathbb{E}}$, it
holds that

$\left\|\pi\left(X_{0}Y^{n}(e-Z_{0})\right)\xi\right\|\leq\max\left\\{r_{1}\|\pi(e-Z_{0})\xi\|,\frac{1}{2^{n-2}}\|\xi\|\right\\}.$

###### Proof.

Fix
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
$n\in\mathbb{N},n\geq 2$ and $\xi\in\operatorname{\mathbb{E}}$. If
$\|\pi(e-Z_{0})\xi\|\leq\frac{1}{2^{n-2}}\|\xi\|$, then

$\left\|\pi\left(X_{0}Y^{n}(e-Z_{0})\right)\xi\right\|\leq\|\pi(e-Z_{0})\xi\|\leq\frac{1}{2^{n-2}}\|\xi\|,$

and we are done.

Assume that $\|\pi(e-Z_{0})\xi\|>\frac{1}{2^{n-2}}\|\xi\|$. Then

$\displaystyle\frac{1}{2^{n-1}}\sum_{m=0}^{2^{n-1}-1}\|\pi((e-z^{m})(e-Z_{0}))\xi\|\geq\left\|\pi\left(\left(e-\frac{1}{2^{n-1}}\sum_{m=0}^{2^{n-1}-1}z^{m}\right)(e-Z_{0})\right)\xi\right\|\geq$
$\displaystyle\|\pi(e-Z_{0})\xi\|-\left\|\pi\left(\left(\frac{1}{2^{n-1}}\sum_{m=0}^{2^{n-1}-1}z^{m}\right)(e-Z_{0})\right)\xi\right\|\geq^{e-Z_{0}=\frac{e-z}{2}}$
$\displaystyle\left\|\pi(e-Z_{0})\xi\right\|-\frac{1}{2^{n-1}}\left\|\pi\left(\frac{e-z^{2^{n-1}}}{2}\right)\xi\right\|\geq$
$\displaystyle\left\|\pi(e-Z_{0})\xi\right\|-\frac{1}{2^{n-1}}\|\xi\|>^{\|\pi(e-Z_{0})\xi\|>\frac{1}{2^{n-2}}\|\xi\|}$
$\displaystyle\left\|\pi(e-Z_{0})\xi\right\|-\frac{1}{2}\left\|\pi(e-Z_{0})\xi\right\|=\frac{1}{2}\left\|\pi(e-Z_{0})\xi\right\|.$

It follows that for $\zeta=\pi(e-Z_{0})\xi$, there is $1\leq m\leq 2^{n-1}-1$
such that $\|\pi(e-z^{m})\zeta\|\geq\frac{1}{2}\|\zeta\|$. Thus, by Lemma 5.4,

$\|\pi\left(X_{0}Y^{n}\right)\zeta\|\leq r_{1}\|\zeta\|,$

i.e.,

$\left\|\pi\left(X_{0}Y^{n}(e-Z_{0})\right)\xi\right\|\leq
r_{1}\|\pi(e-Z_{0})\xi\|,$

as needed. ∎

######  Corollary 5.6.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function. Let $0\leq r_{1}<1$ be
the constant given in Lemma 5.4 above. For every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
every $n\in\mathbb{N},n\geq 2$, every $a_{0},b_{0}\in\mathbb{N}\cup\\{0\\}$
and every $\xi\in\operatorname{\mathbb{E}}$, it holds that

$\left\|\pi\left(\left(X_{a_{0}}\left(\prod_{b=0}^{n-1}Y_{b_{0}+b}\right)\right)(e-Z_{a_{0}+b_{0}})\right)\xi\right\|\leq\max\left\\{r_{1}\|\pi(e-Z_{a_{0}+b_{0}})\xi\|,\frac{1}{2^{n-2}}\|\xi\|\right\\}.$

###### Proof.

Fix $(\pi,\operatorname{\mathbb{E}}),n,a_{0},b_{0}$ as above.

Let $H<H_{3}(\mathbb{Z})$ be the subgroup $H=\langle
x^{2^{a_{0}}},y^{2^{b_{0}}}\rangle$. We note that $H$ is isomorphic to
$H_{3}(\mathbb{Z})$ via the isomorphism $\Phi:H_{3}(\mathbb{Z})\rightarrow H$
induced by $\Phi(x)=x^{2^{a_{0}}},\Phi(y)=y^{2^{b_{0}}}$. Note that (by
extending $\Phi$ linearly)

$\Phi(X_{0})=X_{a_{0}},\Phi(Y^{n})=\prod_{b=0}^{n-1}Y_{b_{0}+b}.$

Also note that

$\Phi(z)=\Phi(x^{-1}y^{-1}xy)=x^{-2^{a_{0}}}y^{-2^{b_{0}}}x^{2^{a_{0}}}y^{2^{b_{0}}}=z^{2^{a_{0}+b_{0}}},$

and thus $\Phi(Z_{0})=Z_{a_{0}+b_{0}}$.

Define a new representation $(\pi_{0},\operatorname{\mathbb{E}})$ of
$H_{3}(\mathbb{Z})$ by $\pi_{0}=\pi\circ\Phi$. Let
$\xi\in\operatorname{\mathbb{E}}$, then

$\displaystyle\left\|\pi\left(\left(X_{a_{0}}\left(\prod_{b=0}^{n-1}Y_{b_{0}+b}\right)\right)(e-Z_{a_{0}+b_{0}})\right)\xi\right\|=\left\|\pi_{0}\left(\left(X_{0}Y^{n}\right)(e-Z_{0})\right)\xi\right\|\leq$
$\displaystyle\max\left\\{r_{1}\|\pi_{0}(e-Z_{0})\xi\|,\frac{1}{2^{n-2}}\|\xi\|\right\\}=\max\left\\{r_{1}\|\pi(e-Z_{a_{0}+b_{0}})\xi\|,\frac{1}{2^{n-2}}\|\xi\|\right\\},$

as needed. ∎

######  Theorem 5.7.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function. There are constants
$0\leq r_{2}<1,C>0$ such that for every $d_{1},d_{2},d_{3}\in\mathbb{N}$ with
$d_{1},d_{2}\leq d_{3},d_{1}+d_{2}\geq d_{3}$, it holds that

$\left\|X^{d_{1}}Y^{d_{2}}\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq
Cr_{2}^{\sqrt{d_{1}+d_{2}-d_{3}}}.$

###### Proof.

Let $r_{1}=r_{1}(\delta_{0})$ be the constant of Corollary 5.6. We will prove
that the inequality stated above holds for
$r_{2}=\max\\{r_{1},\frac{1}{\sqrt{2}}\\}$.

Fix $d_{1},d_{2},d_{3}\in\mathbb{N}$ as above and denote
$t=2\lfloor\frac{\sqrt{d_{1}+d_{2}-d_{3}}}{2}\rfloor$. We note that
$t\leq\sqrt{d_{1}+d_{2}-d_{3}}\leq t+2$ and thus it is enough to prove that
there is a constant $C^{\prime}$ such that

$\left\|X^{d_{1}}Y^{d_{2}}\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq
C^{\prime}r_{2}^{t}.$

Without loss of generality, we can assume that $t\geq 4$ (for $t<4$, the
constant $C^{\prime}$ can be chosen to be large enough such that
$C^{\prime}r_{2}^{4}\geq 2$ and the needed inequality holds trivially).

Denote

$A_{0}=\\{d_{1}-1-kt:0\leq k\leq t-1\\}.$

Note that for every $0\leq k\leq t-1$

$\displaystyle d_{1}-1\geq
d_{1}-1-kt>d_{1}-1-t^{2}+t\geq^{t\leq\sqrt{d_{1}+d_{2}-d_{3}}}$ $\displaystyle
d_{1}-1-(d_{1}+d_{2}-d_{3})+t=d_{3}-d_{2}+t-1\geq^{d_{3}\geq d_{2},t\geq 4}0,$

and thus $A_{0}\subseteq\\{0,...,d_{1}-1\\}$.

Also, denote

$B_{0}=\bigcup_{k=0}^{t-1}\left\\{d_{3}-d_{1}+1+kt+j:0\leq
j\leq\frac{t}{2}-1\right\\},$

(note that $t$ is always even and thus $\frac{t}{2}\in\mathbb{N}$). Note that
for every $0\leq k\leq t-1$ and every $0\leq j\leq\frac{t}{2}-1$ it holds that

$d_{3}-d_{1}+1+kt+j\geq d_{3}-d_{1}+1\geq^{d_{3}\geq d_{1}}0$

and

$\displaystyle d_{3}-d_{1}+1+kt+j\leq
d_{3}-d_{1}+1+t^{2}-\frac{t}{2}\leq^{t\leq\sqrt{d_{1}+d_{2}-d_{3}}}d_{3}-d_{1}+1+(d_{1}+d_{2}-d_{3})-\frac{t}{2}=$
$\displaystyle d_{2}+1-\frac{t}{2}\leq^{t\geq 4}d_{2}-1.$

It follows that $B_{0}\subseteq\\{0,...,d_{2}-1\\}$.

