# Banach Żuk’s criterion for partite complexes with application to random
groups

Izhar Oppenheim

###### Abstract.

We prove a Banach version of Żuk’s criterion for groups acting on partite
simplicial complexes. Using this new criterion we derive a new fixed point
theorem for random groups in the Gromov density model with respect to several
classes of Banach spaces ($L^{p}$ spaces, Hilbertian spaces, uniformly curved
spaces). In particular, we show that for every $p$, a group in the Gromov
density model has asymptotically almost surely property $(FL^{p})$ and give a
sharp lower bound for the growth of the conformal dimension of the boundary of
such group as a function of the parameters of the density model.

##  1\. Introduction

Fixed point properties of groups is a general theme is the study of groups.
For instance, for a discrete group, both amenability and Kazhdan’s property
(T) are equivalent to fixed point properties. This paper will focus on the
study of fixed point properties of a group when acting on (a class of) Banach
spaces. Explicitly, the following definition was given in Bader at el.
[BFGM07] as a generalization of property $(FH)$: Given a Banach space
$\operatorname{\mathbb{E}}$ and a group $G$, we say that $G$ has property
$(F\operatorname{\mathbb{E}})$ if every continuous affine isometric action of
$G$ on $\operatorname{\mathbb{E}}$ has a fixed point. More generally, given a
class of Banach spaces $\mathcal{E}$, we say that $G$ has property
$(F\mathcal{E})$ if it has $(F\operatorname{\mathbb{E}})$ for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$. In this notation, property $(FH)$
is property $(F\mathcal{E}_{\text{Hilbert}})$, where
$\mathcal{E}_{\text{Hilbert}}$ is the class of all Hilbert spaces. Bader at
el. [BFGM07] also studied property $(FL^{p})$ which will be considered in this
article: Using the notation above, property $(FL^{p})$ is property
$(F\mathcal{E}_{L^{p}})$ where $\mathcal{E}_{L^{p}}$ is the class of all
$L^{p}$ spaces.

This paper was motivated by the questions regarding Banach fixed point
properties for random groups. Banach fixed properties for random groups was
studied in [DM19, dLdlS21, Opp20] \- in all these works, the results regarding
random groups in the triangular model were rather strong, e.g., all these
works show that for every $2\leq p<\infty$, a random group in the triangular
model has property $(FL^{p})$ asymptotically almost surely. In contrast, the
results for random groups in the density model were much less satisfactory -
[dLdlS21, Opp20] did not address this model at all and in [DM19] the results
in this model were rather weak (see exact results below). It was conjectured
by Druţu and Mackay [DM19] that property $(FL^{p})$ for every $2\leq p<\infty$
should hold also in the density model. The problem with generalizing the
approach of the author in [Opp20] to the density model was the lack of a Żuk
style criterion in the Banach setting for groups acting on partite complexes
(see more on this below).

The main results of this paper are: First, establishing a Banach Żuk criterion
for groups acting on partite complexes (Theorem 1.3) and second, applying this
criterion for random groups in the density model (Theorem 1.5). As a
consequence, we also derive a sharp bound on the growth of the conformal
dimension of the boundary of a random group in the density model (Theorem
1.8).

###  1.1. Banach Żuk’s criterion for groups acting on partite simplicial
complexes

In [Gar73], Garland showed how to prove vanishing of real cohomology for a
group acting on an affine building via studying the local geometry of the
building. This approach was later generalized by various authors to vanishing
of cohomology with Hilbert coefficients for groups acting on simplicial
complexes - see [Ż96, BSa97, DJ00, DJ02, Opp15]. We recall that vanishing of
the first cohomology with Hilbert coefficients is equivalent to property (FH).
Later, this approach was also applied in the Banach setting in various levels
of effectiveness/success - see [Bou12, Opp14, Now15, Opp17a, Opp17b, DM19,
dLdlS21, Opp20].

The idea behind this approach is the following local to global argument: Given
a group $G$ acting geometrically on a simplicial complex $X$, one can deduce
fixed point properties of $G$ by geometric conditions on the links on $X$. In
dimension $2$, this approach is also known as Żuk’s criterion. Recently, there
were also computer assisted proofs for property (T) and property (FH) that
refined this idea and gave geometric proofs studying the geometry of $2$-balls
around vertices (while Żuk’s criterion studies the geometry of $1$ balls) -
see [KNO19, KKN, Nit20].

In order to explicitly state Żuk’s criterion and our generalization of it, we
will need some terminology of spectral graph theory. Given a finite graph
$(V,E)$, denote $m(v)$ to be the valency of $v\in V$. We also define
$\ell^{2}(V,m)$ as to be the space of functions $\phi:V\rightarrow\mathbb{C}$
with an inner-product

$\langle\phi,\psi\rangle=\sum_{\\{v\\}\in V}=m(v)\phi(v)\overline{\psi(v)}.$

The simple random walk on $(V,E)$ as above is the operator
$A:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$ defined as

$(A\phi)(v)=\sum_{u\in V,\\{u,v\\}\in E}\frac{m(\\{u,v\\})}{m(v)}\phi(u).$

With the inner-product above, this is a self-adjoint operator and its spectrum
is contained in $[-1,1]$. We recall that the space of constant functions is an
eigenspace of $A$ with eigenvalue $1$ and, if $(V,E)$ is connected, then all
other the other eigenvalues of $A$ are strictly less than $1$.

The Hilbert (classical) version of Żuk’s criterion goes as follows:

######  Theorem 1.1 ([BSa97, Ż96] for $n=2$, [Opp15] for all $n\geq 2$).

Let $G$ be a locally compact, unimodular group and $X$ be a pure
$n$-dimensional simplicial complex such that $X$ is gallery connected and all
the $1$-dimensional links of $X$ are connected. Assume that $G$ is acting on
$X$ by simplicial automorphisms and the the action is proper and cocompact. If
for every $1$-dimensional link of $X$, the second largest eigenvalue of the
simple random walk on the link is $<\frac{1}{n+1}$, then $G$ has property
$(FH)$ (and property (T)).

We note that in the Hilbert case, the criteria stated above need only a bound
on the second largest eigenvalue of the links. However, when generalizing this
criterion to the Banach setting, one usually needs to bound the norm of the
random walk operator in order to get a workable version - see [dLdlS21,
Opp20]. Explicitly, for a finite graph $(V,E)$ and a Banach space
$\operatorname{\mathbb{E}}$, we define
$\ell^{2}(V,m;\operatorname{\mathbb{E}})$ to be the space of functions
$\phi:V\rightarrow\operatorname{\mathbb{E}}$ with the norm

$\|\phi\|^{2}=\sum_{v\in V}m(v)|\phi(v)|_{\operatorname{\mathbb{E}}}^{2}.$

Define
$A\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}:\ell^{2}(V,m;\operatorname{\mathbb{E}})\rightarrow\ell^{2}(V,m;\operatorname{\mathbb{E}})$
as above and
$M\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}:\ell^{2}(V,m;\operatorname{\mathbb{E}})\rightarrow\ell^{2}(V,m;\operatorname{\mathbb{E}})$
as

$M\phi\equiv\frac{1}{\sum_{v\in V}m(v)}\phi(v).$

Denote
$\lambda_{(V,E)}^{\operatorname{\mathbb{E}}}=\|(A(I-M))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}$.
Note that in the Hilbert setting (where
$\operatorname{\mathbb{E}}=\mathcal{H}$), bounding
$\lambda_{(V,E)}^{\mathcal{H}}$ is equivalent to bounding the non-trivial
spectrum of the operator from above and below. In [Opp20], the author showed
the following:

######  Theorem 1.2.

[Opp20, Theorem 4.4] Let $G$ be a unimodular, locally compact group that acts
properly and cocompactly on a locally finite, pure $2$-dimensional simply
connected simplicial complex $X$ with connected $1$-dimensional links. For any
reflexive Banach space $\operatorname{\mathbb{E}}$, if

$\max_{v\in X(0)}\lambda_{X_{v}}^{\operatorname{\mathbb{E}}}<\frac{1}{2}$

then $G$ has property $(F\operatorname{\mathbb{E}})$.

The main issue with using the operator norm of the random walk instead to the
second largest eigenvalue is that in some interesting examples the simplicial
complex $X$ is partite. An $n$-dimensional simplicial complex is called
partite (or colorable) if its vertices can be colored in $n+1$ different
colors such that each $n$-simplex has a vertex of every color. In particular,
a graph $(V,E)$ is called bipartite if $V$ can be partitioned into two
disjoint sets $V=S_{1}\cup S_{2}$ called sides such that for each
$\\{u,v\\}\in E$, $|\\{u,v\\}\cap S_{1}|=|\\{u,v\\}\cap S_{2}|=1$, i.e., each
edge has exactly one vertex in each side. We note that for a partite
$n$-dimensional complex, each one dimensional link is a bipartite graph. We
also note that the random walk on bipartite graphs always has the eigenvalue
$-1$ corresponding to the function $\mathbbm{1}_{S_{1}}-\mathbbm{1}_{S_{2}}$
(where $\mathbbm{1}_{S_{i}}$ is the indicator function on $S_{i}$) and thus
for every (non zero dimensional) Banach space $\operatorname{\mathbb{E}}$ we
always have $\lambda_{(V,E)}^{\operatorname{\mathbb{E}}}\geq 1$.

Consider the classical case of $\operatorname{\mathbb{E}}=\mathbb{C}$. For a
bipartite graph $(V,E)$, when $A$ acts on $\ell^{2}(V,m)$, it is standard to
think of the the trivial spectrum of $A$ as $\pm 1$ and the space of trivial
eigenfunctions is the space spanned by the constant function and
$\mathbbm{1}_{S_{1}}-\mathbbm{1}_{S_{2}}$ or equivalently the space of trivial
eigenfunctions is
$\operatorname{span}\\{\mathbbm{1}_{S_{1}},\mathbbm{1}_{S_{2}}\\}$. The
projection on the space of trivial eigenfunctions is can be described
explicitly as follows: We define the following averaging operators
$M_{1},M_{2}:\ell^{2}(V,m)\rightarrow\mathbb{C}$:

$M_{i}\phi=\frac{1}{m(S_{i})}\sum_{u\in S_{i}}\phi(u),$

where $m(S_{i})=\sum_{v\in S_{i}}m(v)$. We also define
$M_{\operatorname{sides}}:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$ by

$M_{\operatorname{sides}}\phi(u)=\begin{cases}M_{1}\phi&u\in S_{1}\\\
M_{2}\phi&u\in S_{2}\end{cases}.$

With this notation, the non-trivial spectrum of $A$ is bounded in the interval
$[-\|A(I-M_{\operatorname{sides}})\|,\|A(I-M_{\operatorname{sides}})\|]$. This
discussion gives rise to the following definition: Let
$\operatorname{\mathbb{E}}$ be a Banach space and $(V,E)$ be a bipartite
graph. Define

$\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}=\|(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}.$

For a partite $n$-dimensional $X$ and an $(n-2)$-simplex $\tau$, we denote
$\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}=\lambda_{X_{\tau},\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}$
where $X_{\tau}$ is the link of $\tau$ (which is a bipartite graph). We these
definitions, we prove the following Theorem:

######  Theorem 1.3 (Żuk type criterion for Banach fixed point property - see
Theorem 3.10 for a more detailed version).

Let $G$ be a locally compact, unimodular group and $X$ be a pure
$n$-dimensional, partite simplicial complex such that $X$ is gallery connected
and all the $1$-dimensional links of $X$ are connected. Assume that $G$ is
acting on $X$ by simplicial automorphisms and the the action is proper and
cocompact. Let $\mathcal{E}_{X}$ be the class of Banach spaces such that for
every $\operatorname{\mathbb{E}}\in\mathcal{E}_{X}$ it holds that

$\max_{\tau\in
X(n-2)}\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}<\frac{1}{8n-3}.$

Assume that $\mathbb{C}\in\mathcal{E}_{X}$, then $G$ property
$(F\mathcal{E}_{X})$.

The idea behind the proof of this Theorem is the following: In [Opp17a], the
author proved a version of this Theorem under the assumption that the
fundamental domain of the action is a single simplex. The main argument there
was bounding the angle between the projections defined by the subgroups
stabilizing the $(n-1)$-faces of the fundamental domain. In this paper, the
idea is to use this angle criterion, but apply it on the projections defined
by the coloring of the simplicial complex. As in [Opp17a], this leads to a
proof of a strengthened version of Banach property (T) (i.e., robust property
(T)) and this property in turn leads to Banach fixed point properties.

###  1.2. Fixed point properties of a random group in the Gromov model

A random group is a group chosen randomly according to some model and one is
interested in the asymptotic properties of such randomly chosen group. The
most famous model is the Gromov density model:

######  Definition 1.4 (Gromov density model).

Let $k\in\mathbb{N},k\geq 2$ and $0\leq d\leq 1$ be constants and
$l\in\mathbb{N}$ be a parameter. A random group is the Gromov density model
$\mathcal{D}(k,l,d)$ is a group $\Gamma=\langle\mathcal{A}|\mathcal{R}\rangle$
where $|\mathcal{A}|=k$ and $\mathcal{R}$ is a set of relators of length $l$
(in $\mathcal{A}\cup\mathcal{A}^{-1}$) randomly chosen from the set

$\\{\mathcal{R}\text{ is a set of cyclically reduced relators of length
}l:|\mathcal{R}|=\lfloor(2k-1)^{dl}\rfloor\\}$

with uniform probability. We denote a random group in this model by
$\Gamma\in\mathcal{D}(k,l,d)$.

For a group property $P$, we say that $P$ holds asymptotically almost surely
(a.a.s.) in $\mathcal{D}(k,l,d)$ if

$\lim_{l\rightarrow\infty}\mathbb{P}(\Gamma\in\mathcal{D}(k,l,d)\text{ has
property }P)=1.$

It was proven [Ż03, KK13, DM19, Ash21b], that for every $d>\frac{1}{3}$
property (T) (and equivalently, property (FH)) holds a.a.s. in
$\mathcal{D}(k,l,d)$ (in most cases, it is assumed that $l$ is divisible by
$3$, but recently Ashcroft [Ash21b] showed how to remove this assumption).
Prior to our work, generalizing this result to $L^{p}$ spaces was met with
some difficulty: Indeed, prior to this work, the state-of-the-art result was
by Druţu and Mackay [DM19] who proved the following: Given $2\leq p<\infty$,
if $k\geq 10\cdot 2^{p}$, then for every $d>\frac{1}{3}$ it holds a.a.s. that
$\mathcal{D}(k,l,d)$ has property $(FL^{p^{\prime}})$ for any $2\leq
p^{\prime}\leq p$. As noted by Druţu and Mackay [DM19] it is natural to expect
that for any $k\geq 2$ and any $d>\frac{1}{3}$ it holds a.a.s. that
$\mathcal{D}(k,l,d)$ has property $(FL^{p})$ for any $2\leq p<\infty$, but
they could not achieve this result.

Below, we use our Theorem 1.3 to vastly improve on the known results for
Banach fixed point properties for random group in the Gromov density model.
Explicitly, we consider the class of uniformly curved Banach spaces (see
Definition 2.9 below) and prove that for every $k\geq 2$ and every
$d>\frac{1}{3}$ it holds a.a.s. that $\mathcal{D}(k,l,d)$ has property
$(F\operatorname{\mathbb{E}})$ for every uniformly curved Banach space. The
class of uniformly curved Banach spaces contains all $L^{p}$ spaces and thus
we prove the fixed point $L^{p}$ property that was conjectured by Druţu and
Mackay. More explicitly, we prove the following:

######  Theorem 1.5 (Banach fixed point properties for the density model - see
Theorem 4.17 for a more detailed version).

Let $\frac{1}{3}<d<\frac{1}{2}$, $k\geq 2$ be constants. For $l$ divisible by
$3$, it holds a.a.s. that $\mathcal{D}(k,l,d)$ has property
$(F\operatorname{\mathbb{E}})$ for any uniformly curved space
$\operatorname{\mathbb{E}}$. In particular, for every $2\leq p<\infty$ it
holds a.a.s. that $\mathcal{D}(k,l,d)$ has property $(FL^{p})$. Explicitly,
there are universal constants $C^{\prime},C^{\prime\prime}$ independent of
$k,l,d$ such that for any

$2\leq p\leq C^{\prime}(d-\frac{1}{3})(\log(2k-1))l-C^{\prime\prime},$

$\mathcal{D}(k,l,d)$ has property $(FL^{p})$ a.a.s. (given that $l$ is
divisible by $3$).

