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 $g, g' \in A_{(\tau , \eta )}$ , if $g \neq g'$ , then $g.\tau \neq g' .\tau $ (it may be that set $A_{(\tau , \eta )}$ is the empty set). Note that by the choice of $D(k), D(l)$ , there are simplices $\sigma , \sigma ' \in D(n)$ such that $\tau \subseteq \sigma $ and $\eta \subseteq \sigma '$ . It follows that for every $g \in A_{(\tau , \eta )}$ , it holds that $g. \tau \subseteq g. \sigma \cap \sigma '$ and thus,

$$ \begin{align*}\vert g. \sigma \cap \sigma ' \vert \geq \vert g. \tau \vert = \vert \tau \vert \geq n-1,\end{align*} $$

(i.e., $A_{(\tau , \eta )} \subseteq K$ ). Thus,

$$ \begin{align*} \sum_{\eta \in D(l)} \frac{1}{\mu (G_{\eta})} \sum_{\tau \in X(k), \tau \subseteq \eta} \psi (\tau, \eta) &= \sum_{\eta \in D(l)} \frac{1}{\mu (G_{\eta})} \sum_{\tau \in D(k)} \sum_{g \in A_{(\tau, \eta)}} \psi (g.\tau, \eta) \\ &\leq \kappa \sum_{\eta \in D(l), \tau \in D(k)} \frac{1}{\mu (G_{\eta})} \sum_{g \in A_{(\tau, \eta)}} \psi (\tau, g^{-1}. \eta) \\ &\leq \kappa^2 \sum_{\eta \in D(l), \tau \in D(k)} \frac{1}{\mu (G_{\eta})} \sum_{g \in A_{(\tau, \eta)}} \frac{1}{\mu (G_{\tau})} \int_{G_{\tau}} \psi (\tau, g' g^{-1}. \eta) d \mu (g')\\ & = \kappa^2 \sum_{\tau \in D(k)} \frac{1}{\mu (G_{\tau})} \sum_{\eta \in D(l)} \frac{1}{\mu (G_{\eta})} \int_{g", \tau \subseteq g".\eta} \psi (\tau, g". \eta) d \mu (g")\\ & =\kappa^2 \sum_{\tau \in D(k)} \frac{1}{\mu (G_{\tau})} \sum_{\eta \in X(l), \tau \subseteq \eta} \psi (\tau, \eta).\\[-47pt] \end{align*} $$

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 $k =\vert \nu \vert -1$ and define

$$ \begin{align*}\psi : \lbrace (\tau, \sigma ) \in X(k) \times X(n) : \tau \subseteq \sigma \rbrace \rightarrow [0, \infty)\end{align*} $$

as

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

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

$$ \begin{align*}\psi (g.\tau, g. \sigma ) \leq \left( \sup_{g \in K} \Vert \pi (g) \Vert^2 \right) \psi (\tau, \sigma )\end{align*} $$

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

$$ \begin{align*} &\Vert \phi \Vert^2 = \sum_{\sigma \in D(n)} \frac{1}{\mu (G_{\sigma})} \vert \phi (\sigma )\vert^2 = \sum_{\sigma \in D(n)} \frac{1}{\mu (G_{\sigma})}\sum_{\tau \in X(k), \tau \subseteq \sigma} \psi (\tau, \sigma) \leq \\ &\quad\left( \sup_{g \in K} \Vert \pi (g) \Vert^2 \right) \sum_{\tau \in D(k)} \frac{1}{\mu (G_{\tau})}\sum_{\sigma \in X(n), \tau \subseteq \sigma} \psi (\tau, \sigma) = \\ &\quad \left( \sup_{g \in K} \Vert \pi (g) \Vert^2 \right) \sum_{\tau \in D(k), \operatorname{type} (\tau) = \nu} \frac{1}{\mu (G_{\tau})}\sum_{\sigma \in X(n), \tau \subseteq \sigma} \vert \phi (\sigma) \vert^2, \end{align*} $$

as needed.

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.

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

$$ \begin{align*} \vert \phi_x (\sigma) \vert = \left\vert \frac{1}{\mu (G_{\sigma}) } \int_{G_{\sigma}} \pi ( g) x d \mu (g) \right\vert \leq \frac{1}{\mu (G_{\sigma} )} \int_{G_{\sigma}} \vert \pi ( g) x \vert d \mu (g) \leq \sup_{g \in G_{\sigma}} \Vert \pi (g) \Vert. \end{align*} $$

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

$$ \begin{align*}\Vert \phi_x \Vert^2 \leq \sum_{\sigma \in D(n)} \frac{1}{\mu (G_{\sigma} )} (\sup_{g \in G_{\sigma}} \Vert \pi (g) \Vert)^2,\end{align*} $$

and the bound on the norm of $R_1$ follows.