Thus,

$\displaystyle\left\|X^{d_{1}}Y^{d_{2}}\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|\left(\prod_{a\in\\{0,...,d_{1}-1\\}\setminus
A_{0}}X_{a}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\left\|\left(\prod_{a\in
A_{0}}X_{a}\right)Y^{d_{2}}\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|\left(\prod_{a\in A_{0}}X_{a}\right)\left(\prod_{b\in
B_{0}}Y_{b}\right)\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\left\|\left(\prod_{b\in\\{0,...,d_{2}-1\\}\setminus
B_{0}}Y_{b}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|\left(\prod_{a\in A_{0}}X_{a}\right)\left(\prod_{b\in
B_{0}}Y_{b}\right)\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}.$

It follows that it is enough to prove that there is a constant $C^{\prime}$
such that

$\left\|\left(\prod_{a\in A_{0}}X_{a}\right)\left(\prod_{b\in
B_{0}}Y_{b}\right)\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq
C^{\prime}r_{2}^{t},$

i.e., it is enough to prove that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{E}_{uc}(\delta_{0})$ and every
$\xi\in\operatorname{\mathbb{E}}$, it holds that

(3) $\left\|\pi\left(\left(\prod_{a\in A_{0}}X_{a}\right)\left(\prod_{b\in
B_{0}}Y_{b}\right)\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right)\xi\right\|\leq
C^{\prime}r_{2}^{t}\|\xi\|.$

Fix $(\pi,\operatorname{\mathbb{E}})\in\mathcal{E}_{uc}(\delta_{0})$ and
$\xi\in\operatorname{\mathbb{E}}$. For $1\leq i\leq t-1$, define the sets

$A_{i}=\\{d_{1}-1-kt:i\leq k\leq t-1\\},$
$B_{i}=\bigcup_{k=i}^{t-1}\left\\{d_{3}-d_{1}+1+kt+j:0\leq
j\leq\frac{t}{2}-1\right\\}.$

Also define $A_{t}=B_{t}=\emptyset$. For $0\leq i\leq t$ denote

$I_{i}=\left\|\pi\left(\left(\prod_{a\in A_{i}}X_{a}\right)\left(\prod_{b\in
B_{i}}Y_{b}\right)\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right)\xi\right\|.$

We claim that in order to prove (3), it is sufficient to show that for every
$0\leq i\leq t-1$ it holds that

(4) $I_{i}\leq 36\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|+r_{1}I_{i+1},$

where $r_{1}$ is the constant of Corollary 5.6. Indeed, if (4) holds, then

$\displaystyle I_{0}\leq
36\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|(1+r_{1}+...+r_{1}^{t-1})+r_{1}^{t}I_{t}\leq$
$\displaystyle\frac{36}{1-r_{1}}\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|+r_{1}^{t}\|\pi(Z^{d_{3}}-Z^{d_{3}+1})\xi\|\leq$
$\displaystyle\frac{36}{1-r_{1}}\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|+2r_{1}^{t}\|\xi\|\leq\left(\frac{36}{1-r_{1}}+2\right)r_{2}^{t},$

as needed.

We are left to prove (4). Fix $0\leq i\leq t-1$. For $0\leq k\leq t-1$, denote

$B_{k}^{\prime}=\left\\{d_{3}-d_{1}+1+kt+j:0\leq j\leq\frac{t}{2}-1\right\\},$

thus $B_{i}=\bigcup_{k=i}^{t-1}B_{k}^{\prime}$. We note that

$d_{3}-\max A_{i+1}-\max
B_{i}^{\prime}=d_{3}-(d_{1}-1-(i+1)t)-(d_{3}-d_{1}+it+\frac{t}{2})=\frac{t}{2}+1.$

Therefore, by Theorem 5.2,

$\displaystyle\left\|\pi\left(\left(\left(\prod_{a\in
A_{i+1}}X_{a}\right)\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)-\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)\left(\prod_{a\in
A_{i+1}}X_{a}\right)\right)Z^{d_{3}}\right)\right\|\leq$ $\displaystyle
8\left(\frac{1}{2}\right)^{d_{3}-\max A_{i+1}-\max
B_{i}^{\prime}}=16\left(\frac{1}{\sqrt{2}}\right)^{t}.$

Thus,

$\displaystyle\left\|\pi\left(\left(\prod_{a\in
A_{i}}X_{a}\right)\left(\prod_{b\in
B_{i}}Y_{b}\right)Z^{d_{3}}-\left(X_{d_{1}-1-it}\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)\right)\left(\prod_{a\in
A_{i+1}}X_{a}\right)\left(\prod_{b\in
B_{i+1}}Y_{b}\right)Z^{d_{3}}\right)\right\|\leq$ $\displaystyle
16\left(\frac{1}{\sqrt{2}}\right)^{t}.$

By Observation 5.1, $Z^{d_{3}+1}=Z^{d_{3}}Z_{d_{3}}$ and thus
$Z^{d_{3}}-Z^{d_{3}+1}=Z^{d_{3}}(e-Z_{d_{3}})$. Using this and the inequality
stated above, we can deduce

$I_{i}=\left\|\pi\left(\left(\prod_{a\in A_{i}}X_{a}\right)\left(\prod_{b\in
B_{i}}Y_{b}\right)Z^{d_{3}}\left(e-Z_{d_{3}}\right)\right)\xi\right\|\leq\left\|\pi\left(\left(X_{d_{1}-1-it}\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)\right)\left(\prod_{a\in
A_{i+1}}X_{a}\right)\left(\prod_{b\in
B_{i+1}}Y_{b}\right)Z^{d_{3}}(e-Z_{d_{3}})\right)\xi\right\|+{16\left(\frac{1}{\sqrt{2}}\right)^{t}\|\pi\left(e-Z_{d_{3}}\right)\xi\|}\leq\left\|\pi\left(\left(X_{d_{1}-1-it}\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)(e-Z_{d_{3}})\right)\left(\prod_{a\in
A_{i+1}}X_{a}\right)\left(\prod_{b\in
B_{i+1}}Y_{b}\right)Z^{d_{3}}\right)\xi\right\|+32\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|=\\\
\left\|\pi\left(X_{d_{1}-1-it}\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)(e-Z_{d_{3}})\right)\pi\left(\left(\prod_{a\in
A_{i+1}}X_{a}\right)\left(\prod_{b\in
B_{i+1}}Y_{b}\right)Z^{d_{3}}\right)\xi\right\|+32\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|.$

Denote $\xi^{\prime}=\pi\left(\left(\prod_{a\in
A_{i+1}}X_{a}\right)\left(\prod_{b\in
B_{i+1}}Y_{b}\right)Z^{d_{3}}\right)\xi$. With this notation, we showed that

$I_{i}\leq\left\|\pi\left(X_{d_{1}-1-it}\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)(e-Z_{d_{3}})\right)\xi^{\prime}\right\|+32\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|.$

We note that

$\left\|\pi(e-Z_{d_{3}})\xi^{\prime}\right\|=\left\|\pi\left(\left(\prod_{a\in
A_{i+1}}X_{a}\right)\left(\prod_{b\in
B_{i+1}}Y_{b}\right)\left(Z^{d_{3}}-Z^{d_{3}+1}\right)\right)\xi\right\|=I_{i+1}.$

Thus, in order to prove (4), we are left to prove that

$\left\|\pi\left(X_{d_{1}-1-it}\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)(e-Z_{d_{3}})\right)\xi^{\prime}\right\|\leq
4\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|+r_{1}\|\pi(e-Z_{d_{3}})\xi^{\prime}\|.$

We recall that

$\displaystyle\left\|\pi\left(X_{d_{1}-1-it}\left(\prod_{b\in
B_{i}^{\prime}}Y_{b}\right)(e-Z_{d_{3}})\right)\xi^{\prime}\right\|=\left\|\pi\left(X_{d_{1}-1-it}\left(\prod_{b=0}^{\frac{t}{2}-1}Y_{(d_{3}-d_{1}+1+it)+b}\right)(e-Z_{d_{3}})\right)\xi^{\prime}\right\|.$

Denoting $a_{0}=d_{1}-1-it,b_{0}=d_{3}-d_{1}+1+it,n=\frac{t}{2}$ and applying
Corollary 5.6 (noting that $a_{0}+b_{0}=d_{3}$ and that we assumed that
$n=\frac{t}{2}\geq 2$) yields that