As a corollary, we give a lower bound on the conformal dimension of the
boundary of a random groups in the density model. Namely, by a Theorem of
Bourdon [Bou16], if for a given $2\leq p$, a hyperbolic group $\Gamma$ has
property $(FL^{p})$, then the conformal dimension of $\partial_{\infty}\Gamma$
is $\geq p$. This readily gives the following:

######  Corollary 1.6.

Let $\frac{1}{3}<d<\frac{1}{2}$, $k\geq 2$ be constants. For $l$ divisible by
$3$ let $\Gamma$ be a random group in the model $\mathcal{D}(k,l,d)$. Then
there are universal constants $C^{\prime},C^{\prime\prime}$ independent of
$k,l,d$ such that it holds a.a.s. that

$C^{\prime}(d-\frac{1}{3})(\log(2k-1))l-C^{\prime\prime}\leq{\rm
Confdim}(\partial_{\infty}\Gamma).$

An upper bound for the conformal dimension was given by Mackay:

######  Proposition 1.7.

[Mac12, Proposition 1.7] Let $\frac{1}{3}<d<\frac{1}{2}$, $k\geq 2$ be
constants and let $\Gamma$ be a random group in the model
$\mathcal{D}(k,l,d)$. Then it holds a.a.s. that

${\rm
Confdim}(\partial_{\infty}\Gamma)\leq\frac{16}{\log(2)}\frac{1}{1-2d}(\log(2k-1))l..$

Combining Corollary 1.6 with Mackay’s upper bound, one can see that our lower
bound for the conformal dimension is in fact sharp:

######  Theorem 1.8.

Let $\frac{1}{3}<d<\frac{1}{2}$, $k\geq 2$ be constants. For $l$ divisible by
$3$, let $\Gamma$ be a random group in the model $\mathcal{D}(k,l,d)$. Then
there are universal constants $C^{\prime},C^{\prime\prime}$ independent of
$k,l,d$ such that it holds a.a.s. that

$C^{\prime}(d-\frac{1}{3})-\frac{C^{\prime\prime}}{(\log(2k-1))l}\leq\frac{{\rm
Confdim}(\partial_{\infty}\Gamma)}{(\log(2k-1))l}\leq\frac{16}{\log(2)}\frac{1}{1-2d}.$

######  Remark 1.9.

After the completion of this manuscript, we were informed about a forthcoming
work by Jordan Frost attaining a similar lower bound on the growth of the
conformal dimension for all $d<\frac{1}{2}$ (and not just
$\frac{1}{3}<d<\frac{1}{2}$ as in our work). We note that Frost methods are
completely different from ours and he does not attain results regarding Banach
fixed point properties.

Our proof of Theorem 1.5 follows the scheme of proof [KK13, Theorem B]: first,
we reduce the problem to another model of random groups in which the Cayley
complex of a random group is a partite simplicial complex and then we apply
our version of Żuk’s criterion (this is not completely straightforward and we
heavily use the ideas of [ALuS15] in the analysis).

###  1.3. Organization

The paper is organized as follows. In Section 2, we cover preliminary
material. In Section 3, we prove Banach Żuk’s criterion for groups acting on
partite complexes, i.e., we prove an extended version of Theorem 1.3. In
Section 4, we use our criterion to prove Banach fixed point properties for
random groups in the density model, i.e., we prove an extended version of
Theorem 1.5.

##  2\. Preliminaries

###  2.1. Robust Banach property (T) and the fixed point property

In [Opp17a], the author introduced the notion of robust Banach property (T) as
a strengthened version of Banach property (T) (as defined is [BFGM07]) that is
weaker than the notion of strong Banach property (T) defined by V. Lafforgue
[Laf09]

Let $G$ be a locally compact group. Let $\mathcal{F}$ be a family of linear
representations on Banach spaces, $\pi:G\rightarrow
B(\operatorname{\mathbb{E}})$, that are continuous with respect to the strong
operator topology. Denote $C_{c}(G)$ to be the compactly supported continuous
functions $f:G\rightarrow\mathbb{C}$. For every $\pi\in\mathcal{F}$, define
$\pi(f)\in B(\operatorname{\mathbb{E}})$ via the Bochner integral

$\pi(f).x=\int_{G}f(g)\pi(g).xdg$

(recall that $f$ is compactly supported and thus this integral converges).
Define the norm $\|.\|_{\mathcal{F}}$ on $C_{c}(G)$ as
$\|f\|_{\mathcal{F}}=\sup_{\pi\in\mathcal{F}}\|\pi(f)\|$.

If $\mathcal{F}$ is closed under complex conjugation (i.e.,
$\pi\in\mathcal{F}\Rightarrow\overline{\pi}\in\mathcal{F}$) and under duality
(i.e., $\pi\in\mathcal{F}\Rightarrow\pi^{*}\in\mathcal{F}$), then
$C_{\mathcal{F}}(G)$ is a Banach algebra with an involution

$f^{*}(g)=\overline{f(g^{-1})},\forall g\in G.$

######  Definition 2.1 (Robust Banach property (T)).

Let $G$ be a compactly generated group and let $K$ be some symmetric compact
set that generates $G$. For a class of Banach spaces $\mathcal{E}$ and a
constant $\beta\geq 1$, denote $\mathcal{F}(\mathcal{E},K,\beta)$ to be the
class of all the continuous representations $\pi$ of $G$ on some
$\operatorname{\mathbb{E}}\in\mathcal{E}$ such that $\sup_{g\in
K}\|\pi(g)\|\leq\beta$.

We say that $G$ has robust Banach property (T) with respect to a class of
Banach spaces $\mathcal{E}$, if there exists $\beta>1$ and a sequence of real
functions $f_{k}\in C_{c}(G)$ such that for every $k$, $\int f_{k}=1$ and such
that the sequence $(f_{k})$ converges in
$C_{\mathcal{F}(\mathcal{E},K,\beta)}$ to $p$ and
$\forall(\pi,\operatorname{\mathbb{E}})\in\mathcal{F}(\mathcal{E},K,\beta)$,
$\pi(p)$ is a projection on $\operatorname{\mathbb{E}}^{\pi(G)}$.

We will call the sequence $f_{k}$ above a Kazhdan projection with respect to
$\mathcal{F}(\mathcal{E},K,\beta)$.

######  Remark 2.2.

The above Definition assumes compact generation. A more general definition can
be found in [Opp17a]. Also, the definition in [Opp17a] assumes that the
functions $f_{k}$ are symmetric and thus (as noted in [dlS19]) $p$ is central.
Since the main focus of this paper is fixed point properties (see below),
centrality of $p$ is not needed and thus omitted from the definition.

Robust Banach property (T) is connected to the fixed point property defined as
follows:

######  Definition 2.3.

For a Banach space $\operatorname{\mathbb{E}}$, we say that $G$ has property
$(F\operatorname{\mathbb{E}})$ if every affine isometric action of $G$ on
$\operatorname{\mathbb{E}}$ has a fixed point. For a class of Banach spaces
$\mathcal{E}$, we say that $G$ has property $(F\mathcal{E})$ if for every
$\operatorname{\mathbb{E}}\in\mathcal{E}$, $G$ has property
$(F\operatorname{\mathbb{E}})$. In particular for $1\leq p\leq\infty$, a group
$G$ is said property $(FL^{p})$ if it has property
$(F\operatorname{\mathbb{E}})$ for every $\operatorname{\mathbb{E}}$ that is
an $L^{p}$-space.

######  Proposition 2.4.

[Opp17a, Proposition 5.9] Let $\mathcal{E}$ be a class of Banach spaces such
that for every $\operatorname{\mathbb{E}}\in\mathcal{E}$ it holds that
$\operatorname{\mathbb{E}}\oplus_{\ell^{2}}\mathbb{C}\in\mathcal{E}$. If $G$
has robust Banach property (T) with respect $\mathcal{E}$, then $G$ has
property $(F\mathcal{E})$.

It is worth noting that under some extra assumptions, then opposite direction
of this Proposition is also true (see [dlS19, Corollary 5.6]), but we will
make no use of this fact.

Last, we state some hereditary properties of property $(F\mathcal{E})$.

######  Proposition 2.5.

If $G$ has property $(F\mathcal{E})$ and $G^{\prime}$ is a quotient of $G$,
then $G^{\prime}$ has property $(F\mathcal{E})$.

###### Proof.

Every isometric action of $G^{\prime}$ on a Banach space
$\operatorname{\mathbb{E}}$ induces an isometric action of $G$ on
$\operatorname{\mathbb{E}}$. ∎

Also, under suitable assumptions, a group $G$ has property $(F\mathcal{E})$ if
and only if a finite index subgroup $H<G$ has property $(F\mathcal{E})$:

######  Proposition 2.6.

[BFGM07, Section 8] Let $G$ be as above and $H<G$ be a closed finite index
subgroup of $G$. For every super-reflexive space, if $H$ has property
$(F\operatorname{\mathbb{E}})$, then $G$ has property
$(F\operatorname{\mathbb{E}})$. Conversely, let $\mathcal{E}$ be a class of
Banach spaces such that every $\operatorname{\mathbb{E}}\in\mathcal{E}$ is
super-reflexive and such that $\mathcal{E}$ is closed under $\ell^{2}$ sums.
Then if $G$ has property $(F\mathcal{E})$ it follows that $H$ has property
$(F\mathcal{E})$.

###  2.2. Vector valued $\ell^{2}$ spaces

Given a finite set $V$, a function $m:V\rightarrow\mathbb{R}_{+}$ and a Banach
space $\operatorname{\mathbb{E}}$, we define the vector valued space
$\ell^{2}(V,m;\operatorname{\mathbb{E}})$ to be the space of functions
$\phi:V\rightarrow\operatorname{\mathbb{E}}$, with the norm

$\|\phi\|_{\ell^{2}(V,m;\operatorname{\mathbb{E}})}=\left(\sum_{v\in
V}m(v)|\phi(v)|^{2}\right)^{\frac{1}{2}},$

where $|.|$ is the norm of $\operatorname{\mathbb{E}}$. We denote
$\ell^{2}(V,m)=\ell^{2}(V,m;\mathbb{C})$ and recall that $\ell^{2}(V,m)$ is
also a Hilbert space with the inner-product

$\langle\phi,\psi\rangle=\sum_{v\in V}m(v)\phi(v)\overline{\psi(v)}.$

Let $T:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$ be a linear operator and
$(T_{v,u})_{u,v\in V}\in M_{|V|}(\mathbb{C})$ be the matrix such that for
every $\phi\in\ell^{2}(V,m)$ it holds that

$(T\phi)(v)=\sum_{u\in V}T_{v,u}\phi(u).$

Define
$T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}:\ell^{2}(V,m;\operatorname{\mathbb{E}})\rightarrow\ell^{2}(V,m;\operatorname{\mathbb{E}})$
by the formula:

$((T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi)(v)=\sum_{u\in
V}T_{v,u}\phi(u),$

for every $\phi\in\ell^{2}(V,m;\operatorname{\mathbb{E}})$. We denote
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}$
to be the operator norm of
$T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}$.

The norm
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}$
is preserved under some operations on $\operatorname{\mathbb{E}}$ \- this is
summed up in the following Lemma:

######  Lemma 2.7.

Let $V$ be a finite set, $T$ a bounded operator on $\ell^{2}(V,m)$ and $C>0$
constant. Let $\mathcal{E}=\mathcal{E}(C)$ be the class of Banach spaces
defined as:

$\mathcal{E}=\\{\operatorname{\mathbb{E}}:\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq
C\\}.$

Then this class is closed under quotients, subspaces, $\ell^{2}$-sums and
ultraproducts of Banach spaces, i.e., preforming any of these operations on
Banach spaces in $\mathcal{E}$ yield a Banach space in $\mathcal{E}$.

###### Proof.

The fact that $\mathcal{E}$ is closed under quotients, subspaces and
ultraproducts of Banach spaces was shown in [Sal16, Lemma 3.1]. The fact that
$\mathcal{E}$ is closed under $\ell^{2}$-sums is straight-forward and left for
the reader. ∎

###  2.3. Uniformly curved Banach spaces

Uniformly curved Banach spaces were introduced by Pisier in [Pis10]:

######  Definition 2.8 (Fully contractive operator).

An operator $T:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$ is called fully
contractive if for every Banach space $\operatorname{\mathbb{E}}$ it holds
that
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq
1$.

######  Definition 2.9 (Uniformly curved space).

Let $\operatorname{\mathbb{E}}$ be a Banach space. The space
$\operatorname{\mathbb{E}}$ is called uniformly curved if for every
$0<\varepsilon\leq 1$ there is $\delta>0$ such that for every space
$\ell^{2}(V,m)$ and every fully contractive linear operator
$T:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$, if
$\|T\|_{B(\ell^{2}(V,m))}\leq\delta$, then
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq\varepsilon$.

Given a monotone increasing function $\omega:(0,1]\rightarrow(0,1]$ such that

$\lim_{t\rightarrow 0^{+}}\omega(t)=0,$

we denote $\mathcal{E}^{\text{u-curved}}_{\omega}$ to be the class of all
uniformly curved Banach spaces $\operatorname{\mathbb{E}}$ such that for every
space $\ell^{2}(V,m)$ and every fully contractive linear operator
$T:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$, if
$\|T\|_{B(\ell^{2}(V,m))}\leq\delta$, then
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq\omega(\delta)$.

Applying Lemma 2.7 on $\mathcal{E}^{\text{u-curved}}_{\omega}$ defined above
yields the following Corollary:

######  Corollary 2.10.

For any monotone increasing function $\omega:(0,1]\rightarrow(0,1]$ such that
$\lim_{t\rightarrow 0^{+}}\omega(t)=0$, the class
$\mathcal{E}^{\text{u-curved}}_{\omega}$ defined above is closed under
quotients, subspaces, $\ell^{2}$-sums and ultraproducts of Banach spaces.

######  Proposition 2.11.

Let $T:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$ be a linear operator and $L\geq
1$, $0<\delta\leq 1$ be constants such that:

  1. (1)

It holds that $\|T\|_{B(\ell^{2}(V,m))}\leq\delta$.

  2. (2)

For every Banach space $\operatorname{\mathbb{E}}$,
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq
L$.

Then for every monotone increasing function $\omega:(0,1]\rightarrow(0,1]$
such that $\lim_{t\rightarrow 0^{+}}\omega(t)=0$ and every
$\operatorname{\mathbb{E}}\in\mathcal{E}^{\text{u-curved}}_{\omega}$,
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq
L\omega(\delta)$.

###### Proof.

We note that $\frac{1}{L}T$ is a fully contractive operator such that

$\|\frac{1}{L}T\|_{B(\ell^{2}(V,m))}\leq\frac{\delta}{L}.$

Thus, by the definition of $\mathcal{E}^{\text{u-curved}}_{\omega}$ it follows
for every $\operatorname{\mathbb{E}}\in\mathcal{E}^{\text{u-curved}}_{\omega}$
that

$\|(\frac{1}{L})T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq\omega(\frac{\delta}{L}),$

and thus

$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq
L\omega(\frac{\delta}{L})\leq L\omega(\delta),$

where the last inequality is due to the fact that $L\geq 1$ and $\omega$ is
monotone increasing. ∎

###  2.4. Strictly $\theta$-Hilbertian spaces

Here we will describe a special class of uniformly curved Banach spaces that
contains all (commutative and non-commutative) $L^{p}$ spaces.

Two Banach spaces
$\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1}$ form a
compatible pair
$(\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1})$ if they are
continuously linear embedded in the same topological vector space. The idea of
complex interpolation is that given a compatible pair
$(\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1})$ and a constant
$0\leq\theta\leq 1$, there is a method to produce a new Banach space
$[\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1}]_{\theta}$ as a
“convex combination” of $\operatorname{\mathbb{E}}_{0}$ and
$\operatorname{\mathbb{E}}_{1}$. We will not review this method here, and the
interested reader can find more information on interpolation in [BL76].

This brings us to consider the following definition due to Pisier [Pis79]: a
Banach space $\operatorname{\mathbb{E}}$ is called strictly
$\theta$-Hilbertian for $0<\theta\leq 1$, if there is a compatible pair
$(\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1})$ with
$\operatorname{\mathbb{E}}_{1}$ a Hilbert space such that
$\operatorname{\mathbb{E}}=[\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1}]_{\theta}$.
Examples of strictly $\theta$-Hilbertian spaces are $L^{p}$ space and non-
commutative $L^{p}$ spaces (see [PX03] for definitions and properties of non-
commutative $L^{p}$ spaces), where in these cases $\theta=\frac{2}{p}$ if
$2\leq p<\infty$ and $\theta=2-\frac{2}{p}$ if $1<p\leq 2$.

For our use, it will be important to bound the norm of an operator of the form
$T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}$ given that
$\operatorname{\mathbb{E}}$ is an interpolation space.

######  Lemma 2.12.

[Sal16, Lemma 3.1] Let
$(\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1})$ be a
compatible pair , $V$ be a finite set, $m:V\rightarrow\mathbb{R}_{+}$ be a
function and $T\in B(\ell^{2}(V,m))$ be an operator. Then for every
$0\leq\theta\leq 1$,

$\|T\otimes\operatorname{id}_{[\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1}]_{\theta}}\|_{B(\ell^{2}(V,m;[\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1}]_{\theta}))}\leq\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}_{0}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}_{0}))}^{1-\theta}\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}_{1}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}_{1}))}^{\theta},$

where $[\operatorname{\mathbb{E}}_{0},\operatorname{\mathbb{E}}_{1}]_{\theta}$
is the interpolation of $\operatorname{\mathbb{E}}_{0}$ and
$\operatorname{\mathbb{E}}_{1}$.

This Lemma has the following Corollary that shows that strictly
$\theta$-Hilbertian spaces are uniformly curved (see also [Sal16, Lemma 3.1]):

######  Corollary 2.13.

Let $\operatorname{\mathbb{E}}$ be a strictly $\theta$-Hilbertian space with
$0<\theta\leq 1$, $V$ be a finite set, $m:V\rightarrow\mathbb{R}_{+}$ be a
function and $0<\delta<1$ be a constant. Assume that $T\in B(\ell^{2}(V,m))$
is a fully contractive operator such that
$\|T\|_{B(\ell^{2}(V,m))}\leq\delta$. Then
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq\delta^{\theta}$.

In other words, if $\operatorname{\mathbb{E}}$ is strictly $\theta$-Hilbertian
space with $0<\theta\leq 1$, then for $\omega(t)=t^{\theta}$, we have that
$\operatorname{\mathbb{E}}\in\mathcal{E}^{\text{u-curved}}_{\omega}$.