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

$$ \begin{align*} \left\vert R_2 \phi \right\vert \leq \frac{1}{\vert D(n) \vert} \sum_{\sigma \in D(n)} \sqrt{ \mu (G_{\sigma} )} \leq \max_{\sigma \in D(n)} \sqrt{\mu (G_{\sigma} )}.\\[-49pt] \end{align*} $$

Proof. Fix $\phi \in \bigcap _{\nu \subseteq \lbrace 0,....,n \rbrace , \vert \nu \vert =n} \operatorname {Im} (P_{\nu }^{\pi })$ . As noted above, $\operatorname {Im} (P_{\nu }^{\pi }) = C (X(n), \pi )_{\nu }$ , and it follows that if $\sigma \cap \sigma ' \in X(n-1)$ , then $\phi (\sigma ) = \phi (\sigma ')$ . Thus, if $\sigma , \sigma ' \in X(n)$ are connected by a gallery, then $\phi (\sigma ) = \phi (\sigma ')$ . By assumption, X is 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 $\operatorname {\mathbb {E}}$ .

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

$$ \begin{align*}\operatorname{Im} (P_{\nu}^{\pi}) \cap \operatorname{Im} (P_{\nu '}^{\pi}) \supseteq \operatorname{Im} (P_{\nu \cap \nu '}^{\pi}).\end{align*} $$

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

$$ \begin{align*}\operatorname{Im} (P_{\nu}^{\pi}) \cap \operatorname{Im} (P_{\nu '}^{\pi}) \subseteq \operatorname{Im} (P_{\nu \cap \nu '}^{\pi}).\end{align*} $$

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

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

$$ \begin{align*} &{P_{\nu \cap \nu '}^{\pi} P_{\nu}^{\pi} \phi (\sigma) = } \\ & \frac{1}{\vert \lbrace \sigma ' \in X(n) : \sigma \sim_{\nu \cap \nu '} \sigma ' \rbrace \vert} \sum_{\sigma ' \sim_{\nu \cap \nu '} \sigma} \frac{1}{\vert \lbrace \sigma " \in X(n) : \sigma ' \sim_{\nu} \sigma " \rbrace \vert} \sum_{\sigma " \sim_{\nu} \sigma '} \phi (\sigma ") = \\ & \frac{1}{\vert \lbrace \sigma ' \in X(n) : \sigma \sim_{\nu \cap \nu '} \sigma ' \rbrace \vert} \sum_{\sigma " \sim_{\nu \cap \nu '} \sigma} \phi (\sigma ") \sum_{\sigma ' \sim_{\nu} \sigma "} \frac{1}{\vert \lbrace \sigma " \in X(n) : \sigma ' \sim_{\nu} \sigma " \rbrace \vert} = P_{\nu \cap \nu '}^{\pi} \phi (\sigma). \end{align*} $$

Therefore, $P_{\nu \cap \nu '}^{\pi } P_{\nu }^{\pi } = P_{\nu \cap \nu '}^{\pi }$ as needed.

Proof. Fix $\nu \subseteq \lbrace 0,..., n \rbrace $ , $\vert \nu \vert = n$ 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

$$ \begin{align*} &\sum_{\sigma \in X(n), \tau \subseteq \sigma} \vert P_{\nu}^{\pi} \phi (\sigma) \vert^2 \leq \sum_{\sigma \in X(n), \tau \subseteq \sigma} \frac{1}{\vert \lbrace \sigma ' \in X(n) : \sigma \sim_{\nu} \sigma ' \rbrace \vert} \sum_{\sigma ' \in X(n), \sigma \sim_{\nu} \sigma '} \vert \phi (\sigma ') \vert^2 = \\ &\sum_{\sigma \in X(n), \tau \subseteq \sigma} \frac{1}{\vert \lbrace \sigma ' \in X(n) : \tau \subseteq \sigma ' \rbrace \vert} \sum_{\sigma ' \in X(n), \tau \subseteq \sigma '} \vert \phi (\sigma ') \vert^2 = \\ &\sum_{\sigma ' \in X(n), \tau \subseteq \sigma '} \vert \phi (\sigma ') \vert^2 \sum_{\sigma \in X(n), \tau \subseteq \sigma} \frac{1}{\vert \lbrace \sigma \in X(n) : \tau \subseteq \sigma \rbrace \vert} = \sum_{\sigma ' \in X(n), \tau \subseteq \sigma '} \vert \phi (\sigma ') \vert^2. \end{align*} $$