$\displaystyle\left\|\pi\left(X_{d_{1}-1-it}\left(\prod_{b=0}^{\frac{t}{2}-1}Y_{(d_{3}-d_{1}+1+it)+b}\right)(e-Z_{d_{3}})\right)\xi^{\prime}\right\|\leq\max\left\\{r_{1}\|\pi(e-Z_{d_{3}})\xi^{\prime}\|,\frac{1}{2^{\frac{t}{2}-2}}\|\xi^{\prime}\|\right\\}\leq$
$\displaystyle
4\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi^{\prime}\|+r_{1}\|\pi(e-Z_{d_{3}})\xi^{\prime}\|\leq
4\left(\frac{1}{\sqrt{2}}\right)^{t}\|\xi\|+r_{1}\|\pi(e-Z_{d_{3}})\xi^{\prime}\|,$

as needed. ∎

######  Corollary 5.8.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be a function. There are constants
$0\leq r<1,C>0$ such that for every $d_{1},d_{2},d_{3},d_{4}\in\mathbb{N}$
such that $d_{1},d_{2}\geq\frac{1}{4}\min\\{d_{3},d_{4}\\}$ and
$d_{1}+d_{2}-\max\\{d_{3},d_{4}\\}\geq\frac{1}{4}\min\\{d_{3},d_{4}\\}$, it
holds that

$\left\|X^{d_{1}}Y^{d_{2}}\left(Z^{d_{3}}-Z^{d_{4}}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq|d_{4}-d_{3}|Cr^{\sqrt{\min\\{d_{3},d_{4}\\}}},$

and

$\left\|Y^{d_{1}}X^{d_{2}}\left(Z^{d_{3}}-Z^{d_{4}}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq|d_{4}-d_{3}|Cr^{\sqrt{\min\\{d_{3},d_{4}\\}}}.$

###### Proof.

Without loss of generality, we will assume that $d_{4}>d_{3}$.

We will start by proving the first inequality.

Let $r_{2},C$ be the constants of Theorem 5.7 and take $r=\sqrt{r_{2}}$. We
note that it is enough to prove that for any $0\leq j\leq d_{4}-d_{3}-1$ it
holds that

$\left\|X^{d_{1}}Y^{d_{2}}\left(Z^{d_{3}+j}-Z^{d_{3}+j+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq
Cr^{\sqrt{d_{3}}}.$

If $d_{1},d_{2}\leq d_{3}+j$ this inequality follow immediately from Theorem
5.7. Otherwise, either $d_{2}>d_{3}+j$ or $d_{1}>d_{3}+j$ (or both) and thus

$\min\\{d_{1},d_{3}+j\\}+\min\\{d_{2},d_{3}+j\\}-d_{3}-j\geq\frac{1}{4}d_{3}.$

In this case, we apply Theorem 5.7 replacing $d_{i}$ with
$\min\\{d_{i},d_{3}+j\\}$ for $i=1,2$:

$\displaystyle\left\|X^{d_{1}}Y^{d_{2}}\left(Z^{d_{3}+j}-Z^{d_{3}+j+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|X^{\min\\{d_{1},d_{3}+j\\}}Y^{\min\\{d_{2},d_{3}+j\\}}\left(Z^{d_{3}+j}-Z^{d_{3}+j+1}\right)\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq
Cr^{\sqrt{d_{3}}},$

as needed.

In order to prove the second inequality, we need to prove that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$
it holds that

$\left\|\pi\left(Y^{d_{2}}X^{d_{1}}\left(Z^{d_{3}}-Z^{d_{4}}\right)\right)\right\|\leq(d_{4}-d_{3})Cr^{\sqrt{d_{3}}}.$

Fix
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$.
Let $\Phi:\rm H_{3}(\mathbb{Z})\rightarrow\rm H_{3}(\mathbb{Z})$ be the
isomorphism induced by $\Phi(x^{-1})=y$ and $\Phi(y)=x$. We note that

$\Phi(z)=\Phi(x^{-1}y^{-1}xy)=yx^{-1}y^{-1}x=[x,y^{-1}]^{-1}=z.$

Extending $\Phi$ linearly yields that

$\Phi(Y^{d_{2}})=X^{d_{2}},\Phi(Z^{d_{3}})=Z^{d_{3}},\Phi(Z^{d_{4}})=Z^{d_{4}},$

and

$\Phi(X^{d_{1}})=\frac{1}{2^{d_{1}}}\sum_{b=-2^{d_{1}}+1}^{0}y^{b}=y^{-2^{d_{1}}+1}\left(\frac{1}{2^{d_{1}}}\sum_{b=0}^{2^{d_{1}}-1}y^{b}\right)=y^{-2^{d_{1}}+1}Y^{d_{1}}.$

Define $\pi_{0}=\pi\circ\Phi$, then
$(\pi_{0},\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$
and by the first inequality proven above it follows that

$\displaystyle
Cr^{\sqrt{d_{3}}}\geq\left\|\pi_{0}\left(X^{d_{1}}Y^{d_{2}}\left(Z^{d_{3}}-Z^{d_{4}}\right)\right)\right\|=$
$\displaystyle\left\|\pi\left(\Phi(X^{d_{1}})\Phi(Y^{d_{2}})\left(\Phi(Z^{d_{3}})-\Phi(Z^{d_{4}})\right)\right)\right\|=$
$\displaystyle\left\|\pi(y^{-2^{d_{1}}+1})\pi\left(Y^{d_{1}}X^{d_{2}}\left(Z^{d_{3}}-Z^{d_{4}}\right)\right)\right\|=$
$\displaystyle\left\|\pi\left(Y^{d_{1}}X^{d_{2}}\left(Z^{d_{3}}-Z^{d_{4}}\right)\right)\right\|,$

as needed. ∎

##  6\. Relative Banach property (T) for
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{UT}_{3}(\mathbb{Z}))$ and
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{LT}_{3}(\mathbb{Z}))$

In this section we will prove our main relative Banach property (T) result
stated in the introduction.

For any $\\{i,j,k\\}=\\{1,2,3\\}$, we denote

$H_{i,k}=\langle
e_{i,j}(1),e_{j,k}(1),e_{i,k}(1)\rangle<\operatorname{SL}_{3}(\mathbb{Z}),$
$\widetilde{H}_{i,k}=\langle
x_{i,j}(1),x_{j,k}(1),x_{i,k}(1)\rangle<\operatorname{St}_{3}(\mathbb{Z}),$

For example, $H_{1,3}$ is the group of uni-upper-triangular matrices
$\operatorname{UT}_{3}(\mathbb{Z})$ that appeared in the introduction.

We will prove the following theorem:

######  Theorem 6.1.

For any function $\delta_{0}:(0,2]\rightarrow(0,1]$ and any $1\leq i,k\leq
3,i\neq k$, the pairs $(\operatorname{SL}_{3}(\mathbb{Z}),H_{i,k})$ and
$(\operatorname{St}_{3}(\mathbb{Z}),\widetilde{H}_{i,k})$ have relative
property $(T^{\operatorname{proj}}_{\mathcal{E}_{uc}(\delta_{0})})$.

In particular, for any function $\delta_{0}:(0,2]\rightarrow(0,1]$, the pairs
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{UT}_{3}(\mathbb{Z}))$ and
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{LT}_{3}(\mathbb{Z}))$ have
relative property $(T^{\operatorname{proj}}_{\mathcal{E}_{uc}(\delta_{0})})$.

Below, we will prove this theorem only for the pair
$(\operatorname{SL}_{3}(\mathbb{Z}),UT_{3}(\mathbb{Z}))$. The proof will only
use the Steinberg relations of $\operatorname{SL}_{3}(\mathbb{Z})$ and thus it
applies verbatim to the pair
$(\operatorname{St}_{3}(\mathbb{Z}),\widetilde{H}_{1,3}(\mathbb{Z}))$
(replacing each $e_{i,j}$ with $x_{i,j}$). The proof for any other $H_{i,k}$
(or $\widetilde{H}_{i,k}$ in the case of the Steinberg group) follows from the
case $UT_{3}(\mathbb{Z})=H_{1,3}$ after permuting the indices.

In order to prove this theorem, we define the following: Let $1\leq i,k\leq
3,i\neq k$ and $d\in\mathbb{N}$. Define
$X_{i,k}^{d}\in\operatorname{Prob}_{c}(\operatorname{SL}_{3}(\mathbb{Z}))$ by

$X_{i,k}^{d}=\frac{1}{2^{d}}\sum_{a=0}^{2^{d}-1}e_{i,k}(a).$

Informally, the idea of the proof is to look a product of all the
$X_{i,k}^{d}$’s and to preform “moves” on this product with a small “norm
cost” (such that this “cost” decrease as $d$ increases). We will consider the
following “moves” for $\\{i,j,k\\}=\\{1,2,3\\}$:

  1. (1)

Switch moves on $H_{i,k}$: Replacing
$X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}$ with
$X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}$ and vice-versa.