###### Proof.

For every Hilbert space $\operatorname{\mathbb{E}}_{1}$ we have that
$\|T\otimes\operatorname{id}_{\operatorname{\mathbb{E}}_{1}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}_{1}))}\leq\delta$
and thus the assertion stated above follows from Lemma 2.12. ∎

Combining Lemma 2.7 with the above Corollary yields:

######  Corollary 2.14.

For a constant $0<\theta_{0}\leq 1$, denote $\mathcal{E}_{\theta_{0}}$ to be
the smallest class of Banach spaces that contains all strictly
$\theta$-Hilbertian Banach spaces for all $\theta_{0}\leq\theta\leq 1$ and is
closed under subspaces, quotients, $\ell^{2}$-sums and ultraproducts of Banach
spaces. Then for every $0<\theta_{0}\leq 1$, we have that
$\mathcal{E}_{\theta_{0}}\subseteq\mathcal{E}^{\text{u-curved}}_{\omega(t)=t^{\theta_{0}}}$.

######  Remark 2.15.

A deep result of Pisier shows that the converse of the above Corollary is
“almost true” if one considers arcwise $\theta_{0}$-Hilbertian spaces (see
definition in [Pis10, Section 6]). Namely, by [Pis10, Corollary 6.7], for
every $\theta_{0}<\theta\leq 1$ it holds that every Banach space in
$\mathcal{E}^{\text{u-curved}}_{\omega(t)=t^{\theta}}$ is a subquotient of an
arcwise $\theta_{0}$-Hilbertian space. We will not define arcwise
$\theta_{0}$-Hilbertian spaces here and we will make no use of this fact.

###  2.5. Angle between projections

The notion of an angle between projection was defined by the author in
[Opp17a] and further developed in [Opp17b] and[Opp18]. Below, we give the
definitions and results on this subject that are needed for this paper.

######  Definition 2.16.

Let $\operatorname{\mathbb{E}}$ be a Banach space and let $P_{1},P_{2}$ be
projections in $B(\operatorname{\mathbb{E}})$. Assume that there is a
projection $P_{1,2}$ on
$\operatorname{Im}(P_{1})\cap\operatorname{Im}(P_{2})$, such that
$P_{1,2}P_{1}=P_{1,2},P_{1,2}P_{2}=P_{1,2}$. Define the cosine of the angle
between $P_{1},P_{2}$ (with respect to $P_{1,2}$) as

$\cos_{P_{1,2}}(\angle(P_{1},P_{2}))=\max\\{\|P_{1}P_{2}-P_{1,2}\|,\|P_{2}P_{1}-P_{1,2}\|\\}.$

We note that the angle between $P_{1},P_{2}$ depends on the choice of
$P_{1,2}$ and different choices yield different angles (see [BKRS20, Example
2]). However, to avoid cumbersome notation, when $P_{1,2}$ is obvious from the
context, we will denote $\cos(\angle(P_{1},P_{2}))$ instead of
$\cos_{P_{1,2}}(\angle(P_{1},P_{2}))$.

######  Theorem 2.17.

[Opp17a, Theorem 3.12] Let $\operatorname{\mathbb{E}}$ be a Banach space and
let $P_{0},...,P_{n}$ be projections in $B(\operatorname{\mathbb{E}})$ ($n\geq
1$). Assume that for every $0\leq j_{1}<j_{2}\leq n$, there is a projection
$P_{j_{1},j_{2}}$ on
$\operatorname{Im}(P_{j_{1}})\cap\operatorname{Im}(P_{j_{2}})$, such that
$P_{j_{1},j_{2}}P_{j_{1}}=P_{j_{1},j_{2}},P_{j_{1},j_{2}}P_{j_{2}}=P_{j_{1},j_{2}}$.
Denote $T=\sum_{k}\frac{1}{n+1}P_{k}$. Assume further that there are constants

$\gamma<\frac{1}{8n-3}\text{ and
}\beta<1+\frac{1-(8n-3)\gamma}{n-1+(3n-1)\gamma},$

such that

$\max_{0\leq j\leq n}\|P_{j}\|\leq\beta$

and

$\max_{0\leq j_{1}<j_{2}<\leq n}\cos(\angle(P_{j_{1}},P_{j_{2}}))\leq\gamma.$

Then there is a projection $T^{\infty}$ on
$\bigcap_{j=0}^{n}\operatorname{Im}(P_{j})$ and constants $0\leq
r(\gamma,\beta)<1,C(\gamma,\beta)>0$, such that for every $i$,

$\|T^{i}-T^{\infty}\|\leq Cr^{i}$

and in particular, $T^{i}$ converges to $T^{\infty}$ in the operator norm.

######  Remark 2.18.

The conditions for the convergence of $T^{i}$ stated above are not optimal and
can be somewhat improved using [Opp18][Theorem 2.2].

###  2.6. Random walks on bipartite finite graphs

Here we collect some facts regrading
$\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}$ defined
in the introduction. To ease the reading, we will repeat the definition and
also repeat several definitions and facts that were already mentioned in the
introduction.

Given a finite graph $(V,E)$, a weight function on $(V,E)$ is a function
$m:E\rightarrow\mathbb{R}_{+}$ and $(V,E)$ with a weight function is called a
weighted graph. Given a weighted graph as above, we define for every $v\in V$,
$m(v)=\sum_{e\in E,v\in e}m(e)$ and for a non-empty subset $U\subseteq V$,
define $m(U)=\sum_{v\in U}m(v)$.

The graph $(V,E)$ is called bipartite if $V$ can be partitioned into two
disjoint sets $V=S_{1}\cup S_{2}$ called sides such that for each
$\\{u,v\\}\in E$, $|\\{u,v\\}\cap S_{1}|=|\\{u,v\\}\cap S_{2}|=1$, i.e., each
edge has exactly one vertex in each side. We note that it follows that
$m(S_{1})=m(S_{2})=\frac{1}{2}m(V)$.

We also define $\ell^{2}(V,m)$ as in section 2.2 above, i.e., $\ell^{2}(V,m)$
is the space of functions $\phi:V\rightarrow\mathbb{C}$ with an inner-product

$\langle\phi,\psi\rangle=\sum_{\\{v\\}\in V}=m(v)\phi(v)\overline{\psi(v)}.$

The random walk on $(V,E)$ as above is the operator
$A:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$ defined as

$(A\phi)(v)=\sum_{u\in V,\\{u,v\\}\in E}\frac{m(\\{u,v\\})}{m(v)}\phi(u).$

In the case where $m$ is constant $1$ on all the edges, then for every vertex
$v$, $m(v)$ is the valency of $v$ and $A$ is called the simple random walk on
$(V,E)$.

We state without proof a few basic facts regarding the random walk operator:

  1. (1)

With the inner-product defined above, $A$ is a self-adjoint operator and the
eigenvalues of $A$ lie in the interval $[-1,1]$.

  2. (2)

The space of constant functions is an eigenspace of $A$ with eigenvalue $1$
and if $(V,E)$ is connected all other the other eigenfunctions of $A$ have
eigenvalues strictly less than $1$.

  3. (3)

The graph $(V,E)$ is bipartite if and only if $-1$ is an eigenvalue of $A$.

Assuming that $(V,E)$ is bipartite with sides $S_{1},S_{2}$, we note that the
spectrum of $A$ is symmetric: Explicitly, if $\lambda$ is an eigenvalue of $A$
with an eigenfunction $\phi$, then

$\phi^{\prime}(u)=\begin{cases}\phi(u)&u\in S_{1}\\\ -\phi(u)&u\in
S_{2}\end{cases}$

is an eigenfunction of $A$ with an eigenvalue $-\lambda$. In particular
$\phi=\mathbbm{1}_{S_{1}}-\mathbbm{1}_{S_{2}}$ is an eigenfunction with the
eigenvalue $-1$.

We define the following averaging operators:
$M_{1},M_{2}:\ell^{2}(V,m)\rightarrow\mathbb{C}$:

$M_{i}\phi=\frac{1}{m(S_{i})}\sum_{u\in S_{i}}\phi(u).$

We also define
$M_{\operatorname{sides}}:\ell^{2}(V,m)\rightarrow\ell^{2}(V,m)$ by

$M_{\operatorname{sides}}\phi(u)=\begin{cases}M_{1}\phi&u\in S_{1}\\\
M_{2}\phi&u\in S_{2}\end{cases}.$

We note that $M_{\operatorname{sides}}$ is the orthogonal projection on the
space of functions
$\operatorname{span}\\{\mathbbm{1}_{V},\mathbbm{1}_{S_{i}}-\mathbbm{1}_{S_{j}}\\}$.

We recall the following definition of spectral expansion:

######  Definition 2.19.

Let $(V,E)$ be a finite connected graph with a weight function $m$ and
$0\leq\lambda<1$ be a constant. The graph $(V,E)$ is called a one-sided
$\lambda$-spectral expander if the spectrum of $A$ is contained in
$[-1,\lambda]\cup\\{1\\}$.

As noted above, for a bipartite graph the spectrum of $A$ is symmetric with an
eigenvalue $-1$ with an eigenfunction
$\mathbbm{1}_{S_{1}}-\mathbbm{1}_{S_{2}}$. Thus, for a connected bipartite
graph $(V,E)$ it holds that the graph is a one-sided $\lambda$-spectral
expander if and only if $\|A(I-M_{\operatorname{sides}})\|\leq\lambda$.

Given a Banach space $\operatorname{\mathbb{E}}$, we consider the operator
$(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}:\ell^{2}(V,m;\operatorname{\mathbb{E}})\rightarrow\ell^{2}(V,m;\operatorname{\mathbb{E}})$
and denote
$\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}=\|(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}$.

######  Claim 2.20.

For every graph $(V,E)$ and every Banach space $\operatorname{\mathbb{E}}$,
$\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}\leq 2$.

###### Proof.

By triangle inequality and linearity,

$\|(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq\\\
\|A\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}+\|A\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\|M_{\operatorname{sides}}\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))},$

and therefore in order to prove the claim, it is enough to show that

$\|A\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq
1\text{ and
}\|M_{\operatorname{sides}}\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|_{B(\ell^{2}(V,m;\operatorname{\mathbb{E}}))}\leq
1.$

Indeed, by the convexity of the function $|.|^{2}$, for every
$\phi\in\ell^{2}(V,m;\operatorname{\mathbb{E}})$,

$\|(A\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi\|^{2}=\\\
\sum_{v\in V}m(v)\left|\sum_{u\in V,\\{u,v\\}\in
E}\frac{m(\\{u,v\\})}{m(v)}\phi(u)\right|^{2}\leq\\\ \sum_{v\in
V}m(v)\sum_{u\in V,\\{u,v\\}\in E}\frac{m(\\{u,v\\})}{m(v)}|\phi(u)|^{2}=\\\
\sum_{u\in V}|\phi(u)|^{2}\sum_{v\in V,\\{u,v\\}\in E}m(\\{u,v\\})=\\\
\sum_{u\in V}m(u)|\phi(u)|^{2}=\|\phi\|^{2},$

and

$\|(M_{\operatorname{sides}}\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi\|^{2}=\\\
\sum_{v\in S_{1}}m(v)\left|\frac{1}{m(S_{1})}\sum_{u\in
S_{i}}m(u)\phi(u)\right|^{2}+\sum_{v\in
S_{2}}m(v)\left|\frac{1}{m(S_{2})}\sum_{u\in
S_{j}}m(u)\phi(u)\right|^{2}\leq\\\ \sum_{v\in
S_{1}}m(v)\frac{1}{m(S_{1})}\sum_{u\in S_{i}}m(u)|\phi(u)|^{2}+\sum_{v\in
S_{2}}m(v)\frac{1}{m(S_{2})}\sum_{u\in S_{2}}m(u)|\phi(u)|^{2}=\\\ \sum_{u\in
S_{1}}m(u)|\phi(u)|^{2}+\sum_{u\in S_{2}}m(u)|\phi(u)|^{2}=\|\phi\|^{2}.$

∎

Combining this Claim with Lemma 2.11 and Corollary 2.13 yields:

######  Corollary 2.21.

Let $(V,E)$ be a connected finite graph with a weight function $m$ and
$0<\lambda<1$ be a constant such that $(V,E)$ is a one-sided
$\lambda$-spectral expander. For every monotone increasing function
$\omega:(0,1]\rightarrow(0,1]$ such that $\lim_{t\rightarrow
0^{+}}\omega(t)=0$ and every
$\operatorname{\mathbb{E}}\in\mathcal{E}^{\text{u-curved}}_{\omega}$, we have
that

$\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}\leq
2\omega(\lambda).$

In particular, for every $0<\theta\leq 1$ and every strictly
$\theta$-Hilbertian space $\operatorname{\mathbb{E}}$, we have that

$\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}\leq
2\lambda^{\theta}.$

The expansion constant
$\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}$ can
also be described as a cosine between projections as in Definition 2.16. In
order to do this, we define $\ell^{2}(E;\operatorname{\mathbb{E}})$ to be the
space of functions $\Phi:E\rightarrow\operatorname{\mathbb{E}}$ with the norm

$\|\Phi\|^{2}=\sum_{\\{u,v\\}\in E}|\Phi(\\{u,v\\})|^{2}.$

For a bipartite graph $(V,E)$ with sides $S_{1},S_{2}$ we define the following
projections on $\ell^{2}(E;\operatorname{\mathbb{E}})$: For $i=1,2$, and
$\\{v_{1},v_{2}\\}\in E$ with $v_{i}\in S_{i}$

$P_{i}\Phi(\\{v_{1},v_{2}\\})=\frac{1}{m(v_{i})}\sum_{\\{v_{i},u\\}\in
E}\Phi(\\{v_{i},u\\}),\forall\Phi\in\ell^{2}(E;\operatorname{\mathbb{E}}).$

We also define

$P_{1,2}\Phi\equiv\frac{1}{|E|}\sum_{\\{u,v\\}\in
E}\Phi(\\{u,v\\}),\forall\Phi\in\ell^{2}(E;\operatorname{\mathbb{E}}).$

Since $P_{1,2}$ is a projection on the space of constant functions, it follows
that $P_{i}P_{1,2}=P_{1,2}$. One can also verify that $P_{1,2}P_{i}=P_{1,2}$
(we leave this to the reader). Thus, we can define $\cos(\angle(P_{1},P_{2}))$
as in Definition 2.16.

######  Proposition 2.22.

For a finite connected bipartite graph $(V,E)$ and a Banach space
$\operatorname{\mathbb{E}}$ it holds that

$\cos(\angle(P_{1},P_{2}))=\lambda_{(V,E),\operatorname{bipartite}}^{\operatorname{\mathbb{E}}},$

where $P_{1},P_{2}$ are the projections defined above.

###### Proof.

For $i=1,2$, define $\ell^{2}_{i}(V,m;\operatorname{\mathbb{E}})$ to be the
subspace of $\ell^{2}(V,m;B)$ composed of functions supported on $S_{i}$. Note
every $\phi\in\ell^{2}(V,m;B)$ can be decomposed to $\phi=\phi_{1}+\phi_{2}$
where $\phi_{i}\in\ell^{2}_{i}(V,m;\operatorname{\mathbb{E}})$ by defining

$\phi_{i}(v)=\begin{cases}\phi(v)&v\in S_{i}\\\ 0&v\notin S_{i}\end{cases}.$

Also note that for this decomposition
$\|\phi\|^{2}=\|\phi_{1}\|^{2}+\|\phi_{2}\|^{2}$ and that

$\|((A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi\|^{2}=\|((A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi_{1}\|^{2}+\|((A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi_{2}\|^{2}.$

Thus for every non-zero $\phi$,

$\frac{\|((A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi\|^{2}}{\|\phi\|^{2}}=\frac{\|((A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi_{1}\|^{2}+\|((A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}})\phi_{2}\|^{2}}{\|\phi_{1}\|^{2}+\|\phi_{2}\|^{2}}.$

It follows that

$\|(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\|=\max_{i=1,2}\|\left.(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\right|_{\ell^{2}_{i}(V,m;\operatorname{\mathbb{E}})}\|.$

Define
$L_{i}:\ell^{2}_{i}(V,m;\operatorname{\mathbb{E}})\rightarrow\ell^{2}(E;\operatorname{\mathbb{E}})$
by

$L_{i}\phi(\\{v_{1},v_{2}\\})=\phi(v_{i}),$

for all $\\{v_{1},v_{2}\\}\in E$ such that $v_{1}\in S_{1},v_{2}\in S_{2}$.
Note that $L_{i}$ is an isometry onto
$\operatorname{Im}(P_{i})\subseteq\ell^{2}(E;\operatorname{\mathbb{E}})$.
Observe that by the definition of all the operators it follows that

$\left.(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\right|_{\ell^{2}_{1}(V,m;\operatorname{\mathbb{E}})}=L_{2}^{-1}P_{2}(I-P_{1,2})L_{1}=L_{2}^{-1}P_{2}(I-P_{1,2})P_{1}L_{1}=L_{2}^{-1}(P_{2}P_{1}-P_{1,2})P_{1}L_{1},$

and

$\left.(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\right|_{\ell^{2}_{2}(V,m;\operatorname{\mathbb{E}})}=L_{1}^{-1}(P_{1}P_{2}-P_{1,2})L_{2}.$

Thus, using the fact that $L_{i}$ are isometries onto
$\operatorname{Im}(P_{i})$ it follows that

$\|\left.(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\right|_{\ell^{2}_{1}(V,m;\operatorname{\mathbb{E}})}\|=\|P_{2}P_{1}-P_{1,2}\|,$

and

$\|\left.(A(I-M_{\operatorname{sides}}))\otimes\operatorname{id}_{\operatorname{\mathbb{E}}}\right|_{\ell^{2}_{2}(V,m;\operatorname{\mathbb{E}})}\|=\|P_{1}P_{2}-P_{1,2}\|,$

as needed.