By Corollary 3.2 and the previous inequality, it holds that

$$ \begin{align*} &\Vert P_{\nu}^{\pi} \phi \Vert^2 \leq \left( \sup_{g \in K} \Vert \pi (g) \Vert^2 \right) \sum_{\tau \in D(n-1), \operatorname{type} (\tau) =\nu} \frac{1}{\mu (G_ \tau)} \sum_{\sigma \in X(n), \tau \subseteq \sigma} \vert P_{\nu}^{\pi} \phi (\sigma) \vert^2 \leq \\ &\quad \left( \sup_{g \in K} \Vert \pi (g) \Vert^2 \right) \sum_{\tau \in D(n-1), \operatorname{type} (\tau) =\nu} \frac{1}{\mu (G_ \tau)}\sum_{\sigma ' \in X(n), \tau \subseteq \sigma '} \vert \phi (\sigma ') \vert^2 \leq \left( \sup_{g \in K} \Vert \pi (g) \Vert^4 \right) \Vert \phi \Vert^2 \end{align*} $$

as needed.

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

$$ \begin{align*}\Vert (P_{\nu}^{\pi} P_{\nu '}^{\pi} - P_{\nu \cap \nu '}^{\pi} ) \phi \Vert \leq \left(\sup_{g \in K} \Vert \pi (g) \Vert^2 \right) \lambda_{\nu \cap \nu '}^{\operatorname{\mathbb{E}}} \Vert \phi \Vert.\end{align*} $$

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

$$ \begin{align*}S_1 = \lbrace v \in X_{\tau} (0) : \operatorname{type} (\lbrace v \rbrace) = \lbrace 0,..., n \rbrace \setminus \nu \rbrace,\end{align*} $$

$$ \begin{align*}S_2 = \lbrace v \in X_{\tau} (0) : \operatorname{type} (\lbrace v \rbrace) = \lbrace 0,..., n \rbrace \setminus \nu ' \rbrace.\end{align*} $$

Let $P_1, P_2, P_{1,2} : \ell ^2 (X_{\tau } (1); \operatorname {\mathbb {E}}) \rightarrow \ell ^2 (X_{\tau } (1); \operatorname {\mathbb {E}})$ be defined in section 2.6; that is,

$$ \begin{align*}P_i \Phi (\lbrace v_1,v_2 \rbrace) = \frac{1}{m (v_i)} \sum_{\lbrace v_i, u \rbrace \in E} \Phi ( \lbrace v_i, u \rbrace ), \forall \Phi \in \ell^2 (X_{\tau} (1); \operatorname{\mathbb{E}}),\end{align*} $$

and

$$ \begin{align*}P_{1,2} \Phi \equiv \frac{1}{\vert X_{\tau} (1) \vert} \sum_{\lbrace u, v \rbrace \in X_{\tau} (1)} \Phi (\lbrace u, v \rbrace), \forall \Phi \in \ell^2 (X_{\tau} (1); \operatorname{\mathbb{E}}).\end{align*} $$

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

$$ \begin{align*}\phi_{\tau} (\lbrace u,v \rbrace) = \phi (\tau \cup \lbrace u,v \rbrace).\end{align*} $$

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

$$ \begin{align*}P_{\nu}^{\pi} \phi (\sigma) = P_1 \phi_{\tau} (\sigma \setminus \tau), P_{\nu '}^{\pi} \phi (\sigma) = P_2 \phi_{\tau} (\sigma \setminus \tau) , P_{\nu \cap \nu '}^{\pi} \phi (\sigma) = P_{1,2} \phi_{\tau} (\sigma \setminus \tau).\end{align*} $$