  2. (2)

Up/down moves on $H_{i,k}$: Replacing
$X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}$ with
$X_{j,k}^{d_{2}}X_{i,k}^{d_{4}}X_{i,j}^{d_{1}}$ where $d_{4}\in\mathbb{N}$
(when $d_{4}>d_{3}$ this will be called an up move and when $d_{4}<d_{3}$ this
will be called a down move).

The following lemma bounds the “norm cost” of these moves:

######  Lemma 6.2.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ and $\\{i,j,k\\}=\\{1,2,3\\}$. For any
$P_{1},P_{1},T\in\operatorname{Prob}_{c}(\operatorname{SL}_{3}(\mathbb{Z}))$
the following holds:

  1. (1)

Switch moves on $H_{i,k}$ have small “norm cost”: For any constants
$d_{1},d_{2},d_{3}\in\mathbb{N}$ with $d_{1}+d_{2}\leq d_{3}-2$ it holds that

$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+8\left(\frac{1}{2}\right)^{d_{3}-(d_{1}+d_{2})},$
$\displaystyle\left\|P_{1}\left(X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+8\left(\frac{1}{2}\right)^{d_{3}-(d_{1}+d_{2})}.$

  2. (2)

Up/down moves on $H_{i,k}$ have small “norm cost”: Let $0\leq r<1,C>0$ be the
constants of Corollary 5.8. For every $d_{1},d_{2},d_{3},d_{4}\in\mathbb{N}$
such that $d_{1},d_{2}\geq\frac{1}{4}\min\\{d_{3},d_{4}\\}$ and
$d_{1}+d_{2}-\max\\{d_{3},d_{4}\\}\geq\frac{1}{4}\min\\{d_{3},d_{4}\\}$, it
holds that

$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{4}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+|d_{4}-d_{3}|Cr^{\sqrt{\min\\{d_{3},d_{4}\\}}},$
$\displaystyle\left\|P_{1}\left(X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{j,k}^{d_{2}}X_{i,k}^{d_{4}}X_{i,j}^{d_{1}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+|d_{4}-d_{3}|Cr^{\sqrt{\min\\{d_{3},d_{4}\\}}}.$

###### Proof.

The bounds of the “norm cost” of switch moves follow directly from Theorem
5.2. We will prove the bound of the first switch move (the proof of second
bound is similar). Let $d_{1},d_{2},d_{3}\in\mathbb{N}$ with $d_{1}+d_{2}\leq
d_{3}-2$, then by Theorem 5.2

$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}-X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right)P_{2}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+\left\|X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}-X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{j,k}^{d_{2}}X_{i,k}^{d_{3}}X_{i,j}^{d_{1}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+8\left(\frac{1}{2}\right)^{d_{3}-(d_{1}+d_{2})}.$

The bounds of the “norm cost” of up/down moves follow directly from Corollary
5.8. We will prove the bound of the first up/down move (the proof of second
bound is similar). Let $d_{1},d_{2},d_{3},d_{4}\in\mathbb{N}$ such that
$d_{1},d_{2}\geq\frac{1}{4}\min\\{d_{3},d_{4}\\}$ and
$d_{1}+d_{2}-\max\\{d_{3},d_{4}\\}\geq\frac{1}{4}\min\\{d_{3},d_{4}\\}$. By
Corollary 5.8,

$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{4}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}-X_{i,j}^{d_{1}}X_{i,k}^{d_{4}}X_{j,k}^{d_{2}}\right)P_{2}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{4}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+\left\|X_{i,j}^{d_{1}}X_{i,k}^{d_{3}}X_{j,k}^{d_{2}}-X_{i,j}^{d_{1}}X_{i,k}^{d_{4}}X_{j,k}^{d_{2}}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|P_{1}\left(X_{i,j}^{d_{1}}X_{i,k}^{d_{4}}X_{j,k}^{d_{2}}\right)P_{2}-T\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+|d_{4}-d_{3}|Cr^{\sqrt{\min\\{d_{3},d_{4}\\}}}.$

∎

For $d\in\mathbb{N}$, we define
$T_{d},S_{d}\in\operatorname{Prob}_{c}(\operatorname{SL}_{3}(\mathbb{Z}))$ as
follows:

$T_{d}=X_{1,2}^{4d}X_{1,3}^{10d}X_{2,3}^{9d}X_{2,1}^{9d}X_{3,1}^{10d}X_{3,2}^{4d},$
$S_{d}=X_{2,3}^{4d}X_{1,3}^{10d}X_{1,2}^{9d}X_{3,2}^{9d}X_{3,1}^{10d}X_{2,1}^{4d}.$

Using the lemma above, we will show that $\|T_{d}-S_{d\pm 1}\|$ is small:

######  Lemma 6.3.

Let $\delta_{0}:(0,2]\rightarrow(0,1]$ be some function. Then there are
constants $0\leq r<1$ and $L>0$ such that for every $d\in\mathbb{N}$ it holds
that

$\left\|T_{d}-S_{d\pm
1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq Ldr^{\sqrt{d}}.$

###### Proof.

We will prove the bound only for $\|T_{d}-S_{d+1}\|$, the proof for
$\|T_{d}-S_{d-1}\|$ is similar. Let $r,C$ be the constants of Corollary 5.8.
Without loss of generality, we can assume that $d>10$ (otherwise, we can
choose $L$ to be large enough such that $Lr^{\sqrt{10}}\geq 2$ and the
inequality holds trivially).

The idea of the proof is to use switch moves and up/down moves to change
$T_{d}$ into $S_{d+1}$ while book-keeping the “norm cost” using Lemma 6.2. We
remark that when preforming up/down moves on $H_{i,k}$, the order of the
product of $X_{i,j}^{d_{1}},X_{j,k}^{d_{2}},X_{i,k}^{d_{3}}$ (where
$\\{i,j,k\\}=\\{1,2,3\\}$) does not matter, since $X_{i,k}^{d_{3}}$ commutes
with $X_{i,j}^{d_{1}}$ and $X_{j,k}^{d_{2}}$.

Step 1 (Up moves on $H_{1,3},H_{3,1}$): We note that we assumed that $d>10$
and thus for $d_{1}=4d,d_{2}=9d,d_{3}=10d,d_{4}=10(d+1)$ the conditions of
Lemma 6.2(2) hold. Preforming up moves on $H_{1,3}$ and $H_{3,1}$, we get by
Lemma 6.2 that

$\displaystyle\left\|T_{d}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}=$
$\displaystyle\left\|\left(X_{1,2}^{4d}X_{1,3}^{10d}X_{2,3}^{9d}\right)\left(X_{2,1}^{9d}X_{3,1}^{10d}X_{3,2}^{4d}\right)-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|\left(X_{1,2}^{4d}X_{1,3}^{10(d+1)}X_{2,3}^{9d}\right)\left(X_{2,1}^{9d}X_{3,1}^{10(d+1)}X_{3,2}^{4d}\right)-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+20Cr^{\sqrt{10d}}.$

Bottom line: The “norm cost” of this step is $20Cr^{\sqrt{10d}}\leq
20Cdr^{\sqrt{d}}$.

Step 2 (Down moves on $H_{2,3},H_{2,1}$): For
$d_{1}=10(d+1),d_{2}=9d,d_{3}=9d,d_{4}=4(d+1)$, the conditions of Lemma 6.2(2)
are fulfilled and thus we can preform the following down move on $H_{2,3}$:

$\displaystyle\left\|X_{1,2}^{4d}\left(X_{1,3}^{10(d+1)}X_{2,3}^{9d}X_{2,1}^{9d}\right)X_{3,1}^{10(d+1)}X_{3,2}^{4d}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|X_{1,2}^{4d}\left(X_{1,3}^{10(d+1)}X_{2,3}^{4(d+1)}X_{2,1}^{9d}\right)X_{3,1}^{10(d+1)}X_{3,2}^{4d}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+(9d-4(d+1))Cr^{\sqrt{4(d+1)}}.$

After that, we preform a down move on $H_{2,1}$ with
$d_{1}=4(d+1),d_{2}=10(d+1),d_{3}=9d,d_{4}=4(d+1)$ (using Lemma 6.2(2) again):

$\displaystyle\left\|X_{1,2}^{4d}X_{1,3}^{10(d+1)}\left(X_{2,3}^{4(d+1)}X_{2,1}^{9d}X_{3,1}^{10(d+1)}\right)X_{3,2}^{4d}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|X_{1,2}^{4d}X_{1,3}^{10(d+1)}\left(X_{2,3}^{4(d+1)}X_{2,1}^{4(d+1)}X_{3,1}^{10(d+1)}\right)X_{3,2}^{4d}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+(9d-4(d+1))Cr^{\sqrt{4(d+1)}}.$

Bottom line: The “norm cost” of this step is
$2(9d-4(d+1))Cr^{\sqrt{4(d+1)}}\leq 10dCr^{\sqrt{d}}$.