∎

###  2.7. Weighted simplicial complexes

Given a set $V$, an abstract simplicial complex $X$ with a vertex set $V$ is a
family of subsets $X\subseteq 2^{V}$ such that if $\tau\in X$ and
$\eta\subseteq\tau$, then $\eta\in X$. We will denote $X(k)$ to be the sets in
$X$ of cardinality $k+1$. A simplicial complex $X$ is called $n$-dimensional
if $X(n+1)=\emptyset$. An $n$-dimensional simplicial complex $X$ is called
pure $n$-dimensional if for every $\tau$ in $X$, there is $\sigma\in X(n)$
such that $\tau\subseteq\sigma$.

A pure $n$-dimensional simplicial complex $X$ is called gallery connected if
for every $\sigma,\sigma^{\prime}\in X(n)$, there is a finite sequence
$\sigma_{1},...,\sigma_{l}\in X(n)$ such that
$\sigma=\sigma_{1},\sigma^{\prime}=\sigma_{l}$ and for every $1\leq i\leq
l-1$, $\sigma_{i}\cap\sigma_{i+1}\in X(n-1)$. Below, we will always assume
that $X$ is pure $n$-dimensional and gallery connected.

We define the following weight function
$m:\bigcup_{k=0}^{n}\operatorname{X}(k)\rightarrow\mathbb{R}\cup\\{\infty\\}$
inductively as follows:

$\forall\sigma\in\operatorname{X}(n),m(\sigma)=1,$

For $0\leq k\leq n-1$ and $\tau\in X(k)$,

$m(\tau)=\sum_{\sigma\in\operatorname{X}(k+1),\tau\subseteq\sigma}m(\sigma).$

More explicitly,

$\forall\tau\in\operatorname{X}(k),m(\tau)=(n-k)!|\\{\sigma\in\operatorname{X}(n):\tau\subseteq\sigma\\}|.$

Given a simplex $\tau\in X$, the link of $\tau$ is the subcomplex of $X$,
denoted $X_{\tau}$, that is defined as

$X_{\tau}=\\{\eta\in X:\tau\cap\eta=\emptyset,\tau\cup\eta\in X\\}.$

We define a weight function $m_{\tau}$ on $X_{\tau}$ by

$m_{\tau}(\eta)=m(\tau\cup\eta).$

Below we will only be interested in the $1$-dimensional links of $X$: We note
that if $\tau\in X(k)$ and $X$ is pure $n$-dimensional, then $X_{\tau}$ is
pure $(n-k-1)$-dimensional. In particular, if $\tau\in X(n-2)$, then
$X_{\tau}$ is a graph. Thus, we will refer to all links of the form $X_{\tau}$
where $\tau\in X(n-2)$ as the $1$-dimensional links of $X$. For
$1$-dimensional links, the weight function is easy to describe: For a
$1$-dimensional link $X_{\tau}$, we note that this weight function assigns the
weight $1$ for each edge in the link $X_{\tau}$ and for each every $\\{u\\}\in
X_{\tau}(0)$, $m_{\tau}(\\{u\\})$ is the valency of $\\{u\\}$ in $X_{\tau}$.

Given a pure $n$-dimensional simplicial complex $X$, we call $X$ partite if
there is a partition of the vertex set $V$, $V_{0}\sqcup...\sqcup V_{n}=V$
such that for every $\sigma\in X(n)$ and every $0\leq i\leq n$ it holds that
$|\sigma\cap V_{i}|=1$, i.e., every $n$-dimensional simplex has exactly one
vertex in each of the sets $V_{0},...,V_{n}$. A partite simplicial complex is
also sometimes called colorable, since we can think of the partition
$V_{0},...,V_{n}$ as a coloring of the vertex sets with $n+1$ colors, such
that each $n$-dimensional simplex has vertices with all the colors (or
equivalently, each $n$-dimensional simplex do not have two vertices with the
same color). For an $n$-dimensional partite simplicial complex, we define a
type function $\operatorname{type}:X\rightarrow 2^{\\{0,...,n\\}}$ by

$\operatorname{type}(\tau)=\\{i:\exists v\in\tau\cap V_{i}\\}.$

##  3\. Żuk’s criterion for partite links

Let $X$ be a partite pure $n$-dimensional simplicial complex such that $X$ is
gallery connected and the $1$-dimensional links of $X$ are connected finite
graphs. Also let $G$ be a locally compact, unimodular group acting on $X$ such
that the action is cocompact and for every $\tau\in X(n-2)\cup X(n-1)\cup
X(n)$ the subgroup stabilizing $\tau$, denoted $G_{\tau}$, is an open compact
subgroup. We also assume that the action of $G$ is type-preserving, i.e., for
every $0\leq k\leq n$ and every $\tau\in X(k)$, it holds for every $g\in G$
that $\operatorname{type}(\tau)=\operatorname{type}(g.\tau)$. Last, let $\pi$
be a continuous representation $\pi$ of $G$ on a Banach space
$\operatorname{\mathbb{E}}$.

Note that by the assumption that the action of $G$ is type preserving it
follows for each $\sigma\in X(k)$ that if $g.\sigma=\sigma$, then for each
vertex $v\in\sigma$ it holds that $g.v=v$. For every $n-2\leq k\leq n$, we
choose a fundamental domain $D(k)$ of the action of $G$ on $X(k)$ such that
the following holds: for every $0\leq k_{1}<k_{2}\leq n$ and every $\tau\in
D(k_{1})$ there is $\sigma\in D(k_{2})$ such that $\tau\subseteq\sigma$.

Define $C(X(n),\pi)$ to be the space of maps
$\phi:X(n)\rightarrow\operatorname{\mathbb{E}}$ that are equivariant with
respect to $\pi$, i.e., for every $\sigma\in X(n)$ and every $g\in G$,
$\pi(g)\phi(\sigma)=\phi(g.\sigma)$. Define a norm on $C(X(n),\pi)$ by

$\|\phi\|^{2}=\sum_{\sigma\in
D(n)}\frac{1}{\mu(G_{\sigma})}|\phi(\sigma)|^{2},$

where $G_{\sigma}$ is the subgroup stabilizing $\sigma$ and $\mu$ is the Haar
measure of $G$.

Define $K\subseteq G$ to be the set

$K=\\{g\in G:\exists\sigma,\sigma^{\prime}\in
D(n),|g.\sigma\cap\sigma^{\prime}|\geq n-1\\}.$

Note that $K$ is a compact symmetric set that generates $G$ (the last fact is
due to the assumption that $X$ is gallery connected).

The following Lemma is a variant of [DJ00, Lemma 3.3]:

######  Lemma 3.1.

Let $n-2\leq k<l\leq n$ and

$\psi:\\{(\tau,\eta)\in X(k)\times
X(l):\tau\subseteq\eta\\}\rightarrow[0,\infty).$

Assume that there is a constant $\kappa\geq 1$ such that for every $g\in K$
and every $(\tau,\eta)\in X(k)\times X(l),\tau\subseteq\eta$ it holds that
$\psi(g.\tau,g.\eta)\leq\kappa\psi(\tau,\eta)$. Then

$\sum_{\eta\in D(l)}\frac{1}{\mu(G_{\eta})}\sum_{\tau\in
X(k),\tau\subseteq\eta}\psi(\tau,\eta)\leq\kappa^{2}\sum_{\tau\in
D(k)}\frac{1}{\mu(G_{\tau})}\sum_{\eta\in
X(k),\tau\subseteq\eta}\psi(\tau,\eta)$

and

$\sum_{\eta\in D(l)}\frac{1}{\mu(G_{\eta})}\sum_{\tau\in
X(k),\tau\subseteq\eta}\psi(\tau,\eta)\geq\frac{1}{\kappa^{2}}\sum_{\tau\in
D(k)}\frac{1}{\mu(G_{\tau})}\sum_{\eta\in
X(k),\tau\subseteq\eta}\psi(\tau,\eta).$

###### Proof.

Note that since $K$ is symmetric, it follows that
$\psi(g.\tau,g.\eta)\geq\frac{1}{\kappa}\psi(\tau,\eta)$ for every $g\in K$
and every $(\tau,\eta)\in X(k)\times X(l),\tau\subseteq\eta$. Thus the proofs
of both inequalities are similar (only using reverse inequalities) and we will
only prove the first one.

For every $\tau\in D(k),\eta\in D(l)$, we fix a maximal set
$A_{(\tau,\eta)}\subseteq G$ such that $\forall g\in A_{(\tau,\eta)}$,
$g.\tau\subseteq\eta$ and for $g,g^{\prime}\in A_{(\tau,\eta)}$, if $g\neq
g^{\prime}$, then $g.\tau\neq g^{\prime}.\tau$ (it may be that set
$A_{(\tau,\eta)}$ is the empty set). Note that by the choice of $D(k),D(l)$,
it follows that $A_{(\tau,\eta)}\subseteq K$. Thus

$\sum_{\eta\in D(l)}\frac{1}{\mu(G_{\eta})}\sum_{\tau\in
X(k),\tau\subseteq\eta}\psi(\tau,\eta)=\sum_{\eta\in
D(l)}\frac{1}{\mu(G_{\eta})}\sum_{\tau\in D(k)}\sum_{g\in
A_{(\tau,\eta)}}\psi(g.\tau,\eta)\leq\kappa\sum_{\eta\in D(l),\tau\in
D(k)}\frac{1}{\mu(G_{\eta})}\sum_{g\in
A_{(\tau,\eta)}}\psi(\tau,g^{-1}.\eta)\leq\kappa^{2}\sum_{\eta\in D(l),\tau\in
D(k)}\frac{1}{\mu(G_{\eta})}\sum_{g\in
A_{(\tau,\eta)}}\frac{1}{\mu(G_{\tau})}\int_{G_{\tau}}\psi(\tau,g^{\prime}g^{-1}.\eta)d\mu(g^{\prime})=\kappa^{2}\sum_{\tau\in
D(k)}\frac{1}{\mu(G_{\tau})}\sum_{\eta\in
D(l)}\frac{1}{\mu(G_{\eta})}\int_{g^{\prime\prime},\tau\subseteq
g^{\prime\prime}.\eta}\psi(\tau,g^{\prime\prime}.\eta)d\mu(g^{\prime\prime})=\kappa^{2}\sum_{\tau\in
D(k)}\frac{1}{\mu(G_{\tau})}\sum_{\eta\in
X(l),\tau\subseteq\eta}\psi(\tau,\eta).$

∎

######  Corollary 3.2.

Let $\nu\subseteq\\{0,...,n\\}$ such that $|\nu|=n-1$ or $|\nu|=n$. For every
$\phi\in C(X(n),\pi)$ it holds that

$\|\phi\|^{2}\leq\left(\sup_{g\in K}\|\pi(g)\|^{2}\right)\sum_{\tau\in
D(|\nu|-1),\operatorname{type}(\tau)=\nu}\frac{1}{\mu(G_{\tau})}\sum_{\sigma\in
X(n),\tau\subseteq\eta}|\phi(\sigma)|^{2},$

and

$\|\phi\|^{2}\geq\left(\sup_{g\in K}\|\pi(g)\|^{2}\right)^{-1}\sum_{\tau\in
D(|\nu|-1),\operatorname{type}(\tau)=\nu}\frac{1}{\mu(G_{\tau})}\sum_{\sigma\in
X(n),\tau\subseteq\eta}|\phi(\sigma)|^{2},$

###### Proof.

The proofs of both the inequalities are similar and we will prove only the
first one. Let $\phi\in C(X(n),\pi)$ and $\nu$ as above. Denote $k=|\nu|-1$
and define

$\psi:\\{(\tau,\sigma)\in X(k)\times
X(n):\tau\subseteq\sigma\\}\rightarrow[0,\infty)$

as

$\psi(\tau,\sigma)=\begin{cases}|\psi(\sigma)|^{2}&\operatorname{type}(\tau)=\nu\\\
0&\operatorname{type}(\tau)\neq\nu\end{cases},$

for all $(\tau,\sigma)\in X(k)\times X(n)$ with $\tau\subseteq\sigma$. Note
that for every $g\in K$ it holds that

$\psi(g.\tau,g.\sigma)\leq\left(\sup_{g\in
K}\|\pi(g)\|\right)\psi(\tau,\sigma)$

(recall that the action is type preserving). Thus, by Lemma 3.1 it follows
that

$\|\phi\|^{2}=\sum_{\sigma\in
D(n)}\frac{1}{\mu(G_{\sigma})}|\phi(\sigma)|^{2}=\sum_{\sigma\in
D(n)}\frac{1}{\mu(G_{\sigma})}\sum_{\tau\in
X(k),\tau\subseteq\sigma}\psi(\tau,\sigma)\leq\\\ \left(\sup_{g\in
K}\|\pi(g)\|^{2}\right)\sum_{\tau\in
D(k)}\frac{1}{\mu(G_{\tau})}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}\psi(\tau,\sigma)=\\\ \left(\sup_{g\in
K}\|\pi(g)\|^{2}\right)\sum_{\tau\in
D(k),\operatorname{type}(\tau)=\nu}\frac{1}{\mu(G_{\tau})}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|\phi(\sigma)|^{2},$

as needed. ∎

For $x\in\operatorname{\mathbb{E}}$, define
$\phi_{x}:D(n)\rightarrow\operatorname{\mathbb{E}}$ by

$\phi_{x}(\sigma)=\pi\left(\frac{\mathbbm{1}_{G_{\sigma}}}{\mu(G_{\sigma})}\right)x,\forall\sigma\in
D(n),$

where $\mathbbm{1}_{G_{\sigma}}$ is the indicator function of $G_{\sigma}$.
Observe that $\phi_{x}(\sigma)\in\operatorname{\mathbb{E}}^{\pi(G_{\sigma})}$
for every $\sigma\in D(n)$. Extend $\phi_{x}$ to $X(n)$ as follows: for every
$g\in G$ and every $\sigma\in D(n)$, define

$\phi_{x}(g.\sigma)=\pi(g)\phi_{x}(\sigma)=\pi\left(\frac{\mathbbm{1}_{g^{-1}G_{\sigma}}}{\mu(g^{-1}G_{\sigma})}\right)x=\pi\left(\frac{\mathbbm{1}_{g^{-1}G_{\sigma}}}{\mu(G_{\sigma})}\right)x.$

######  Proposition 3.3.

The map $\phi_{x}$ is well-defined and equivariant, i.e., $\phi_{x}\in
C(X(n),\pi)$.

###### Proof.

To show that $\phi_{x}$ is well-defined, we need to show that for every
$\sigma\in D(n)$ and every $g_{1},g_{2}\in G$, if $g_{1}.\sigma=g_{2}.\sigma$,
then $\pi(g_{1})\phi_{x}(\sigma)=\pi(g_{2})\phi_{x}(\sigma)$ or equivalently,
that $\pi(g_{2}^{-1}g_{1})\phi_{x}(\sigma)=\phi_{x}(\sigma)$. This equality
follows from the fact that $g_{2}^{-1}g_{1}\in G_{\sigma}$ and that
$\phi_{x}(\sigma)\in\operatorname{\mathbb{E}}^{\pi(G_{\sigma})}$.

The fact that $\phi_{x}$ is equivariant readily follows from its definition. ∎

Define linear maps $R_{1},R_{2}$ as follows:

$R_{1}:\operatorname{\mathbb{E}}\rightarrow C(X(n),\pi),R_{1}x=\phi_{x},$
$R_{2}:C(X(n),\pi)\rightarrow\operatorname{\mathbb{E}},R_{2}\phi=\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}\phi(\sigma).$

######  Proposition 3.4.

The maps $R_{1},R_{2}$ are bounded and

$\|R_{1}\|\leq(\max_{\sigma\in D(n)}\sup_{g\in
G_{\sigma}}\|\pi(g)\|)\sqrt{\sum_{\sigma\in D(n)}\frac{1}{\mu(G_{\sigma})}},$
$\|R_{2}\|\leq\max_{\sigma\in D(n)}\sqrt{\mu(G_{\sigma})}.$

###### Proof.

Note that for every $x\in\operatorname{\mathbb{E}}$ with $|x|=1$ and every
$\sigma\in D(n)$,

$|\phi_{x}(\sigma)|=\left|\frac{1}{\mu(G_{\sigma})}\int_{G_{\sigma}}\pi(g)xd\mu(g)\right|\leq\frac{1}{\mu(G_{\sigma})}\int_{G_{\sigma}}|\pi(g)x|d\mu(g)\leq\sup_{g\in
G_{\sigma}}\|\pi(g)\|.$

Thus for $x\in\operatorname{\mathbb{E}}$, with $|x|=1$, it follows that

$\|\phi_{x}\|^{2}\leq\sum_{\sigma\in D(n)}\frac{1}{\mu(G_{\sigma})}(\sup_{g\in
G_{\sigma}}\|\pi(g)\|)^{2},$

and the bound on the norm of $R_{1}$ follows.

Next, let $\phi\in C(X(n),\pi)$ with $\|\phi\|=1$. Then for every $\sigma\in
D(n)$, $|\phi(\sigma)|^{2}\leq\mu(G_{\sigma})$ and it follows that

$\left|R_{2}\phi\right|\leq\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}\sqrt{\mu(G_{\sigma})}\leq\max_{\sigma\in D(n)}\sqrt{\mu(G_{\sigma})}.$

∎

For $\emptyset\neq\nu\subseteq\\{0,....,n\\}$ define an equivalence relation
$\sim_{\nu}$ on $X(n)$ as $\sigma\sim_{\nu}\sigma^{\prime}$ if
$|\sigma\cap\sigma^{\prime}|\geq|\nu|$ and there is $\tau\in X$ such that
$\operatorname{type}(\tau)=\nu$ and $\tau\subseteq\sigma\cap\sigma^{\prime}$.
Note that by the assumption that the $1$-dimensional links of $X$ are finite
graphs, it follows that for every $\emptyset\neq\nu\subseteq\\{0,...,n\\}$,
$|\nu|\geq n-1$ and every $\sigma\in X(n)$, the set $\\{\sigma^{\prime}\in
X(n):\sigma\sim_{\nu}\sigma^{\prime}\\}$ is finite. For
$\emptyset\neq\nu\subseteq\\{0,...,n\\}$, $|\nu|\geq n-1$, define a projection
$P_{\nu}^{\pi}:C(X(n),\pi)\rightarrow C(X(n),\pi)$ by

$P_{\nu}^{\pi}\phi(\sigma)=\frac{1}{|\\{\sigma^{\prime}\in
X(n):\sigma\sim_{\nu}\sigma^{\prime}\\}|}\sum_{\sigma^{\prime}\sim_{\nu}\sigma}\phi(\sigma^{\prime}),$

(verifying that $P_{\nu}^{\pi}\phi$ is equivariant with respect to $\pi$ is
straight-forward and left for the reader). Denote

$C(X(n),\pi)_{\nu}=\\{\phi\in
C(X(n),\pi):\forall\sigma,\sigma^{\prime},\sigma\sim_{\nu}\sigma^{\prime}\Rightarrow\phi(\sigma)=\phi(\sigma^{\prime})\\},$

and note that $P_{\nu}^{\pi}$ is a projection on $C(X(n),\pi)_{\nu}$.

######  Lemma 3.5.

The space
$\bigcap_{\nu\subseteq\\{0,....,n\\},|\nu|=n}\operatorname{Im}(P_{\nu}^{\pi})$
is the space of all the constant maps $\phi\equiv x_{0}$ where $x_{0}$ is a
fixed point of the action of $G$ on $\operatorname{\mathbb{E}}$.

###### Proof.

Fix
$\phi\in\bigcap_{\nu\subseteq\\{0,....,n\\},|\nu|=n}\operatorname{Im}(P_{\nu}^{\pi})$.
As noted above $\operatorname{Im}(P_{\nu}^{\pi})=C(X(n),\pi)_{\nu}$ and it
follows that if $\sigma\cap\sigma^{\prime}\in X(n-1)$, then
$\phi(\sigma)=\phi(\sigma^{\prime})$. Thus if $\sigma,\sigma^{\prime}\in X(n)$
are connected by a gallery, then $\phi(\sigma)=\phi(\sigma^{\prime})$. By
assumption, $X$ is a gallery connected and thus $\phi$ is a constant map,
i.e., there is $x_{0}\in\operatorname{\mathbb{E}}$ such that $\phi\equiv
x_{0}$. Since $\phi$ is equivariant it follows that $x_{0}$ is a fixed point
of the action of $G$ on $\operatorname{\mathbb{E}}$. ∎

######  Proposition 3.6.

For every $\nu,\nu^{\prime}\subseteq\\{0,...,n\\}$ such that
$|\nu|=|\nu^{\prime}|=n$ and $\nu\neq\nu^{\prime}$ it holds that

$\operatorname{Im}(P_{\nu}^{\pi})\cap\operatorname{Im}(P_{\nu^{\prime}}^{\pi})=\operatorname{Im}(P_{\nu\cap\nu^{\prime}}^{\pi})$

and

$P_{\nu\cap\nu^{\prime}}^{\pi}P_{\nu}^{\pi}=P_{\nu}^{\pi}P_{\nu\cap\nu^{\prime}}^{\pi}=P_{\nu\cap\nu^{\prime}}^{\pi}.$

###### Proof.

Note that since $\nu\cap\nu^{\prime}\subseteq\nu,\nu^{\prime}$, it follows
that $C(X(n),\pi)_{\nu\cap\nu^{\prime}}\subseteq
C(X(n),\pi)_{\nu},C(X(n),\pi)_{\nu^{\prime}}$. Thus,

$\operatorname{Im}(P_{\nu}^{\pi})\cap\operatorname{Im}(P_{\nu^{\prime}}^{\pi})\supseteq\operatorname{Im}(P_{\nu\cap\nu^{\prime}}^{\pi}).$

For the reverse inclusion, note for every $\tau\in X(n-2)$ with
$\operatorname{type}(\tau)=\nu\cap\nu^{\prime}$, the link $X_{\tau}$ is
connected and thus we can argue as in the proof of Lemma 3.5 above for the set
$\\{\sigma:\tau\subseteq\sigma\\}$ and deduce that if
$\phi\in\operatorname{Im}(P_{\nu}^{\pi})\cap\operatorname{Im}(P_{\nu^{\prime}}^{\pi})$,
then $\phi$ is constant on $\tau$. This is true for every $\tau\in X(n-2)$
with $\operatorname{type}(\tau)=\nu\cap\nu^{\prime}$ and thus

$\operatorname{Im}(P_{\nu}^{\pi})\cap\operatorname{Im}(P_{\nu^{\prime}}^{\pi})\subseteq\operatorname{Im}(P_{\nu\cap\nu^{\prime}}^{\pi}).$

Second, since $P_{\nu\cap\nu^{\prime}}^{\pi},P_{\nu}^{\pi}$ are projections
and
$\operatorname{Im}(P_{\nu\cap\nu^{\prime}}^{\pi})\subseteq\operatorname{Im}(P_{\nu}^{\pi})$,
it follows that
$P_{\nu}^{\pi}P_{\nu\cap\nu^{\prime}}^{\pi}=P_{\nu\cap\nu^{\prime}}^{\pi}$.

Last, note that for every two $\sigma,\sigma^{\prime}\in X(n)$, if
$\sigma\sim_{\nu}\sigma^{\prime}$, then
$\sigma\sim_{\nu\cap\nu^{\prime}}\sigma^{\prime}$. Thus, for every $\phi\in
C(X(n),\pi)$ and every $\sigma\in X(n)$ it holds that

${P_{\nu\cap\nu^{\prime}}^{\pi}P_{\nu}^{\pi}\phi(\sigma)=}\\\
\frac{1}{|\\{\sigma^{\prime}\in
X(n):\sigma\sim_{\nu\cap\nu^{\prime}}\sigma^{\prime}\\}|}\sum_{\sigma^{\prime}\sim_{\nu\cap\nu^{\prime}}\sigma}\frac{1}{|\\{\sigma^{\prime\prime}\in
X(n):\sigma^{\prime}\sim_{\nu}\sigma^{\prime\prime}\\}|}\sum_{\sigma^{\prime\prime}\sim_{\nu}\sigma^{\prime}}\phi(\sigma^{\prime\prime})=\\\
\frac{1}{|\\{\sigma^{\prime}\in
X(n):\sigma\sim_{\nu\cap\nu^{\prime}}\sigma^{\prime}\\}|}\sum_{\sigma^{\prime\prime}\sim_{\nu\cap\nu^{\prime}}\sigma}\phi(\sigma^{\prime\prime})\sum_{\sigma^{\prime}\sim_{\nu}\sigma^{\prime\prime}}\frac{1}{|\\{\sigma^{\prime\prime}\in
X(n):\sigma^{\prime}\sim_{\nu}\sigma^{\prime\prime}\\}|}=P_{\nu\cap\nu^{\prime}}^{\pi}\phi(\sigma).$

Therefore
$P_{\nu\cap\nu^{\prime}}^{\pi}P_{\nu}^{\pi}=P_{\nu\cap\nu^{\prime}}^{\pi}$ as
needed. ∎

The above Proposition implies that we can define
$\cos(\angle(P_{\nu}^{\pi},P_{\nu^{\prime}}^{\pi}))=\cos_{P_{\nu\cap\nu^{\prime}}^{\pi}}(\angle(P_{\nu}^{\pi},P_{\nu^{\prime}}^{\pi}))$
for every $\nu,\nu^{\prime}\subseteq\\{0,...,n\\},\nu\neq\nu^{\prime}$ with
$|\nu|=|\nu^{\prime}|=n$.

######  Lemma 3.7.

For every $\nu\subseteq\\{0,...,n\\}$, $|\nu|=n$ it holds that

$\|P_{\nu}^{\pi}\|\leq\sup_{g\in K}\|\pi(g)\|^{2}.$

###### Proof.

Fix $\nu\subseteq\\{0,...,n\\}$, $|\nu|=n$ and fix some $\phi\in C(X(n),\pi)$.

For every $\tau\in D(n-1)$ with $\operatorname{type}(\tau)=\nu$, it follows by
the convexity of the norm that

$\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|P_{\nu}^{\pi}\phi(\sigma)|^{2}\leq\sum_{\sigma\in
X(n),\tau\subseteq\sigma}\frac{1}{|\\{\sigma^{\prime}\in
X(n):\sigma\sim_{\nu}\sigma^{\prime}\\}|}\sum_{\sigma^{\prime}\in
X(n),\sigma\sim_{\nu}\sigma^{\prime}}|\phi(\sigma^{\prime})|^{2}=\\\
\sum_{\sigma\in X(n),\tau\subseteq\sigma}\frac{1}{|\\{\sigma^{\prime}\in
X(n):\tau\subseteq\sigma^{\prime}\\}|}\sum_{\sigma^{\prime}\in
X(n),\tau\subseteq\sigma^{\prime}}|\phi(\sigma^{\prime})|^{2}=\\\
\sum_{\sigma^{\prime}\in
X(n),\tau\subseteq\sigma^{\prime}}|\phi(\sigma^{\prime})|^{2}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}\frac{1}{|\\{\sigma\in
X(n):\tau\subseteq\sigma\\}|}=\\\ \sum_{\sigma^{\prime}\in
X(n),\tau\subseteq\sigma^{\prime}}|\phi(\sigma^{\prime})|^{2}.$

By Corollary 3.2 and the previous inequality, it holds that

$\|P_{\nu}^{\pi}\phi\|^{2}\leq\left(\sup_{g\in
K}\|\pi(g)\|^{2}\right)\sum_{\tau\in
D(n-1),\operatorname{type}(\tau)=\nu}\frac{1}{\mu(G_{\tau})}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|P_{\nu}^{\pi}\phi(\sigma)|^{2}\leq\left(\sup_{g\in
K}\|\pi(g)\|^{2}\right)\sum_{\tau\in
D(n-1),\operatorname{type}(\tau)=\nu}\frac{1}{\mu(G_{\tau})}\sum_{\sigma^{\prime}\in
X(n),\tau\subseteq\sigma^{\prime}}|\phi(\sigma^{\prime})|^{2}\leq\left(\sup_{g\in
K}\|\pi(g)\|^{4}\right)\|\phi\|^{2}$

as needed. ∎

We note that for every $\tau\in X(n-2)$, $X_{\tau}$ is a bipartite graph and
we denote
$\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}=\lambda_{X_{\tau},\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}$.

######  Lemma 3.8.

For every $\nu,\nu^{\prime}\subseteq\\{0,...,n\\}$ such that
$|\nu|=|\nu^{\prime}|=n$ and $\nu\neq\nu^{\prime}$, denote

$\lambda_{\nu\cap\nu^{\prime}}^{\operatorname{\mathbb{E}}}=\max_{\tau\in
D(n-2),\operatorname{type}(\tau)=\nu\cap\nu^{\prime}}\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}.$

Then it holds that

$\cos(\angle(P_{\nu}^{\pi},P_{\nu^{\prime}}^{\pi}))\leq\left(\sup_{g\in
K}\|\pi(g)\|^{2}\right)\lambda_{\nu\cap\nu^{\prime}}^{\operatorname{\mathbb{E}}}.$

###### Proof.

Let $\phi\in C(X(n),\pi)$ be some map. Without loss of generality, it is
enough to show that

$\|(P_{\nu}^{\pi}P_{\nu^{\prime}}^{\pi}-P_{\nu\cap\nu^{\prime}}^{\pi})\phi\|\leq\left(\sup_{g\in
K}\|\pi(g)\|^{2}\right)\lambda_{\nu\cap\nu^{\prime}}^{\operatorname{\mathbb{E}}}\|\phi\|.$

For every $\tau\in D(n-2)$ with
$\operatorname{type}(\tau)=\nu\cap\nu^{\prime}=\\{2,...,n\\}$ the link
$X_{\tau}$ is a bipartite graph with sides

$S_{1}=\\{v\in
X_{\tau}(0):\operatorname{type}(\\{v\\})=\\{0,...,n\\}\setminus\nu\\},$
$S_{2}=\\{v\in
X_{\tau}(0):\operatorname{type}(\\{v\\})=\\{0,...,n\\}\setminus\nu^{\prime}\\}.$

Let
$P_{1},P_{2},P_{1,2}:\ell^{2}(X_{\tau}(1);\operatorname{\mathbb{E}})\rightarrow\ell^{2}(X_{\tau}(1);\operatorname{\mathbb{E}})$
defined in section 2.6, i.e.,

$P_{i}\Phi(\\{v_{1},v_{2}\\})=\frac{1}{m(v_{i})}\sum_{\\{v_{i},u\\}\in
E}\Phi(\\{v_{i},u\\}),\forall\Phi\in\ell^{2}(X_{\tau}(1);\operatorname{\mathbb{E}}),$

and

$P_{1,2}\Phi\equiv\frac{1}{|X_{\tau}(1)|}\sum_{\\{u,v\\}\in
X_{\tau}(1)}\Phi(\\{u,v\\}),\forall\Phi\in\ell^{2}(X_{\tau}(1);\operatorname{\mathbb{E}}).$

Define the localization of $\phi$ on $X_{\tau}$ to be the function
$\phi_{\tau}\in\ell^{2}(X_{\tau}(1);\operatorname{\mathbb{E}})$ defined as

$\phi_{\tau}(\\{u,v\\})=\phi(\tau\cup\\{u,v\\}).$

One can verify that for every $\sigma\in X(n)$ with $\tau\subseteq\sigma$ it
holds that

$P_{\nu}\phi(\sigma)=P_{1}\phi_{\tau}(\sigma\setminus\tau),P_{\nu^{\prime}}\phi(\sigma)=P_{2}\phi_{\tau}(\sigma\setminus\tau),P_{\nu\cap\nu^{\prime}}\phi(\sigma)=P_{1,2}\phi_{\tau}(\sigma\setminus\tau).$

Let $\tau\in D(n-2)$ with $\operatorname{type}(\tau)=\nu\cap\nu^{\prime}$. By
Proposition 2.22 it follows that

$\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|(P_{\nu}^{\pi}P_{\nu^{\prime}}^{\pi}-P_{\nu\cap\nu^{\prime}}^{\pi})\phi(\sigma)|^{2}=\\\
\sum_{\\{u,v\\}\in
X_{\tau}(1)}|(P_{1}P_{2}-P_{1,2})\phi_{\tau}(\\{u,v\\})|^{2}\leq\\\
\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}\sum_{\\{u,v\\}\in
X_{\tau}(1)}|\phi_{\tau}(\\{u,v\\})|^{2}\leq\\\
\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|\phi(\sigma)|^{2}\leq\lambda_{\nu\cap\nu^{\prime}}^{\operatorname{\mathbb{E}}}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|\phi(\sigma)|^{2}.$

By Corollary 3.2 (applied twice) and the above computation,

$\|(P_{\nu}^{\pi}P_{\nu^{\prime}}^{\pi}-P_{\nu\cap\nu^{\prime}}^{\pi})\phi\|^{2}\leq\\\
{\left(\sup_{g\in K}\|\pi(g)\|^{2}\right)\sum_{\tau\in
D(n-2),\operatorname{type}(\tau)=\nu\cap\nu^{\prime}}\frac{1}{\mu(G_{\tau})}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|(P_{\nu}^{\pi}P_{\nu^{\prime}}^{\pi}-P_{\nu\cap\nu^{\prime}}^{\pi})\phi(\sigma)|^{2}}\leq\\\
{\left(\sup_{g\in K}\|\pi(g)\|^{2}\right)\sum_{\tau\in
D(n-2),\operatorname{type}(\tau)=\nu\cap\nu^{\prime}}\frac{1}{\mu(G_{\tau})}\lambda_{\nu\cap\nu^{\prime}}^{\operatorname{\mathbb{E}}}\sum_{\sigma\in
X(n),\tau\subseteq\sigma}|\phi(\sigma)|^{2}}\leq\left(\sup_{g\in
K}\|\pi(g)\|^{4}\right)\lambda_{\nu\cap\nu^{\prime}}^{\operatorname{\mathbb{E}}}\|\phi\|^{2},$

as needed. ∎

######  Lemma 3.9.

Let $\pi$ be a representation of $G$ and define
$T_{\pi}=\frac{1}{n+1}\sum_{\nu\subseteq\\{0,...,n\\},|\nu|=n}P_{\nu}^{\pi}$.
For every $k\in\mathbb{N}$, there is a real valued positive function $f_{k}\in
C_{c}(G)$ such that $\int_{G}f_{k}d\mu=1$ (i.e., $f_{k}$ is a compactly
support probability function) and

$R_{2}T_{\pi}^{k}R_{1}=\pi(f_{k}).$

###### Proof.

We start by noting that for every $\sigma\in D(n)$ and every $k\in\mathbb{N}$,
there is a probability function $f^{k}_{\sigma}:X(n)\rightarrow[0,1]$ such
that $f_{\sigma}$ is supported on a ball around $\sigma$, and

$T_{\pi}^{k}\phi(\sigma)=\sum_{\sigma^{\prime}\in
X(n)}f^{k}_{\sigma}(\sigma^{\prime})\phi(\sigma^{\prime})$

for every $\phi\in C(X(n),\pi)$. Recall that for every
$x\in\operatorname{\mathbb{E}}$, $R_{1}x=\phi_{x}$ where
$\phi_{x}(\sigma)=\pi(\frac{\mathbbm{1}_{G_{\sigma}}}{\mu(G_{\sigma})})x$.
Also recall that $R_{2}\phi=\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}\phi(\sigma)$. Thus for every $x\in\operatorname{\mathbb{E}}$