Step 3 (Switch moves on $H_{1,3},H_{3,1}$): Preforming switch moves on
$H_{1,3}$ and $H_{3,1}$ with $d_{1}=4d,d_{2}=4(d+1),d_{3}=10(d+1)$, we get by
Lemma 6.2(1) that

$\displaystyle\left\|\left(X_{1,2}^{4d}X_{1,3}^{10(d+1)}X_{2,3}^{4(d+1)}\right)\left(X_{2,1}^{4(d+1)}X_{3,1}^{10(d+1)}X_{3,2}^{4d}\right)-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|\left(X_{2,3}^{4(d+1)}X_{1,3}^{10(d+1)}X_{1,2}^{4d}\right)\left(X_{3,2}^{4d}X_{3,1}^{10(d+1)}X_{2,1}^{4(d+1)}\right)-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+16\left(\frac{1}{2}\right)^{2(d+1)}.$

Bottom line: The “norm cost” of this step is
$16\left(\frac{1}{2}\right)^{2(d+1)}\leq 16dr^{\sqrt{d}}$ (by the choice of
$r_{2}$ in the proof of Theorem 5.7 it follows that $r\geq\frac{1}{2}$).

Step 4 (Up moves on $H_{1,2},H_{3,2}$): For
$d_{1}=10(d+1),d_{2}=4d,d_{3}=4d,d_{4}=9(d+1)$, the conditions of Lemma 6.2(2)
are fulfilled and thus we can preform the following up move on $H_{1,2}$:

$\displaystyle\left\|X_{2,3}^{4(d+1)}\left(X_{1,3}^{10(d+1)}X_{1,2}^{4d}X_{3,2}^{4d}\right)X_{3,1}^{10(d+1)}X_{2,1}^{4(d+1)}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|X_{2,3}^{4(d+1)}\left(X_{1,3}^{10(d+1)}X_{1,2}^{9(d+1)}X_{3,2}^{4d}\right)X_{3,1}^{10(d+1)}X_{2,1}^{4(d+1)}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+(9(d+1)-4d)Cr^{\sqrt{4d}}.$

After that, we preform an up move on $H_{3,2}$ with
$d_{1}=9(d+1),d_{2}=10(d+1),d_{3}=4d,d_{4}=9(d+1)$ (using Lemma 6.2(2) again):

$\displaystyle\left\|X_{2,3}^{4(d+1)}X_{1,3}^{10(d+1)}\left(X_{1,2}^{9(d+1)}X_{3,2}^{4d}X_{3,1}^{10(d+1)}\right)X_{2,1}^{4(d+1)}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq$
$\displaystyle\left\|X_{2,3}^{4(d+1)}X_{1,3}^{10(d+1)}\left(X_{1,2}^{9(d+1)}X_{3,2}^{9(d+1)}X_{3,1}^{10(d+1)}\right)X_{2,1}^{4(d+1)}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+(9(d+1)-4d)Cr^{\sqrt{4d}}=$
$\displaystyle\left\|S_{d+1}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}+(9(d+1)-4d)Cr^{\sqrt{4d}}=(9(d+1)-4d)Cr^{\sqrt{4d}}.$

Bottom line: The “norm cost” of this step is $2(9(d+1)-4d)Cr^{\sqrt{4d}}\leq
12dCr^{\sqrt{d}}$.

Using the bounds of the norm costs at each step, we deduce that

$\left\|T_{d}-S_{d+1}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq
20Cdr^{\sqrt{d}}+10dCr^{\sqrt{d}}+16dr^{\sqrt{d}}+12dCr^{\sqrt{d}}=(42C+16)dr^{\sqrt{d}},$

as needed. ∎

After this, we can prove Theorem 6.1:

###### Proof.

Fix $\delta_{0}:(0,2]\rightarrow(0,1]$.

Define $h_{d}\in\operatorname{Prob}_{c}(\operatorname{SL}_{3}(\mathbb{Z}))$ by

$h_{d}=\begin{cases}T_{d}&d\text{ is odd}\\\ S_{d}&d\text{ is
even}\end{cases}.$

By Lemma 6.3, there are $L>0,0\leq r<1$ such that for every $d$,

$\|h_{d}-h_{d+1}\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq
dLr^{\sqrt{d}}.$

Thus $\\{h_{d}\\}_{d\in\mathbb{N}}$ is a Cauchy sequence with respect to
$\|.\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}$ and it has a limit that we
will denote $f\in C_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}$. We note that
for every odd $d$,

$\|(e-e_{1,2}(1))h_{d}\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq\left\|(e-e_{1,2}(1))X_{1,2}^{4d}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq\\\
\left\|\frac{1}{2^{4d}}(e-e_{1,2}(2^{4d}))\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq\frac{1}{2^{4d-1}}.$

Therefore $\|(e-e_{1,2}(1))f\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}=0$.
This implies that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
$\pi(e_{1,2}(1)f)=\pi(f)$ and thus
$\operatorname{Im}(\pi(f))\subseteq\operatorname{\mathbb{E}}^{\pi(\langle
e_{1,2}(1)\rangle)}$. Similarly, for every even $d$,

$\|(e-e_{2,3}(1))h_{d}\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq\left\|(e-e_{1,2}(1))X_{2,3}^{4d}\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq\\\
\left\|\frac{1}{2^{4n}}(e-e_{1,2}(2^{4d}))\right\|_{\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))}\leq\frac{1}{2^{4d-1}},$

and thus for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
$\operatorname{Im}(\pi(f))\subseteq\operatorname{\mathbb{E}}^{\pi(\langle
e_{2,3}(1)\rangle)}$. It follows that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{U}(\mathcal{E}_{uc}(\delta_{0}))$,
$\operatorname{Im}(\pi(f))\subseteq\operatorname{\mathbb{E}}^{\pi(\langle
e_{1,2}(1),e_{2,3}(1)\rangle)}=\operatorname{\mathbb{E}}^{\pi(\operatorname{UT}_{3}(\mathbb{Z}))}$
as needed. ∎

As a corollary, we get Theorem 1.11 that appeared in the introduction:

######  Corollary 6.4.

The pairs
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{UT}_{3}(\mathbb{Z}))$ and
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{LT}_{3}(\mathbb{Z}))$ have
relative property $(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex
Banach space $\operatorname{\mathbb{E}}$.

###### Proof.

The proof readily follows from Theorem 6.1 and Proposition 3.4. ∎

##  7\. Banach property (T) for $\operatorname{SL}_{3}(\mathbb{Z})$

In this section, we will prove our main result regarding the Banach property
(T) of $\operatorname{SL}_{3}(\mathbb{Z})$.

We fix the following terminology: an elementary subgroup of
$\operatorname{SL}_{n}(\mathbb{Z})$ is a subgroup of the form
$E_{i,j}=\\{e_{i,j}(a):a\in\mathbb{Z}\\}$ for some $1\leq i,j\leq n,i\neq j$.
A theorem of Carter and Keller is that these subgroups boundedly generate
$\operatorname{SL}_{n}(\mathbb{Z})$:

######  Theorem 7.1.

[CK83, Main Theorem] Let $n\geq 3$. The group
$\operatorname{SL}_{n}(\mathbb{Z})$ is boundedly generated by all the
elementary subgroups.

This allows us to prove the following theorem:

######  Theorem 7.2.

For every super-reflexive Banach space $\operatorname{\mathbb{E}}$, the group
$\operatorname{SL}_{3}(\mathbb{Z})$ has property
$(T_{\operatorname{\mathbb{E}}})$.

###### Proof.

As noted above in section 3.1, it is enough to prove the result for uniformly
convex Banach spaces. Let $\operatorname{\mathbb{E}}$ be some uniformly convex
Banach space.

Denote $\operatorname{UT}_{3}(\mathbb{Z})$ and
$\operatorname{LT}_{3}(\mathbb{Z})$ be the subgroups of uni-upper-triangular
and uni-lower-triangular matrices defined above.