$R_{2}T_{\pi}^{k}R_{1}x=\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}T_{\pi}^{k}\phi_{x}(\sigma)=\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}\sum_{\sigma^{\prime}\in\operatorname{supp}(f^{k}_{\sigma})}f^{k}_{\sigma}(\sigma^{\prime})\phi_{x}(\sigma^{\prime})=\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}\sum_{\sigma^{\prime}\in\operatorname{supp}(f^{k}_{\sigma})}f^{k}_{\sigma}(\sigma^{\prime})\pi(\frac{\mathbbm{1}_{G_{\sigma^{\prime}}}}{\mu(G_{\sigma}^{\prime})})x=\pi\left(\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}\sum_{\sigma^{\prime}\in\operatorname{supp}(f^{k}_{\sigma})}f^{k}_{\sigma}(\sigma^{\prime})\frac{\mathbbm{1}_{G_{\sigma^{\prime}}}}{\mu(G_{\sigma}^{\prime})}\right)x$

and we take

$f_{k}=\frac{1}{|D(n)|}\sum_{\sigma\in
D(n)}\sum_{\sigma^{\prime}\in\operatorname{supp}(f^{k}_{\sigma})}f^{k}_{\sigma}(\sigma^{\prime})\frac{\mathbbm{1}_{G_{\sigma^{\prime}}}}{\mu(G_{\sigma}^{\prime})}.$

∎

######  Theorem 3.10 (Żuk type criterion).

Let $G$ be a locally compact, unimodular group and $X$ be a pure
$n$-dimensional, partite simplicial complex such that $X$ is gallery connected
and all the $1$-dimensional links of $X$ are connected. Assume that $G$ is
acting on $X$ by simplicial automorphisms and the the action is cocompact and
that for every $\tau\in X(n-2)\cup X(n-1)\cup X(n)$, $G_{\tau}$ is an open
compact subgroup. Let $\mathcal{E}_{X}$ be the class of Banach spaces such
that for every $\operatorname{\mathbb{E}}\in\mathcal{E}_{X}$ it holds that

$\max_{\tau\in
D(n-2)}\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}<\frac{1}{8n-3}.$

Also, for $\varepsilon>0$, let $\mathcal{E}_{X,\varepsilon}$ be the class of
Banach spaces such that for every
$\operatorname{\mathbb{E}}\in\mathcal{E}_{X}$ it holds that

$\max_{\tau\in
D(n-2)}\lambda_{\tau,\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}\leq\frac{1}{8n-3}-\varepsilon.$

Then for every $\varepsilon>0$, $G$ has robust property (T) with respect to
$\mathcal{E}_{X,\varepsilon}$. Also, if $\mathbb{C}\in\mathcal{E}_{X}$, then
$G$ property $(F\mathcal{E}_{X})$.

###### Proof.

Note that by Lemma 2.7, $\mathcal{E}_{X}$ is closed under $\ell_{2}$ sums.
Thus if $\mathbb{C}\in\mathcal{E}_{X}$ it follows from Proposition 2.4 that if
$G$ has robust property (T) with respect to $\mathcal{E}_{X,\varepsilon}$,
then $G$ has property $(F\mathcal{E}_{X,\varepsilon})$. Therefore if we show
that $G$ has robust property (T) with respect to $\mathcal{E}_{X,\varepsilon}$
for every $\varepsilon>0$, it will follow that if
$\mathbb{C}\in\mathcal{E}_{X}$ then $G$ has property $(F\mathcal{E}_{X})$,
since $\mathcal{E}_{X}=\bigcup_{\varepsilon>0}\mathcal{E}_{X,\varepsilon}$.

Fix $\varepsilon>0$. Denote

$\beta_{1}=\sqrt{\frac{\frac{1}{8n-3}-\frac{\varepsilon}{2}}{\frac{1}{8n-3}-\varepsilon}},$
$\gamma=\frac{1}{8n-3}-\frac{\varepsilon}{2},$

and

$\beta_{2}=\sqrt{1+\frac{1}{2}\frac{1-(8n-3)\gamma}{n-1+(3n-1)\gamma}}.$

Further denote $\beta=\min\\{\beta_{1},\beta_{2}\\}$ and note that $\beta>1$.

Recall that $K\subseteq G$ is a compact symmetric generating set of $G$. We
will show that there is a Kazhdan projection with respect to
$\mathcal{F}(\mathcal{E}_{X,\varepsilon},K,\beta)$. Explicitly, let $f_{k}\in
C_{c}(G)$ be the probability functions in Lemma 3.9, i.e., the functions such
that for every representation $\pi$,

$\pi(f_{k})=R_{2}T_{\pi}^{k}R_{1},$

where $T=\frac{1}{n+1}\sum_{\nu\subseteq\\{0,...,n\\},|\nu|=n}P_{\nu}^{\pi}$.
We will show that $(f_{k})$ converges in
$C_{\mathcal{F}(\mathcal{E}_{X,\varepsilon},K,\beta)}$ to $p$ and
$\forall(\pi,\operatorname{\mathbb{E}})\in\mathcal{F}(\mathcal{E},K,\beta)$,
$\pi(p)$ is a projection on $\operatorname{\mathbb{E}}^{\pi(G)}$.

Fix $\pi\in\mathcal{F}(\mathcal{E}_{X,\varepsilon},K,\beta)$. By Lemma 3.7 and
the choice of $\beta_{2}$, it holds for every $\nu\subseteq\\{0,...,n\\}$,
$|\nu|=n$ that

$\|P_{\nu}^{\pi}\|\leq\beta_{2}^{2}<1+\frac{1-(8n-3)\gamma}{n-1+(3n-1)\gamma}.$

By Lemma 3.8 and the choice of $\beta_{1}$, it holds for every
$\nu,\nu^{\prime}\subseteq\\{0,...,n\\},\nu\neq\nu^{\prime}$,
$|\nu|=|\nu^{\prime}|=n$ that

$\cos(\angle(P_{\nu}^{\pi},P_{\nu^{\prime}}^{\pi}))\leq\beta_{1}^{2}\left(\frac{1}{8n-3}-\varepsilon\right)=\gamma<\frac{1}{8n-3}.$

It follows that the set of projections
$\\{P_{\nu}^{\pi}:\nu\subseteq\\{0,...,n\\},|\nu|=n\\}$ fulfil the conditions
of Theorem 2.17.

Denote as above
$T_{\pi}=\frac{1}{n+1}\sum_{\nu\subseteq\\{0,...,n\\},|\nu|=n}P_{\nu}^{\pi}$.
By Theorem 2.17 there is a projection $T_{\pi}^{\infty}$ on
$\bigcap_{\nu\subseteq\\{0,...,n\\},|\nu|=n}\operatorname{Im}(P_{\nu}^{\pi})$
and constants $0\leq r(\gamma,\beta)<1,C(\gamma,\beta)>0$, such that for every
$k$,

$\|T_{\pi}^{k}-T_{\pi}^{\infty}\|\leq Cr^{k}.$

Thus, it holds for every $k$ that

$\|R_{2}T_{\pi}^{\infty}R_{1}-\pi(f_{k})\|\leq\|R_{2}\|\|R_{1}\|Cr^{k}\leq^{\text{Proposition
}\ref{bound R_1, R_2 prop}}\\\ \left(\beta\sqrt{\sum_{\sigma\in
D(n)}\frac{1}{\mu(G_{\sigma})}}\max_{\sigma\in
D(n)}\sqrt{\mu(G_{\sigma})}\right)Cr^{k}.$

This shows that for every
$\pi\in\mathcal{F}(\mathcal{E}_{X,\varepsilon},K,\beta)$, the sequence
$\pi(f_{k})$ converges in norm and the rate of convergence is bounded
independently of $\pi$. Thus, the sequence $(f_{k})$ converges in
$C_{\mathcal{F}(\mathcal{E}_{X,\varepsilon},K,\beta)}$.

It remains to show that for every
$(\pi,\operatorname{\mathbb{E}})\in\mathcal{F}(\mathcal{E}_{X,\varepsilon},K,\beta)$,
the operator $R_{2}T_{\pi}^{\infty}R_{1}$ is a projection on
$\operatorname{\mathbb{E}}^{\pi(G)}$. First, let
$x\in\operatorname{\mathbb{E}}^{\pi(G)}$. By the definition of $\phi_{x}$,
$R_{1}x=\phi_{x}$ is the constant function $\phi_{x}\equiv x$. By Lemma 3.5,
$T_{\pi}^{\infty}$ is a projection on the subspace of constant functions in
$C(X(n),\pi)$ and thus $T_{\pi}^{\infty}\phi_{x}=\phi_{x}$. By the definition
of $R_{2}$, $R_{2}\phi_{x}=x$ (since $\phi_{x}\equiv x$). Combining all of the
above, it follows that for every $x\in\operatorname{\mathbb{E}}^{\pi(G)}$,
$R_{2}T^{\infty}R_{1}x=x$.

Second, for every $x\in\operatorname{\mathbb{E}}$,
$T_{\pi}^{\infty}R_{1}x=T_{\pi}^{\infty}\phi_{x}$ and by Lemma 3.5 it follows
that $T_{\pi}^{\infty}\phi_{x}$ is a constant equivariant function in
$C(X(n),\pi)$, i.e., there is $x_{0}\in\operatorname{\mathbb{E}}^{\pi(G)}$
such that $T_{\pi}^{\infty}\phi_{x}\equiv x_{0}$. Thus,
$R_{2}T_{\pi}^{\infty}\phi_{x}=x_{0}\in\operatorname{\mathbb{E}}^{\pi(G)}$ and
it follows that
$\operatorname{Im}(R_{2}T_{\pi}^{\infty}R_{1})\subseteq\operatorname{\mathbb{E}}^{\pi(G)}$.
Therefore, we can conclude that $R_{2}T^{\infty}R_{1}$ is a projection on
$\operatorname{\mathbb{E}}^{\pi(G)}$ as needed. ∎

##  4\. Application to random groups in the Gromov density model

The aim of the Section is to apply of Banach Żuk criterion in the setting of
random groups in the Gromov density model and prove an extended version of
Theorem 1.5 that appeared in the introduction. Some definitions below already
appeared in the introduction and we recall them for completeness.

A random group is a group chosen randomly according to some model and one is
interested in the asymptotic properties of such randomly chosen group. The
most famous model is the Gromov density model:

######  Definition 4.1 (Gromov density model).

Let $k\in\mathbb{N},k\geq 2$ and $0\leq d\leq 1$ be constants and
$l\in\mathbb{N}$ be a parameter. A random group is the Gromov density model
$\mathcal{D}(k,l,d)$ is a group $\Gamma=\langle\mathcal{A}|\mathcal{R}\rangle$
where $|\mathcal{A}|=k$ and $\mathcal{R}$ is a set of relators of length $l$
(in $\mathcal{A}\cup\mathcal{A}^{-1}$) randomly chosen from the set

$\\{\mathcal{R}\text{ is a set of cyclically reduced relators of length
}l:|\mathcal{R}|=\lfloor(2k-1)^{dl}\rfloor\\}$

with uniform probability. We denote a random group in this model by
$\Gamma\in\mathcal{D}(k,l,d)$.

For a group property $P$, we say that $P$ holds asymptotically almost surely
(a.a.s.) in $\mathcal{D}(k,l,d)$ if

$\lim_{l\rightarrow\infty}\mathbb{P}(\Gamma\in\mathcal{D}(k,l,d)\text{ has
property }P)=1.$

We will show that for certain class of Banach spaces $\mathcal{E}$ property
$(F\mathcal{E})$ holds a.a.s. in $\mathcal{D}(k,l,d)$ (see exact formulation
below).

A closely related model to the density model is the Binomial model:

######  Definition 4.2 (Gromov binomial model).

Let $k\in\mathbb{N},k\geq 2$ be a constant, $\rho:\mathbb{N}\rightarrow[0,1]$
be a function and $l$ be a parameter. A random group is the Gromov binomial
model $\mathcal{B}(k,l,\rho)$ is a group
$\Gamma=\langle\mathcal{A}|\mathcal{R}\rangle$ where $|\mathcal{A}|=k$ and
$\mathcal{R}$ is a set of relators of cyclically reduced length $l$ (in
$\mathcal{A}\cup\mathcal{A}^{-1}$) where each relator in $\mathcal{R}$ chosen
independently with probability $\rho(l)$. We denote a random group in this
model by $\Gamma\in\mathcal{B}(k,l,\rho)$.

For a group property $P$, we say that $P$ holds asymptotically almost surely
(a.a.s.) in $\mathcal{B}(k,l,\rho)$ if

$\lim_{l\rightarrow\infty}\mathbb{P}(\Gamma\in\mathcal{B}(k,l,\rho)\text{ has
property }P)=1.$

For a group property $P$ we say that $P$ is monotone increasing if it is
preserved under quotients. For our purposes it will be important that
Proposition 2.5 implies that property $(F\mathcal{E})$ is monotone increasing.
The following Proposition appears in [JLuR00] (as part of a systematic study
of random sets in both models):

######  Proposition 4.3.

[JLuR00, Proof of Proposition 1.13] Let $P$ be a monotone increasing group
property and $k\geq 2,0<d<1$ be constants. For $C\geq 1$, we denote

$\rho_{C}(l)=(2k-1)^{-(1-d)l}-C(2k-1)^{-(\frac{3}{2}-\frac{d}{2})l}.$

Then

$\mathbb{P}(\Gamma\in\mathcal{D}(k,l,d)\text{ has property
}P)\geq\mathbb{P}(\Gamma\in\mathcal{B}(k,l,\rho_{C})\text{ has property
}P)-\frac{3}{C}.$

The following Corollary readily follows:

######  Corollary 4.4.

Let $P$ be a monotone increasing group property and $k\geq 2,0<d<1$ be
constants. Denote

$\rho(l)=\frac{1}{2}(2k-1)^{-(1-d)l}.$

Then

$\mathbb{P}(\Gamma\in\mathcal{D}(k,l,d)\text{ has property
}P)\geq\mathbb{P}(\Gamma\in\mathcal{B}(k,l,\rho)\text{ has property
}P)-\frac{6}{(2k-1)^{\frac{l}{2}}}.$

Following the ideas of [KK13] and assuming that $l$ is divisible by $3$, we
introduce the model $\mathcal{B}^{\prime}(k,l,\rho)$ as a variant of
$\mathcal{B}(k,l,\rho)$ defined as follows: Let $W_{\frac{l}{3}}^{\prime}$ be
reduced words in $\mathcal{A}\cup\mathcal{A}^{-1}$ of length $\frac{l}{3}$
with a first and last letter in $\mathcal{A}$. Define $W^{\prime}_{l}$ to be
words in $\mathcal{A}\cup\mathcal{A}^{-1}$ of length $l$ that are
concatenations of $3$ words in $W_{\frac{l}{3}}^{\prime}$.

######  Definition 4.5.

Let $k\geq 2$ be a constant and $\rho:\mathbb{N}\rightarrow[0,1]$ be a
function. A random group is the model $\mathcal{B}^{\prime}(k,l,\rho)$ is a
group $\Gamma=\langle\mathcal{A}|\mathcal{R}\rangle$ where $|\mathcal{A}|=k$
and $\mathcal{R}$ is a set of relators chosen from $W_{l}^{\prime}$
independently with probability $\rho(l)$.

######  Proposition 4.6.

Let $P$ be a monotone increasing group property, $k\geq 2$ be a constant and
$\rho:\mathbb{N}\rightarrow[0,1]$ be a function. If a random group in
$\mathcal{B}^{\prime}(k,l,\rho)$ has $P$ a.a.s., then a random group in
$\mathcal{B}(k,l,\rho)$ has $P$ a.a.s. More explicitly,

$\mathbb{P}(\Gamma\in\mathcal{B}(k,l,\rho)\text{ has property
}P)\geq\mathbb{P}(\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\rho)\text{ has
property }P).$

###### Proof.

Denote $W_{l}$ to be the set of cyclically reduced words of length $l$ in
$\mathcal{A}\cup\mathcal{A}^{-1}$. The relators of
$\Gamma\in\mathcal{B}(k,l,\rho)$ are chosen independently from $W_{l}$ and
thus one can start by first choosing relators from $W_{l}^{\prime}$ and then
from $W_{l}\setminus W_{l}^{\prime}$. Thus it is clear that
$\Gamma\in\mathcal{B}(k,l,\rho)$ is a quotient of
$\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\rho)$ and by the assumption that
$P$ is monotone increasing

$\mathbb{P}(\Gamma\in\mathcal{B}(k,l,\rho)\text{ has
}P)\geq\mathbb{P}(\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\rho)\text{ has
}P).$

∎

The above Proposition shows that in order to prove property $(F\mathcal{E})$
holds a.a.s. for $\mathcal{B}(k,l,\rho)$ (and thus for $\mathcal{D}(k,l,d)$
for $\rho(l)=\frac{1}{2}(2k-1)^{-(1-d)l}$ by Corollary 4.4) it is sufficient
to prove that property $(F\mathcal{E})$ holds a.a.s. for
$\mathcal{B}^{\prime}(k,l,\rho)$.

Last, again following [KK13], we define the model $\mathcal{M}^{+}(m,\rho)$ as
follows:

######  Definition 4.7 (The positive triangular model).

Let $m\in\mathbb{N}$ be a parameter and $\rho:\mathbb{N}\rightarrow[0,1]$ be a
function. A random group is the model $\mathcal{M}^{+}(m,\rho)$ is a group
$\Gamma=\langle S|R\rangle$ where $|S|=m,S\cap S^{-1}=\emptyset$ and $R$ is a
set of length $3$ relators in the letters of $S$ (not using the letters of
$S^{-1}$) chosen independently with probability $\rho(m)$.

Fix some $k\geq 2$ and $\mathcal{A}$ with $|\mathcal{A}|=k$. Let
$W_{\frac{l}{3}}^{\prime},W_{l}^{\prime}$ be as above (with respect to
$\mathcal{A}$). For $m=|W_{\frac{l}{3}}^{\prime}|$ and
$S=\\{s_{1},...,s_{m}\\}$ fix a bijection $\varphi:S\rightarrow
W_{\frac{l}{3}}^{\prime}$. This bijection extends to a bijection
$\varphi:\\{s_{i_{1}}s_{i_{2}}s_{i_{3}}:1\leq i_{1},i_{2},i_{3}\leq
m\\}\rightarrow W_{l}^{\prime}$. Thus, for
$\Gamma\in\mathcal{M}^{+}(m,\rho),\Gamma=\langle S|R\rangle$ the map
$\varphi:\Gamma\rightarrow\langle\mathcal{A}|\varphi(R)\rangle$ extends to a
homomorphism. As the following Lemma states, $\varphi(\Gamma)$ is a finite
index subgroup of $\langle\mathcal{A}|\varphi(R)\rangle$.

######  Lemma 4.8.

[KK13, Lemma 3.15] For every $\Gamma\in\mathcal{M}^{+}(m,\rho)$,
$\varphi(\Gamma)$ is a finite index subgroup of
$\langle\mathcal{A}|\varphi(R)\rangle$.

######  Remark 4.9.

The random group models considered in [KK13] are the density models, but the
proof of [KK13, Lemma 3.15] passes verbatim to the binomial models.

All the discussion above reduces the problem of proving $(F\mathcal{E})$ to
$\mathcal{D}(k,l,d)$ to proving $(F\mathcal{E})$ to $\mathcal{M}^{+}(m,\rho)$
(with the correct choice of $\rho$):

######  Theorem 4.10.

Let $\frac{1}{3}<d<1$, $k\geq 2$ be constants and $\mathcal{E}_{l}$ be a class
of Banach spaces (this class may change as $l$ increases). Define
$m(l)=\frac{1}{2}(2k-1)^{\frac{l}{3}}$ and $\rho:\mathbb{N}\rightarrow[0,1]$
to be $\rho(m)=\frac{1}{4m^{3(1-d)}}$. For every $l$ divisible by $3$,

$\mathbb{P}(\Gamma\in\mathcal{D}(k,l,d)\text{ has property
}(F\mathcal{E}_{l}))\geq{\mathbb{P}(\Gamma\in\mathcal{M}^{+}(m(l),\rho)\text{
has property }(F\mathcal{E}_{l}))-\frac{6}{(2k-1)^{\frac{l}{2}}}.}$

###### Proof.

By Corollary 4.4, for $\rho_{1}(l)=\frac{1}{2}(2k-1)^{-(1-d)l}$ it hold that

$\mathbb{P}(\Gamma\in\mathcal{D}(k,l,d)\text{ has property
}(F\mathcal{E}_{l}))\geq\mathbb{P}(\Gamma\in\mathcal{B}(k,l,\rho_{1})\text{
has property }(F\mathcal{E}_{l}))-\frac{6}{(2k-1)^{\frac{l}{2}}}.$

By Proposition 4.6,

$\mathbb{P}(\Gamma\in\mathcal{B}(k,l,\rho_{1})\text{ has property
}(F\mathcal{E}_{l}))\geq\mathbb{P}(\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\rho_{1})\text{
has property }(F\mathcal{E}_{l})).$

Thus we are left to show that for every $l$ divisible by $3$ is holds that

$\mathbb{P}(\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\rho_{1})\text{ has
property }P)\geq{\mathbb{P}(\Gamma\in\mathcal{M}^{+}(m(l),\rho)\text{ has
property }(F\mathcal{E}_{l})),}$

with $m(l)=\frac{1}{2}(2k-1)^{\frac{l}{3}}$ and
$\rho(m)=\frac{1}{4m^{3(1-d)}}$.

Assume that $l$ is divisible by $3$. By Lemma 4.8, for every
$\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\frac{1}{2}(2k-1)^{-(1-d)l})$
there is a group $\Gamma\in\mathcal{M}^{+}(m_{1},\rho_{2})$ with
$m_{1}(l)=|W_{\frac{l}{3}}^{\prime}|=k^{2}(2k-1)^{\frac{l}{3}-2}$ and
$\rho_{2}(m_{1})=\frac{1}{2}(2k-1)^{-(1-d)l}$ such that $\varphi(\Gamma)$ is a
finite index subgroup in $\Gamma^{\prime}$ and $\Gamma$, $\Gamma^{\prime}$
occur in the same probability.

By Proposition 2.6, if $\Gamma$ has property $(F\mathcal{E}_{l})$, then
$\Gamma^{\prime}$ has property $(F\mathcal{E}_{l})$. Using the fact that
property $(F\mathcal{E}_{l})$ is monotone increasing it follows that we can
replace $m_{1}$ by

$m=\frac{1}{2}(2k-1)^{\frac{l}{3}}\geq m_{1},$

i.e.,

$\mathbb{P}(\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\rho_{1})\text{ has
property
}P)\geq{\mathbb{P}(\Gamma\in\mathcal{M}^{+}(m_{1},\rho_{2}(m_{1}))\text{ has
property
}(F\mathcal{E}_{l}))}\geq{\mathbb{P}(\Gamma\in\mathcal{M}^{+}(m,\rho_{2}(m_{1}))\text{
has property }(F\mathcal{E}_{l})).}$

Note that

$\rho_{2}(m_{1})=\frac{1}{2}(2k-1)^{-(1-d)l}=\frac{1}{2}\frac{1}{((2k-1)^{\frac{l}{3}})^{3(1-d)}}=\frac{1}{2}\frac{1}{2m^{3(1-d)}}=\frac{1}{4m^{3(1-d)}}.$

It follows that for $\rho(m)=\frac{1}{4m^{3(1-d)}}$ it holds that

$\mathbb{P}(\Gamma^{\prime}\in\mathcal{B}^{\prime}(k,l,\rho_{1})\text{ has
property }P)\geq{\mathbb{P}(\Gamma\in\mathcal{M}^{+}(m,\rho(m))\text{ has
property }(F\mathcal{E}_{l})),}$

as needed. ∎

Theorem 4.10 reduces the proof of Banach fixed point properties in the Gromov
density to proving Banach fixed properties in the positive triangular model.
This model is convenient for us, since its Cayley complex is a two-dimensional
simplicial complex with bipartite links and thus we can apply our Theorem 3.10
above. For this, we will need to bound the second eigenvalue of the random
walk on the link of the Cayley complex of a group in
$\mathcal{M}^{+}(m,\rho)$. Our approach for bounding the second eigenvalue is
heavily based on [ALuS15] and [dLdlS21] and we claim very little originality
here (a few adaptations were needed in order to provide a sharp result in the
triangular positive model). In order to bound the second eigenvalue, we will
first need some general lemmata regarding graphs and the bipartite Erdös-Rényi
graph.

######  Lemma 4.11.

Let $(V,E_{1}),...,(V,E_{k})$ be connected graphs on the same vertex set $V$.
Assume that $E_{1},...,E_{k}$ are pairwise disjoint and that there are
constants $d,0<\varepsilon<1$ such that for every $i$, the degree of a vertex
in $(V,E_{i})$ is between $d(1-\varepsilon)$ and $d(1+\varepsilon)$. Assume
there is $\lambda<1$ such that the second eigenvalue of the random walk on
each $(V,E_{i})$ is less than $\lambda$. Then the second eigenvalue of the
random walk on $(V,E_{1}\cup...\cup E_{k})$ is less that
$\lambda+\frac{16\varepsilon^{2}}{(1-\varepsilon)^{4}}$.

###### Proof.

For $v\in V$, denote $m_{i}(v)$ to be the degree of $v$ in $(V,E_{i})$ and
$m(v)$ to be the degree of $v$ in $(V,E_{1}\cup...E_{k})$. We need to prove
that for $\phi:V\rightarrow\mathbb{R}$, if $\sum_{v\in V}m(v)\phi(v)=0$, then

$\sum_{v\in
V}m(v)(M\phi(v))\phi(v)\leq\left(\lambda+\frac{16\varepsilon^{2}}{(1-\varepsilon)^{4}}\right)\sum_{v\in
V}m(v)|\phi(v)|^{2},$

where $M$ is the random walk operator on $(V,E_{1}\cup...\cup E_{k})$.
Equivalently, we need to show that for the normalized Laplacian $I-M$ it holds
for $\phi$ as above that

$\sum_{v\in
V}m(v)((I-M)\phi(v))\phi(v)\geq\left(1-\lambda-\frac{16\varepsilon^{2}}{(1-\varepsilon)^{4}}\right)\sum_{v\in
V}m(v)|\phi(v)|^{2}.$

Recall that

$\sum_{v\in V}m(v)((I-M)\phi(v))\phi(v)=\sum_{\\{u,v\\}\in E_{1}\cup...\cup
E_{k}}|\phi(u)-\phi(v)|^{2}$

where double edges (if they exist) are added several times according to their
multiplicity. Similarly, if we denote $M_{i}$ to be the random walk operator
on $(V,E_{i})$ then for every $i$

$\sum_{v\in V}m(v)((I-M_{i})\phi(v))\phi(v)=\sum_{\\{u,v\\}\in
E_{i}}|\phi(u)-\phi(v)|^{2}.$

Thus

$\sum_{v\in V}m(v)((I-M)\phi(v))\phi(v)=\sum_{\\{u,v\\}\in E_{1}\cup...\cup
E_{k}}|\phi(u)-\phi(v)|^{2}=\sum_{i=1}^{k}\sum_{\\{u,v\\}\in
E_{i}}|\phi(u)-\phi(v)|^{2}=\sum_{i=1}^{k}\sum_{v\in
V}m(v)((I-M_{i})\phi(v))\phi(v)\geq(1-\lambda)\sum_{i=1}^{k}\sum_{v\in
V}m_{i}(v)\left|\phi(v)-\frac{1}{m_{i}(V)}\sum_{u\in
V}m_{i}(u)\phi(u)\right|^{2}=(1-\lambda)\sum_{i=1}^{k}\sum_{v\in
V}m_{i}(v)|\phi(v)|^{2}-(1-\lambda)\sum_{i=1}^{k}\sum_{v\in
V}m_{i}(v)\left|\sum_{u\in
V}\frac{m_{i}(u)}{m_{i}(V)}\phi(u)\right|^{2}=.(1-\lambda)\sum_{v\in
V}m(v)|\phi(v)|^{2}-(1-\lambda)\sum_{i=1}^{k}m_{i}(V)\left|\sum_{u\in
V}\frac{m_{i}(u)}{m_{i}(V)}\phi(u)\right|^{2}.$

Thus it is sufficient to prove that

$\sum_{i=1}^{k}m_{i}(V)\left|\sum_{u\in
V}\frac{m_{i}(u)}{m_{i}(V)}\phi(u)\right|^{2}\leq\frac{16\varepsilon^{2}}{(1-\varepsilon)^{4}}\sum_{v\in
V}m(v)|\phi(v)|^{2}.$

Fix $1\leq i\leq k$. Note that for every $u\in V$,

$\frac{(1-\varepsilon)d}{k(1+\varepsilon)d}\leq\frac{m_{i}(u)}{m(u)}\leq\frac{(1+\varepsilon)d}{k(1-\varepsilon)d}.$

Thus

$\left(\frac{1-\varepsilon}{1+\varepsilon}\right)^{2}\leq\frac{m(u)}{m_{i}(u)}\frac{m_{i}(V)}{m(V)}\leq\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^{2}.$

Therefore

$\left|1-\frac{m(u)}{m_{i}(u)}\frac{m_{i}(V)}{m(V)}\right|\leq\max\left\\{\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^{2}-1,1-\left(\frac{1-\varepsilon}{1+\varepsilon}\right)^{2}\right\\}=\frac{4\varepsilon}{(1-\varepsilon)^{2}}.$

It follows that

$\sum_{i=1}^{k}m_{i}(V)\left|\sum_{u\in
V}\frac{m_{i}(u)}{m_{i}(V)}\phi(u)\right|^{2}=\sum_{i=1}^{k}m_{i}(V)\left|\sum_{u\in
V}\left(\frac{m_{i}(u)}{m_{i}(V)}-\frac{m(u)}{m(V)}\right)\phi(u)\right|^{2}={\sum_{i=1}^{k}m_{i}(V)\left|\sum_{u\in
V}\frac{m_{i}(u)}{m_{i}(V)}\left(1-\frac{m(u)}{m_{i}(u)}\frac{m_{i}(V)}{m(V)}\right)\phi(u)\right|^{2}}\leq{\sum_{i=1}^{k}m_{i}(V)\sum_{u\in
V}\frac{m_{i}(u)}{m_{i}(V)}\left|\left(1-\frac{m(u)}{m_{i}(u)}\frac{m_{i}(V)}{m(V)}\right)\phi(u)\right|^{2}}\leq{\sum_{i=1}^{k}\sum_{u\in
V}m_{i}(u)\left(\frac{4\varepsilon}{(1-\varepsilon)^{2}}\right)^{2}|\phi(u)|^{2}}=\frac{16\varepsilon^{2}}{(1-\varepsilon)^{4}}\sum_{u\in
V}m(u)|\phi(u)|^{2},$

as needed.

∎

######  Lemma 4.12.

Let $(V,E_{1}),(V,E_{2})$ be finite graphs on the same vertex set $V$ and
$0\leq\lambda<1$, $0\leq\varepsilon<1$ be a constants. Denote $m_{i}(v)$ to be
the degree of $v$ in $(V,E_{i})$. Assume that $(V,E_{1}\cup E_{2})$ is
connected and that the second eigenvalue of the random walk on it is less than
$\lambda$. If for every $v\in V$,
$\frac{m_{2}(v)}{m_{1}(v)}\leq\varepsilon<\frac{1-\lambda}{4}$, then
$(V,E_{1})$ is connected and the second eigenvalue of the random walk on it is
less than $\lambda+4\varepsilon$.

The proof is very similar to the proof of Lemma 4.11 and thus we allow
ourselves to omit some details.

###### Proof.

Denote $M_{i},i=1,2$ to be the random walk operator on $(V,E_{i})$. Let
$\phi:V\rightarrow\mathbb{R}$ such that $\sum_{v}m_{1}(v)\phi(v)=0$. We need
to show that

$\sum_{v\in
V}m_{1}(v)(M_{1}\phi(v))\phi(v)\leq\left(\lambda+4\varepsilon\right)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}$

or equivalently that

$\sum_{v\in
V}m_{1}(v)((I-M_{1})\phi(v))\phi(v)\geq\left(1-\lambda-4\varepsilon\right)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}.$

Observe that

$\sum_{v\in V}m(v)((I-M)\phi(v))\phi(v)=\sum_{v\in
V}m_{1}(v)((I-M_{1})\phi(v))\phi(v)+\sum_{v\in
V}m_{2}(v)((I-M_{2})\phi(v))\phi(v)\leq\sum_{v\in
V}m_{1}(v)((I-M_{1})\phi(v))\phi(v)+2\sum_{v\in
V}m_{2}(v)|\phi(v)|^{2}\leq\sum_{v\in
V}m_{1}(v)((I-M_{1})\phi(v))\phi(v)+2\varepsilon\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}.$

Thus,

$\sum_{v\in V}m_{1}(v)((I-M_{1})\phi(v))\phi(v)\geq\sum_{v\in
V}m(v)((I-M)\phi(v))\phi(v)-2\varepsilon\sum_{v\in V}m_{1}(v)|\phi(v)|^{2}$

and we are left to prove that

$\sum_{v\in V}m(v)((I-M)\phi(v))\phi(v)\geq(1-\lambda-2\varepsilon)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}.$

Note that for every $u\in V$,
$\frac{1}{1+\varepsilon}\leq\frac{m_{1}(u)}{m(u)}\leq 1$ and

$\frac{1}{1+\varepsilon}\leq\frac{m(V)}{m(u)}\frac{m_{1}(u)}{m_{1}(V)}\leq
1+\varepsilon.$

Thus

$\sum_{v\in V}m(v)((I-M)\phi(v))\phi(v)\geq(1-\lambda)\sum_{v\in
V}m(v)|\phi(v)|^{2}-(1-\lambda)m(V)\left|\sum_{u\in
V}\frac{m(u)}{m(V)}\phi(u)\right|^{2}\geq(1-\lambda)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}-m(V)\left|\sum_{u\in
V}\frac{m(u)}{m(V)}\phi(u)\right|^{2}=(1-\lambda)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}-m(V)\left|\sum_{u\in
V}\frac{m(u)}{m(V)}\left(1-\frac{m(V)}{m(u)}\frac{m_{1}(u)}{m_{1}(V)}\right)\phi(u)\right|^{2}\leq(1-\lambda)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}-\varepsilon^{2}\sum_{u\in
V}m(u)|\phi(u)|^{2}\leq(1-\lambda)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}-\varepsilon^{2}(1+\varepsilon)\sum_{u\in
V}m_{1}(u)|\phi(u)|^{2}\leq(1-\lambda)\sum_{v\in
V}m_{1}(v)|\phi(v)|^{2}-2\varepsilon\sum_{u\in V}m_{1}(u)|\phi(u)|^{2},$

as needed. ∎

For $n\in\mathbb{N}$ and $\rho:\mathbb{N}\rightarrow[0,1]$, let
$\mathbb{G}_{\operatorname{bipartite}}(n,\rho)$ be the bipartite Erdös-Rényi
graph defined as follows: The vertex set of $V$ is $S_{1}\cup S_{2}$ where
$S_{1},S_{2}$ are disjoint sets of size $n$. We randomly choose edges in the
set $\\{\\{u,v\\}:u\in S_{1},v\in S_{2}\\}$ where each edge is chosen
independently with probability $\rho(n)$.