By Theorem 7.1, $\operatorname{UT}_{3}(\mathbb{Z})$ and
$\operatorname{LT}_{3}(\mathbb{Z})$ boundedly generate
$\operatorname{SL}_{3}(\mathbb{Z})$ and by Corollary 6.4,
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{UT}_{3}(\mathbb{Z}))$ and
$(\operatorname{SL}_{3}(\mathbb{Z}),\operatorname{LT}_{3}(\mathbb{Z}))$ both
have relative property $(T_{\operatorname{\mathbb{E}}})$. Thus, by Theorem
4.3, $\operatorname{SL}_{3}(\mathbb{Z})$ has property
$(T_{\operatorname{\mathbb{E}}})$.

∎

######  Remark 7.3.

At this point, it is also possible to give a bounded generation proof that
shows that for that for every super-reflexive Banach space
$\operatorname{\mathbb{E}}$ and every $n\geq 3$, the group
$\operatorname{SL}_{n}(\mathbb{Z})$ has property
$(T_{\operatorname{\mathbb{E}}})$ via induction on $n$ (using the
$\operatorname{SL}_{3}(\mathbb{Z})$ case as the basis of the induction). We
will not do this here, since this result will follow from our general
treatment of Banach property $(T)$ for simple Lie groups below.

##  8\. Banach property $(T)$ for simple Lie groups

The aim of the section is to prove Theorem 1.3 that appeared in the
introduction. We start by stating Howe-Moore’s Theorem for reflexive Banach
spaces:

######  Theorem 8.1.

[Howe-Moore’s Theorem for reflexive Banach spaces [Vee79]] Let
$\operatorname{\mathbb{E}}$ be a reflexive Banach space and $G$ be a connected
simple real Lie group with a finite center. Then for every continuous linear
isometric representation $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ such
that $\operatorname{\mathbb{E}}^{\pi(G)}=\\{0\\}$ it holds for every
$\xi\in\operatorname{\mathbb{E}}$ and $\eta\in\operatorname{\mathbb{E}}^{*}$
that

$\lim_{g\rightarrow\infty}\langle\pi(g)\xi,\eta\rangle=0.$

######  Corollary 8.2.

Let $\operatorname{\mathbb{E}}$ be a uniformly convex Banach space and $G$ be
a connected simple real Lie group with a finite center. For every unbounded
subgroup $H<G$ and every continuous linear isometric representation
$\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ it holds that
$\operatorname{\mathbb{E}}^{\pi(G))}=\operatorname{\mathbb{E}}^{\pi(H)}$ and
that
$\operatorname{\mathbb{E}}^{\prime}(\pi)=\operatorname{\mathbb{E}}^{\prime}(\left.\pi\right|_{H})$.

###### Proof.

Let
$\xi\in\operatorname{\mathbb{E}}^{\pi(H)}\cap\operatorname{\mathbb{E}}^{\prime}(\pi)$
and denote $\pi^{\prime}$ the restriction of $\pi$ to
$\operatorname{\mathbb{E}}^{\prime}(\pi)$. Fix $h_{n}\in H$ tending to
infinity. By Howe-Moore it follows for every
$\eta\in\operatorname{\mathbb{E}}^{*}$ that

$\langle\xi,\eta\rangle=\lim_{n}\langle\pi(h_{n})\xi,\eta\rangle=0,$

thus $\xi=0$. This shows that
$\operatorname{\mathbb{E}}^{\pi(G)}=\operatorname{\mathbb{E}}^{\pi(H)}$.
Similarly,
$\operatorname{\mathbb{E}}^{\pi^{*}(G)}=\operatorname{\mathbb{E}}^{\pi^{*}(H)}$.
Recall that $\operatorname{\mathbb{E}}^{\prime}$ and
$\operatorname{\mathbb{E}}^{\prime}(\left.\pi\right|_{H})$ are the
annihilators of $\operatorname{\mathbb{E}}^{\pi^{*}(G)}$ and
$\operatorname{\mathbb{E}}^{\pi^{*}(H)}$, and thus
$\operatorname{\mathbb{E}}^{\prime}(\pi)=\operatorname{\mathbb{E}}^{\prime}(\left.\pi\right|_{H})$.
∎

######  Corollary 8.3.

Let $\operatorname{\mathbb{E}}$ be a uniformly convex Banach space and $G$ be
a connected simple real Lie group with a finite center. For every unbounded
subgroup $H<G$, if $H$ has property $(T_{\operatorname{\mathbb{E}}})$, then so
does $G$.

###### Proof.

By Observation 3.2, it is enough to show that every continuous isometric
representation $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ with
$\operatorname{\mathbb{E}}^{\pi(G)}=\\{0\\}$ does not have almost invariant
vectors. Fix $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ with
$\operatorname{\mathbb{E}}^{\pi(G)}=\\{0\\}$.

By Corollary 8.2, it holds that $\operatorname{\mathbb{E}}^{\pi(H)}=\\{0\\}$.
Thus it follows that $\left.\pi\right|_{H}$ does not have ($H$-)almost
invariant vectors and as a result $\pi$ does not have ($G$-)almost invariant
vectors. ∎

An immediate consequence of this corollary is the following:

######  Corollary 8.4.

The group $\operatorname{SL}_{3}(\mathbb{R})$ has property
$(T_{\operatorname{\mathbb{E}}})$ for uniformly convex Banach space
$\operatorname{\mathbb{E}}$.

###### Proof.

For every uniformly convex Banach space $\operatorname{\mathbb{E}}$, combining
Theorem 7.2 and Corollary 8.3 yields that $\operatorname{SL}_{3}(\mathbb{R})$
has property $(T_{\operatorname{\mathbb{E}}})$. ∎

A standard argument allows us to use this corollary in order to prove Theorem
1.3 that appeared in the introduction :

######  Theorem 8.5.

Let $G$ be a connected simple real Lie group with a finite center and
$\mathfrak{g}$ the Lie algebra of $G$. If $\mathfrak{g}$ contains
$\mathfrak{sl}_{3}(\mathbb{R})$ as a Lie sub-algebra, then $G$ all its
lattices have property $(T_{\operatorname{\mathbb{E}}})$ for every uniformly
convex Banach space $\operatorname{\mathbb{E}}$.

###### Proof.

We note since $Z(G)$ is finite, it follows that if $\rm Ad(G)=G/Z(G)$ has
property $(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach
space $\operatorname{\mathbb{E}}$, then $G$ has property
$(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach space
$\operatorname{\mathbb{E}}$. Thus we can replace $G$ with $\rm Ad(G)=G/Z(G)$
and assume that $G$ is algebraic.

Fix a uniformly convex Banach space $\operatorname{\mathbb{E}}$. By Corollary
8.3, in order to prove that $G$ has property $(T_{\operatorname{\mathbb{E}}})$
it is enough to show that there is an unbounded subgroup $G$ such that has
property $(T_{\operatorname{\mathbb{E}}})$.

By our assumption on $\mathfrak{g}$, the group $G$ contains a subgroup $H$
whose (algebraic) simply connected covering (see [Mar91, Definition 1.4.12])
is isomorphic to $\operatorname{SL}_{3}(\mathbb{R})$. Thus there is
$\varphi:\operatorname{SL}_{3}(\mathbb{R})\rightarrow H$ such that the
cokernel of $\varphi$ is finite. Since property
$(T_{\operatorname{\mathbb{E}}})$ is inherited by quotients, it follows from
Corollary 8.4 that $\varphi(H)$ has property $(T_{\operatorname{\mathbb{E}}})$
as needed.

Last, note that by Corollary 3.8, if $G$ has property
$(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach space
$\operatorname{\mathbb{E}}$, then every lattice $\Gamma<G$ has property
$(T_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach space
$\operatorname{\mathbb{E}}$. ∎

##  9\. Applications

###  9.1. Banach fixed point property for simple Lie groups

Let $\operatorname{\mathbb{E}}$ be a Banach space and $G$ be a topological
group. An affine isometric action of $G$ on $\operatorname{\mathbb{E}}$ is a
continuous homomorphism
$\rho:G\rightarrow\operatorname{Isom}_{aff}(\operatorname{\mathbb{E}})$, where
$\operatorname{Isom}_{aff}(\operatorname{\mathbb{E}})$ denotes the group of
affine isometric automorphisms of $\operatorname{\mathbb{E}}$. We recall such
that $\rho$ is of the form

$\rho(g)\xi=\pi(g)\xi+c(g),\forall\xi\in\operatorname{\mathbb{E}}$

where $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ is a continuous
isometric linear representation that is called the linear part of $\rho$ and
$c:G\rightarrow\operatorname{\mathbb{E}}$ is a continuous $1$-cocycle into
$\pi$, i.e., it is a continuous map such that for every $g,h\in G$,

$c(gh)=c(g)+\pi(g)c(h).$

The group $G$ is said to have property $(F_{\operatorname{\mathbb{E}}})$ if
every affine isometric action of $G$ on $\operatorname{\mathbb{E}}$ admits a
fixed point.

Below, we will prove Theorem 1.5 that states that a simple Lie group whose Lie
algebra contains $\mathfrak{sl}_{4}(\mathbb{R})$ has property
$(F_{\operatorname{\mathbb{E}}})$ for every super-reflexive Banach space. For
this we will state the following results:

######  Theorem 9.1.

Metric Mautner phenomenon, [BG17, Theorem 1.3] Let $G$ be a connected simple
real Lie group with a finite center. Assume that $G$ acts isometrically and
continuously on a metric space $(\mathbb{X},d)$. If there is a non-compact
point stabilizer (of the action of $G$ on $\mathbb{X}$), then the action of
$G$ on $\mathbb{X}$ admits a fixed point.

######  Lemma 9.2.

[BFGM07, Lemma 5.6] Let $H_{1}\times H_{2}$ be a topological group that acts
by a continuous affine isometric action $\rho$ on a uniformly convex Banach
space $\operatorname{\mathbb{E}}$. If the linear part of $\rho$ restricted to
$H_{1}$ does not have almost invariant vectors, then the action of $H_{2}$ on
$\operatorname{\mathbb{E}}$ admits a fixed point.

Last, using [BFGM07] we can show that for higher rank Lie groups property
$(F_{\operatorname{\mathbb{E}}})$ for every uniformly convex
$\operatorname{\mathbb{E}}$ is inherited to (and from) passing to lattices:

######  Proposition 9.3.

Let $G$ be a connected higher rank simple real Lie group and $\Gamma<G$ a
lattice. The group $G$ has property $(F_{\operatorname{\mathbb{E}}})$ for
every uniformly convex Banach space if and only if the group $\Gamma$ has
property $(F_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach
space $\operatorname{\mathbb{E}}$.

###### Proof.

In [BFGM07, Proposition 8.8] it was shown that for any locally compact group
$G$, any lattice $\Gamma<G$ and any uniformly convex Banach space
$\operatorname{\mathbb{E}}$ the following holds:

  * •

If $\Gamma$ has property $(F_{\operatorname{\mathbb{E}}})$, then $G$ has
property $(F_{\operatorname{\mathbb{E}}})$.

  * •

If $G$ has property $(F_{L^{2}(G/\Gamma;\operatorname{\mathbb{E}})})$ and
$\Gamma$ is $2$-integrable (see [BFGM07, Definition 8.2]), then $\Gamma$ has
property $(F_{\operatorname{\mathbb{E}}})$.

Thus it readily follows that if $\Gamma$ has property
$(F_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach space
$\operatorname{\mathbb{E}}$, then so does $G$.

In the other direction, we use the following two facts: First, Shalom [Sha00]
showed that every lattice in a connected higher rank simple Lie groups in
$2$-integrable. Second, by Theorem 2.4,
$L^{2}(G/\Gamma;\operatorname{\mathbb{E}})$ is uniformly convex. ∎

Our starting point towards proving Theorem 1.5 it to prove
$\operatorname{SL}_{4}(\mathbb{R})$ has property
$(F_{\operatorname{\mathbb{E}}})$ for every super-reflexive Banach space
$\operatorname{\mathbb{E}}$:

######  Theorem 9.4.

For every super-reflexive Banach space $\operatorname{\mathbb{E}}$, the group
$\operatorname{SL}_{4}(\mathbb{R})$ has property
$(F_{\operatorname{\mathbb{E}}})$.

###### Proof.

By [BFGM07, Proposition 2.13] it is enough to consider uniformly convex Banach
spaces.

Fix some uniformly convex Banach space $\operatorname{\mathbb{E}}$.

Let $\rho:\operatorname{SL}_{4}(\mathbb{R})\rightarrow\rm
Isom_{aff}(\operatorname{\mathbb{E}})$ be a continuous affine isometric action
of $\operatorname{SL}_{4}(\mathbb{R})$ on $\operatorname{\mathbb{E}}$ with a
linear part $\pi$. As in [BFGM07, 5.c], the fact that
$\operatorname{SL}_{4}(\mathbb{R})$ has a compact abelianization allows us to
assume that
$\operatorname{\mathbb{E}}^{\pi(\operatorname{SL}_{4}(\mathbb{R}))}=\\{0\\}$.

Define $H<\operatorname{SL}_{4}(\mathbb{R})$ to be the subgroup

$H=\left\\{\left(\begin{matrix}&0\\\ A&0\\\ &0\\\ 0\hskip 7.22743pt0\hskip
7.22743pt0&\frac{1}{\rm det(A)}\end{matrix}\right):A\in\rm
GL_{3}(\mathbb{R})\right\\}.$

We note that $H\cong\rm
GL_{3}(\mathbb{R})\cong\operatorname{SL}_{3}(\mathbb{R})\times\mathbb{R}^{*}$.
By Corollary 8.2,
$\operatorname{\mathbb{E}}^{\pi(\operatorname{SL}_{3}(\mathbb{R}))}=\\{0\\}$.
Thus, by Corollary 8.4, the representation
$\left.\pi\right|_{\operatorname{SL}_{3}(\mathbb{R})}$ does not have almost
invariant vectors and by Lemma 9.2, the $\rho$-action of $\mathbb{R}^{*}$ on
$\operatorname{\mathbb{E}}$ has a fixed point. It follows from Theorem 9.1
that the $\rho$-action of $\operatorname{SL}_{4}(\mathbb{R})$ on
$\operatorname{\mathbb{E}}$ has a fixed point as needed. ∎

######  Theorem 9.5.

Let $G$ be a connected simple real Lie group with a finite center and
$\mathfrak{g}$ be the Lie algebra of $G$. If $\mathfrak{g}$ contains
$\mathfrak{sl}_{4}(\mathbb{R})$ as a Lie sub-algebra, then $G$ and any lattice
$\Gamma<G$ have property $(F_{\operatorname{\mathbb{E}}})$ for every super-
reflexive Banach space $\operatorname{\mathbb{E}}$.

###### Proof.

By [BFGM07, Proposition 2.13] it is enough to consider uniformly convex Banach
spaces.

We note that since $Z(G)$ is finite, if $\rm Ad(G)=G/Z(G)$ has property
$(F_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach space
$\operatorname{\mathbb{E}}$, then $G$ has property
$(F_{\operatorname{\mathbb{E}}})$ for every uniformly convex Banach space
$\operatorname{\mathbb{E}}$. Thus we can replace $G$ with $\rm Ad(G)=G/Z(G)$
and assume that $G$ is algebraic.

By Proposition 9.3, if $G$ has property $(F_{\operatorname{\mathbb{E}}})$ for
every uniformly convex Banach space $\operatorname{\mathbb{E}}$, then every
lattice of $G$ has property $(F_{\operatorname{\mathbb{E}}})$ for every
uniformly convex Banach space $\operatorname{\mathbb{E}}$. Thus, it is enough
to prove that $G$ has property $(F_{\operatorname{\mathbb{E}}})$ for every
uniformly convex Banach space $\operatorname{\mathbb{E}}$.

Fix a uniformly convex Banach space $\operatorname{\mathbb{E}}$ and a
continuous affine isometric action of $G$ on $\operatorname{\mathbb{E}}$.

By our assumption on $\mathfrak{g}$, the group $G$ contains a subgroup $H$
whose (algebraic) simply connected covering is isomorphic to
$\operatorname{SL}_{4}(\mathbb{R})$. Thus there is
$\varphi:\operatorname{SL}_{4}(\mathbb{R})\rightarrow H$ such that the
cokernel of $\varphi$ is finite. Observe that property
$(F_{\operatorname{\mathbb{E}}})$ is preserved under passing to quotients and
thus by Theorem 9.4 it follows that
$\varphi(\operatorname{SL}_{4}(\mathbb{R}))$ has property
$(F_{\operatorname{\mathbb{E}}})$. It follows that the action of
$\varphi(\operatorname{SL}_{4}(\mathbb{R}))$ on $\operatorname{\mathbb{E}}$
has a fixed point, so there is a non-compact point stabilizer of the action of
$G$ on $\operatorname{\mathbb{E}}$. By Theorem 9.1 it follows that the $G$
action on $\operatorname{\mathbb{E}}$ has a fixed point. ∎

###  9.2. Super-expanders

We start by recalling the definition of Mendel and Naor [MN14] for super-
expanders.