######  Theorem 4.13.

[Ash21a, Theorem A] If $\rho(n)\geq\frac{\log^{6}(n)}{n}$, then with
probability tending to $1$ as $n$ grows to infinity the graph
$\mathbb{G}_{\operatorname{bipartite}}(n,\rho)$ is connected and its second
largest eigenvalue is less than $\frac{8}{\sqrt{n\rho(n)}}$.

######  Lemma 4.14.

Let $0\leq\alpha<1$ be a constant. Assume $\rho(n)\geq\frac{C}{n^{\alpha}}$
for some positive constant $C$. Then

$\lim_{n}\mathbb{P}(|m(v)-n\rho(n)|\geq\frac{n\rho(n)}{n^{\frac{1-\alpha}{3}}}\text{
for some }v\text{ in }\mathbb{G}_{\operatorname{bipartite}}(n,\rho))=0.$

###### Proof.

For every vertex $v$, $m(v)$ is a binomial random variable. For every
$0<\delta<1$ it holds by Chernoff bound that

$\mathbb{P}(m(v)\geq(1+\delta)n\rho(n))\leq\exp(-\frac{\delta^{2}}{3}n\rho(n)),$
$\mathbb{P}(m(v)\leq(1-\delta)n\rho)\leq\exp(-\frac{\delta^{2}}{3}n\rho(n)).$

Thus if we choose $\delta=\frac{1}{n^{\frac{1-\alpha}{3}}}$, we have that for
every large $n$

$\mathbb{P}\left(|m(v)-n\rho(n)|\geq\frac{n\rho(n)}{n^{\frac{1-\alpha}{3}}}\right)\leq
2\exp\left(-\frac{n\rho(n)}{3n^{\frac{2(1-\alpha)}{3}}}\right)\leq
2\exp\left(-\frac{Cn^{\frac{1-\alpha}{3}}}{3}\right).$

By union bound, the probability the some vertex $v$ has
$|m(v)-n\rho(n)|\geq\frac{n\rho(n)}{n^{\frac{1-\alpha}{3}}}$ is less than
$4n\exp\left(-\frac{Cn^{\frac{1-\alpha}{3}}}{3}\right)$ and this tends to $0$
as $n$ tends to infinity. ∎

After this set-up, we argue as in [ALuS15]: Let
$\Gamma\in\mathcal{M}^{+}(m,\rho)$. The relations of $\Gamma$ are of the form
$s_{i}s_{j}s_{k}$ and thus the Cayley complex of $\Gamma$ is a two-dimensional
simplicial complex on which $\Gamma$ acts transitively on the vertices. The
link of $e\in\Gamma$ is a bipartite graph with sides $S_{0}=S,S_{1}=S^{-1}$,
where each relation $s_{i}s_{j}s_{k}$ gives rise to $3$ edges
$\\{s_{j},s_{i}^{-1}\\},\\{s_{k},s_{j}^{-1}\\},\\{s_{i},s_{k}^{-1}\\}$. Thus
the the link is composed of $3$ bipartite graphs $L_{1},L_{2},L_{3}$ on $S\cup
S^{-1}$:

  1. (1)

The graph $L_{1}$ is composed of all the edges $\\{s_{j},s_{i}^{-1}\\}$ where
$s_{i}s_{j}s_{k}$ is a relation.

  2. (2)

The graph $L_{2}$ is composed of all the edges $\\{s_{k},s_{j}^{-1}\\}$ where
$s_{i}s_{j}s_{k}$ is a relation.

  3. (3)

The graph $L_{3}$ is composed of all the edges $\\{s_{i},s_{k}^{-1}\\}$ where
$s_{i}s_{j}s_{k}$ is a relation.

Note that we allow double edges, i.e., if
$s_{i}s_{j}s_{k},s_{i}s_{j}s_{k^{\prime}}$ are two relations of $\Gamma$, then
the edge $\\{s_{j},s_{i}^{-1}\\}$ will appear twice.

The idea of [ALuS15] is that each of these graphs behave like
$\mathbb{G}_{\operatorname{bipartite}}(m,\rho^{\prime})$ where
$\rho^{\prime}\geq\rho$ and thus by Lemma 4.11 and Theorem 4.13, we can bound
the second eigenvalue of the link. However, some analysis is needed to account
for double edges and this will require Lemma 4.14. To sum up, we will prove
the following:

######  Theorem 4.15.

Let $\frac{3}{2}<\alpha<2$, $C>0$ be constants and
$\rho(m)=\frac{C}{m^{\alpha}}$. For a random group
$\Gamma\in\mathcal{M}^{+}(m,\rho)$, it holds a.a.s. that link of each vertex
in the Cayley complex of $\Gamma$ is a connected graph and the second
eigenvalue of the random walk on it is
$<\frac{10}{\sqrt{C}m^{1-\frac{\alpha}{2}}}$.

###### Proof.

For clarity, in this proof we will denote the degree of a vertex by $w(v)$ and
not $m(v)$, since the letter $m$ already appears as a parameter of the random
group.

We first claim that a.a.s., for every $i=1,2,3$ each edge in $L_{i}$ appears
with multiplicity $\leq 3$. Indeed, the probability that a specific edge
appear $4$ times or more is bounded by $m^{4}\rho(m)^{4}$. By union bound, the
probability that some edge appears $4$ times or more is bounded by
$m^{6}\rho(m)^{6}\leq C^{4}m^{6-4\alpha}$ and this tends to $0$ since
$\alpha>\frac{3}{2}$.

Second, we claim that a.a.s, for every for every $i=1,2,3$ each vertex in
$L_{i}$ has at most $K=\lfloor\frac{1}{2\alpha-3}\rfloor+1$ double/triple
edges connected to it. Indeed, the probability that a specific vertex has $K$
double/triple edges connected to it is bounded by $m^{3K}\rho(m)^{2K}$. By
union bound, the probability that some vertex has $K$ double/triple edges
connected to it is bounded by $2m^{3K+1}\rho(m)^{2K}\leq
2C^{2K}m^{3K+1-2K\alpha}$ and this tends to $0$ since $K>\frac{1}{2\alpha-3}$.

Let $L_{i}^{\prime}$ be the graph obtained from $L_{i}$ by deleting multiple
edges. Then a.a.s., for every $v\in S\cup S^{-1}$, the degree of $v$ in
$L_{i}^{\prime}$ differ than the degree of $v$ in $L_{i}$ by at most $2K$
(which is constant and does not depend on $m$).

Each of the $L_{i}^{\prime}$ is a random bipartite Erdös-Rényi graph
$\mathbb{G}_{\operatorname{bipartite}}(m,\rho^{\prime})$ with

$\rho^{\prime}(m)=1-(1-\rho(m))^{m}\geq\frac{C}{m^{\alpha-1}}$

for every large $m$.

By Theorem 4.13, it holds a.a.s. for every $i=1,2,3$ that $L_{i}^{\prime}$ is
connected and the second eigenvalue of the random walk operator on it is
bounded from above by
$\frac{8}{\sqrt{m\rho^{\prime}(m)}}\leq\frac{8}{\sqrt{C}m^{1-\frac{\alpha}{2}}}$.
By Lemma 4.14, for every $i$ and every vertex in $L_{i}^{\prime}$, it holds
a.a.s. that

$|w_{i}^{\prime}(v)-m\rho^{\prime}(m)|\leq\frac{m\rho^{\prime}(m)}{m^{\frac{2-\alpha}{3}}},$

where $w_{i}^{\prime}$ denotes the degree of $v$ in $L_{i}^{\prime}$. Thus, by
Lemma 4.11, it holds a.a.s. that for every large $m$ the second largest
eigenvalue of the random walk on $L_{1}^{\prime}\cup L_{2}^{\prime}\cup
L_{3}^{\prime}$ is bounded from above by

$\frac{8}{\sqrt{C}m^{1-\frac{\alpha}{2}}}+\frac{20}{m^{\frac{4}{3}(1-\frac{\alpha}{2})}}<\frac{9}{\sqrt{C}m^{1-\frac{\alpha}{2}}}.$

Note that by the discussion above $L_{1}\cup L_{2}\cup L_{3}$ is a union of
$L_{1}^{\prime}\cup L_{2}^{\prime}\cup L_{3}^{\prime}$ and the graph $H$ of
the deleted edges. Also note that a.a.s. it holds that every vertex $v$, $H$
has at most $6K=6\lfloor\frac{1}{2\alpha-3}\rfloor+6$ edges connected to it.
Therefore if we denote $w^{\prime}$ to be the degree of a vertex is
$L_{1}^{\prime}\cup L_{2}^{\prime}\cup L_{3}^{\prime}$ and $w^{\prime\prime}$
to be the degree of a vertex in $H$, it holds a.a.s. for every large $m$ that

$\frac{w^{\prime\prime}(v)}{w^{\prime}(v)}\leq\frac{6K}{3(m\rho^{\prime}(m)-\frac{m\rho^{\prime}(m)}{m^{\frac{2-\alpha}{3}}})}\leq\frac{4K}{Cm^{2-\alpha}}.$

Thus, from Lemma 4.12 it follows that a.a.s. the second largest eigenvalue of
the random walk on $L_{1}\cup L_{2}\cup L_{3}$ is bounded from above by

$\frac{9}{\sqrt{C}m^{1-\frac{\alpha}{2}}}+\frac{16K}{Cm^{2-\alpha}}<\frac{10}{\sqrt{C}m^{1-\frac{\alpha}{2}}}$

as needed. ∎

After this, we can prove the following Theorem:

######  Theorem 4.16.

Let $\frac{3}{2}<\alpha<2,C>0$ be constants and
$\rho(m)=\frac{C}{m^{\alpha}}$. Define a class of Banach spaces
$\mathcal{E}_{\mathcal{M}^{+}}(\alpha,C,m)$ as the union of all the classes
$\mathcal{E}^{\text{u-curved}}_{\omega}$ for which
$\omega(\frac{10}{\sqrt{C}m^{1-\frac{\alpha}{2}}})\leq\frac{1}{26}$.

Then $\mathcal{M}^{+}(m,\rho)$ has property
$(F\mathcal{E}_{\mathcal{M}^{+}}(\alpha,C,m))$ a.a.s. In particular:

  * •

For any uniformly curved space $\operatorname{\mathbb{E}}$,
$\mathcal{M}^{+}(m,\rho)$ has property $(F\operatorname{\mathbb{E}})$ a.a.s.

  * •

For
$\theta_{0}=\frac{\log(26)}{(1-\frac{\alpha}{2})\log(m)+\frac{1}{2}\log(C)-\log(10)}$,
$\mathcal{M}^{+}(m,\rho)$ has property $(F\mathcal{E}_{\theta_{0}})$ a.a.s.

  * •

For $2\leq
p\leq\frac{2(1-\frac{\alpha}{2})\log(m)+\log(C)-2\log(10)}{\log(26)}$,
$\mathcal{M}^{+}(m,\rho)$ has property $(FL^{p})$ a.a.s.

###### Proof.

Let $\Gamma\in\mathcal{M}^{+}(m,\rho)$ be a random group. Denote by $X$ the
Cayley complex of $X$. This is a $2$-dimensional simplicial complex on which
$\Gamma$ acts freely and transitively on vertices and the links of $X$ are
bipartite graphs. By Theorem 4.15 it holds a.a.s. that the second largest
eigenvalue of the random walk on the link of a vertex in $X$ is
$<\frac{10}{\sqrt{C}m^{1-\frac{\alpha}{2}}}$. Thus by Corollary 2.21 it holds
a.a.s. that for every
$\operatorname{\mathbb{E}}\in\mathcal{E}_{\mathcal{M}^{+}}(\alpha,C,m)$ and
every $v\in X(0)$

$\lambda_{X_{v},\operatorname{bipartite}}^{\operatorname{\mathbb{E}}}<2\omega\left(\frac{10}{\sqrt{C}m^{1-\frac{\alpha}{2}}}\right)\leq\frac{1}{13}.$

Thus by Theorem 3.10 (noting that
$\mathbb{C}\in\mathcal{E}_{\mathcal{M}^{+}}(\alpha,C,m)$) it holds that a.a.s.
that $\Gamma$ has property $(F\mathcal{E}_{\mathcal{M}^{+}}(\alpha,C,m))$.

Next, we derive the particular statements stated above.

First, for every uniformly curved space $\operatorname{\mathbb{E}}$ it holds
that $\operatorname{\mathbb{E}}\in\mathcal{M}^{+}(m,\rho)$ given that $m$ is
large enough. Thus for every uniformly curved space it holds a.a.s. that
$\mathcal{M}^{+}(m,\rho)$ has property $(F\operatorname{\mathbb{E}})$.

Second, by Corollary 2.14 it holds for every $0<\theta_{0}\leq 1$ that
$\mathcal{E}_{\theta_{0}}\subseteq\mathcal{E}^{\text{u-curved}}_{\omega(t)=t^{\theta_{0}}}$.
Thus if

(1)
$\left(\frac{10}{\sqrt{C}m^{1-\frac{\alpha}{2}}}\right)^{\theta_{0}}=\frac{1}{26}$

it follows that
$\mathcal{E}_{\theta_{0}}\subseteq\mathcal{E}_{\mathcal{M}^{+}}(\alpha,C,m)$
and therefore a.a.s. $\mathcal{M}^{+}(m,\rho)$ has property
$(F\mathcal{E}_{\theta_{0}})$. The equation (1) is equivalent to

$\theta_{0}=\frac{\log(26)}{(1-\frac{\alpha}{2})\log(m)+\frac{1}{2}\log(C)-\log(10)}$

as needed.

Last, for every $2\leq p<\infty$, every $L^{p}$ space is
$\frac{2}{p}$-strictly Hilbertian. Thus for
$\theta_{0}=\frac{\log(26)}{(1-\frac{\alpha}{2})\log(m)+\frac{1}{2}\log(C)-\log(10)}$
as above, if $\frac{2}{p}\geq\theta_{0}$ it follows that

$\frac{2}{p}\leq\frac{\log(26)}{(1-\frac{\alpha}{2})\log(m)+\frac{1}{2}\log(C)-\log(10)}$

it follows that a.a.s. $\mathcal{M}^{+}(m,\rho)$ has property $(FL^{p})$. The
condition $\frac{2}{p}\geq\theta_{0}$ is equivalent to

$p\leq\frac{2(1-\frac{\alpha}{2})\log(m)+\log(C)-2\log(10)}{\log(26)}$

as needed.

∎

Combining the above Theorem and Theorem 4.10 allows us to deduced fixed point
properties for random groups in the Gromov density model:

######  Theorem 4.17.

Let $\frac{1}{3}<d<\frac{1}{2}$, $k\geq 2$ be constants. Define a class of
Banach spaces $\mathcal{E}_{\mathcal{D}}(k,l,d)$ as the union of all the
classes $\mathcal{E}^{\text{u-curved}}_{\omega}$ for which
$\omega\left(\frac{20\sqrt{2}}{(2k-1)^{\frac{l}{6}(3d-1)}}\right)\leq\frac{1}{26}$.

Then for $l$ divisible by $3$, $\mathcal{D}(k,l,d)$ has property
$(F\mathcal{E}_{\mathcal{D}}(k,l,d))$ a.a.s. In particular:

  * •

For any uniformly curved space $\operatorname{\mathbb{E}}$,
$\mathcal{D}(k,l,d)$ has property $(F\operatorname{\mathbb{E}})$ a.a.s. given
that $l$ is divisible by $3$.

  * •

For
$\theta_{0}=\frac{\log(26)}{\frac{l}{6}(3d-1)\log(2k-1)-\log(20\sqrt{2})}$,
$\mathcal{D}(k,l,d)$ has property $(F\mathcal{E}_{\theta_{0}})$ a.a.s. given
that $l$ is divisible by $3$.

  * •

For $2\leq
p\leq\frac{l(d-\frac{1}{3})\log(2k-1)-2\log(20\sqrt{2})}{\log(26)}$,
$\mathcal{D}(k,l,d)$ has property $(FL^{p})$ a.a.s. given that $l$ is
divisible by $3$.

###### Proof.

Let $m=m(k,l)=\frac{1}{2}(2k-1)^{\frac{l}{3}}$ and
$\rho(m)=\frac{1}{4}\frac{1}{m^{3(1-d)}}$. We can write
$\rho(m)=C\frac{1}{m^{\alpha}}$ for $C=\frac{1}{4},\alpha=3(1-d)$. Then with
this $m$ and $\rho$, the class $\mathcal{E}_{\mathcal{M}^{+}}(\alpha,C,m)$
defined in Theorem 4.16 contains the class $\mathcal{E}_{\mathcal{D}}(k,l,d)$
defined above. By Theorem 4.10 for $l$ divisible by $3$, $\mathcal{D}(k,l,d)$
has property $(F\mathcal{E}_{\mathcal{D}}(k,l,d))$ a.a.s.

The particular statements are proven as in Theorem 4.16 and are left for the
reader. ∎

######  Remark 4.18.

By the proof of Theorem 4.10 a similar Theorem can also be proved in the
Gromov binomial model and this is left for the reader.

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

E-mail address<EMAIL_ADDRESS>