Let $\operatorname{\mathbb{E}}$ be a Banach space and
$\\{(V_{i},E_{i})\\}_{i\in\mathbb{N}}$ be a sequence of finite graphs with
uniformly bounded degree, such that $\lim_{i}|V_{i}|=\infty$. We say that
$\\{(V_{i},E_{i})\\}_{i\in\mathbb{N}}$ has a Poincaré inequality with respect
to $\operatorname{\mathbb{E}}$ if there are constants $p,\gamma\in(0,\infty)$
such that for every $i\in\mathbb{N}$ and every
$\phi:V_{i}\rightarrow\operatorname{\mathbb{E}}$ we have

$\frac{1}{|V_{i}|^{2}}\sum_{(u,v)\in V_{i}\times
V_{i}}\|\phi(u)-\phi(v)\|^{p}\leq\frac{\gamma}{|V_{i}|}\sum_{(x,y)\in
E_{i}}\|\phi(x)-\phi(y)\|^{p}.$

The sequence $\\{(V_{i},E_{i})\\}_{i\in\mathbb{N}}$ is called a super-expander
family if it has a Poincaré inequality with respect to every super-reflexive
Banach space (or equivalently for every uniformly convex Banach space).

For Cayley graphs, the following proposition of Lafforgue gives a relation
between Poincaré inequality of Cayley graphs and Banach property $(T)$:

######  Proposition 9.6.

[Laf08, Proposition 5.2] Let $\Gamma$ be a finitely generated discrete group
and let $\\{N_{i}\\}_{i\in\mathbb{N}}$ be a sequence of finite index normal
subgroups of $\Gamma$ such that $\bigcap_{i}N_{i}=\\{1\\}$. Also let
$\operatorname{\mathbb{E}}$ be a Banach space. If $\Gamma$ has Banach property
$(T_{\operatorname{\mathbb{E}}^{\prime}})$ for
$\operatorname{\mathbb{E}}^{\prime}=\ell^{2}(\bigcup_{i}G/N_{i};\operatorname{\mathbb{E}})$,
then for every fixed finite symmetric generating set $S$, the family of Cayley
graphs of $\\{(G/N_{i},S/N_{i})\\}_{i\in\mathbb{N}}$ has a Poincaré inequality
with respect to $\operatorname{\mathbb{E}}$.

A consequence of this proposition and Theorem 1.4 implies the following
theorem that appeared in the introduction (Theorem 1.6):

######  Theorem 9.7.

Let $n\geq 3$ and let $S$ be a finite generating set of
$\operatorname{SL}_{n}(\mathbb{Z})$ (e.g., $S=\\{e_{i,j}(\pm 1):1\leq i,j\leq
n,i\neq j\\}$). Let
$\Phi_{i}:\operatorname{SL}_{n}(\mathbb{Z})\rightarrow\operatorname{SL}_{n}(\mathbb{Z}/i\mathbb{Z})$
be the natural surjective homomorphism. Then the family of Cayley graphs of
$\\{(\operatorname{SL}_{n}(\mathbb{Z}/i\mathbb{Z}),\Phi_{i}(S))\\}_{i\in\mathbb{N}}$
is a super-expander family.

###### Proof.

Let $\operatorname{\mathbb{E}}$ be some uniformly convex Banach space. By
Theorem 2.3, $\ell^{2}(\bigcup_{i}G/N_{i};\operatorname{\mathbb{E}})$ is a
uniformly convex Banach space. Thus by Theorem 1.4 and Proposition 9.6, the
family
$\\{(\operatorname{SL}_{n}(\mathbb{Z}/i\mathbb{Z}),S)\\}_{i\in\mathbb{N}}$ is
a $\operatorname{\mathbb{E}}$-expander family. ∎

It was shown in [Vig19, dLV19, FNvL19, Saw20] that one can construct super-
expanders using warped cones arising from an action of a Banach property (T)
group on a compact manifold. Combining this machinery with our Theorem 1.4
also leads to a construction of super-expanders as we will briefly now
explain. Let $(M,d_{M})$ be a compact Riemannian manifold and $\Gamma$ be a
finitely generated group with finite symmetric generating set $S$. Assume that
$\Gamma$ acts on $M$ by Lipschitz homeomorphisms. For $t>0$, define the
$t$-level warped cone denoted $(M,d_{\Gamma}^{t})$ to be the metric space such
that $d_{\Gamma}^{t}$ is the largest metric satisfying:

  * •

$d_{\Gamma}^{t}(x,y)\leq td_{M}(x,y)$ for every $x,y\in M$.

  * •

$d_{\Gamma}^{t}(x,s.x)\leq 1$ for every $x\in M$ and $s\in S$.

######  Remark 9.8.

The metric $d_{\Gamma}^{t}$ is dependent on the choice of the generating set,
but this dependence will be irrelevant with respect to our application below,
because of the following fact (see [Roe05]): For metrics $d_{\Gamma}^{t}$ and
$(d_{\Gamma}^{t})^{\prime}$ that correspond to generating sets $S$ and
$S^{\prime}$, it holds that $d_{\Gamma}^{t}$ and $(d_{\Gamma}^{t})^{\prime}$
are Lipschitz equivalent.

The following theorem is a straight-forward implication of Sawicki’s main
result in [Saw20]:

######  Theorem 9.9.

[Saw20, Theorem 1.1] Let $(M,d_{M})$ be a compact Riemannian manifold and
$\Gamma$ be a finitely generated group acting on $M$ by Lipschitz
homeomorphisms. If $\Gamma$ has property $(T_{\operatorname{\mathbb{E}}})$ for
every super-reflexive Banach space $\operatorname{\mathbb{E}}$, then for every
increasing sequence $\\{t_{i}\\}_{i\in\mathbb{N}}\subseteq\mathbb{R}_{>0}$
tending to infinity, the family
$\\{(M,d^{t_{i}}_{\Gamma})\\}_{i\in\mathbb{N}}$ is quasi-isometric to a super-
expander.

Combining this theorem with Theorem 1.4 leads to the following theorem stated
in the introduction:

######  Theorem 9.10.

Let $n\geq 3$ and let $(M,d_{M})$ be a compact Riemannian manifold such that
$\operatorname{SL}_{n}(\mathbb{Z})$ acts on $M$ by Lipschitz homeomorphisms.
For every increasing sequence
$\\{t_{i}\\}_{i\in\mathbb{N}}\subseteq\mathbb{R}$ tending to infinity, the
family
$\\{(M,d^{t_{i}}_{\operatorname{SL}_{n}(\mathbb{Z})})\\}_{i\in\mathbb{N}}$ is
quasi-isometric to a super-expander.

###  9.3. Property $(FF_{\operatorname{\mathbb{E}}})$ for
$\operatorname{SL}_{n}(\mathbb{R}),\operatorname{SL}_{n}(\mathbb{Z})$

When $\operatorname{\mathbb{E}}$ is reflexive it follows for the Ryll-
Nardzewski fixed-point Theorem that $G$ has property
$(F_{\operatorname{\mathbb{E}}})$ if and only if for every isometric linear
representation $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ it holds that
every continuous $1$-cocycle into $\pi$ is bounded. This lead to the stronger
notion of property $(FF_{\operatorname{\mathbb{E}}})$ defined by Mimura
[Mim11] as a Banach version of Monod’s [Mon01] property (TT): Given a
continuous isometric linear representation $\pi:G\rightarrow
O(\operatorname{\mathbb{E}})$, a continuous quasi-$1$-cocycle into $\pi$ is a
continuous map $c:G\rightarrow\operatorname{\mathbb{E}}$ such that

$\sup_{g,h\in G}\|c(gh)-(c(g)+\pi(g)c(h))\|<\infty.$

A group $G$ is said to have property property
$(FF_{\operatorname{\mathbb{E}}})$ if for every continuous isometric linear
representation $\pi:G\rightarrow O(\operatorname{\mathbb{E}})$ it holds that
every continuous quasi-$1$-cocycle into $\pi$ is bounded.

The following result of de Laat, Mimura and de la Salle allows one to deduce
property $(FF_{\operatorname{\mathbb{E}}})$ from property
$(T_{\operatorname{\mathbb{E}}})$:

######  Theorem 9.11.

[dLMdlS16, Section 5] Let $n\geq 3$ and $\operatorname{\mathbb{E}}$ be a
super-reflexive Banach space. For $R=\mathbb{Z},\mathbb{R}$, if
$\operatorname{SL}_{n}(R)$ has property $(T_{\operatorname{\mathbb{E}}})$,
then $\operatorname{SL}_{n+2}(R)$ has property
$(FF_{\operatorname{\mathbb{E}}})$.

Combining this theorem with Theorem 1.4 yields the following corollary that
appeared in the introduction (Corollary 1.8):

######  Corollary 9.12.

For every $n\geq 5$ and every super-reflexive Banach space
$\operatorname{\mathbb{E}}$, the groups
$\operatorname{SL}_{n}(\mathbb{Z}),\operatorname{SL}_{n}(\mathbb{R})$ have
property $(FF_{\operatorname{\mathbb{E}}})$.

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