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

$$ \begin{align*} &\sum_{\sigma \in X(n), \tau \subseteq \sigma} \vert (P_{\nu}^{\pi} P_{\nu '}^{\pi} - P_{\nu \cap \nu '}^{\pi} ) \phi (\sigma) \vert^2 = \sum_{\lbrace u,v \rbrace \in X_{\tau} (1)} \vert (P_1 P_{2} - P_{1,2} ) \phi_{\tau} (\lbrace u,v \rbrace) \vert^2 \leq \\ &\lambda_{\tau, \operatorname{bipartite}}^{\operatorname{\mathbb{E}}} \sum_{\lbrace u,v \rbrace \in X_{\tau} (1)} \vert \phi_{\tau} (\lbrace u,v \rbrace) \vert^2 \leq \lambda_{\tau, \operatorname{bipartite}}^{\operatorname{\mathbb{E}}} \sum_{\sigma \in X(n), \tau \subseteq \sigma} \vert \phi (\sigma) \vert^2 \leq \lambda_{\nu \cap \nu '}^{\operatorname{\mathbb{E}}} \sum_{\sigma \in X(n), \tau \subseteq \sigma} \vert \phi (\sigma) \vert^2. \end{align*} $$

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

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

as needed.

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

$$ \begin{align*}T_{\pi}^k \phi (\sigma) = \sum_{\sigma ' \in X(n)} f^k_{\sigma} (\sigma ') \phi (\sigma ')\end{align*} $$

for every $\phi \in C (X(n), \pi )$ .

Define $h: X(n) \rightarrow C_c (G)$ as follows: for every $\sigma ' \in X(n)$ , choose $\sigma " \in D(n)$ and $g \in G$ such that $g. \sigma " = \sigma '$ and define

From the definition of $\phi _x$ and Proposition 3.3, it follows that h is well-defined and that for every $\sigma ' \in X(n)$ , $\phi _x (\sigma ') = \pi (h (\sigma ')) x.$

Recall that $R_2 \phi = \frac {1}{\vert D(n) \vert } \sum _{\sigma \in D(n)} \phi (\sigma )$ . Thus, for every $x \in \operatorname {\mathbb {E}}$ ,

$$ \begin{align*} R_2 T_{\pi}^k R_1 x = \frac{1}{\vert D(n) \vert} \sum_{\sigma \in D(n)} T_{\pi}^k \phi_x (\sigma) = \frac{1}{\vert D(n) \vert} \sum_{\sigma \in D(n)} \sum_{\sigma ' \in \operatorname{supp} ( f^k_{\sigma})} f^k_{\sigma} (\sigma ') \phi_x (\sigma ') = \\ \frac{1}{\vert D(n) \vert} \sum_{\sigma \in D(n)} \sum_{\sigma ' \in \operatorname{supp} ( f^k_{\sigma})} f^k_{\sigma} (\sigma ') \pi ( h (\sigma ')) x = \pi \left(\frac{1}{\vert D(n) \vert} \sum_{\sigma \in D(n)} \sum_{\sigma ' \in \operatorname{supp} ( f^k_{\sigma})} f^k_{\sigma} (\sigma ') h (\sigma ') \right)x \end{align*} $$

and we take

$$ \begin{align*}f_k = \frac{1}{\vert D(n) \vert} \sum_{\sigma \in D(n)} \sum_{\sigma ' \in \operatorname{supp} ( f^k_{\sigma})} f^k_{\sigma} (\sigma ') h (\sigma ').\\[-49pt] \end{align*} $$

Proof. Note that

$$ \begin{align*}\mathcal{E}_X = \bigcup_{0 < \varepsilon < \frac{1}{8n-3}} \mathcal{E}_{X, \varepsilon},\end{align*} $$

and thus if we show that G has property $(F \mathcal {E}_{X,\varepsilon })$ for every $0< \varepsilon < \frac {1}{8n-3}$ , it will follow that G has property $(F \mathcal {E}_{X})$ . Below, we will show that for every $0< \varepsilon < \frac {1}{8n-3}$ , it holds that G has robust property (T) with respect to $\mathcal {E}_{X, \varepsilon }$ and property $(F \mathcal {E}_{X,\varepsilon })$ .

Fix $0< \varepsilon < \frac {1}{8n-3}$ . If $\mathcal {E}_{X, \varepsilon } = \lbrace \lbrace \underline {0} \rbrace \rbrace $ , then robust property (T) with respect to $\mathcal {E}_{X, \varepsilon }$ and property $(F \mathcal {E}_{X,\varepsilon })$ holds trivially. Thus, we will assume below that $\mathcal {E}_{X, \varepsilon }$ contains a non-trivial Banach space.

Note that by Lemma 2.7, $\mathcal {E}_{X, \varepsilon }$ is closed under $\ell _2$ sums. It is also obvious that by its definition, $\mathcal {E}_X$ is closed under passing to subspaces, and thus $\mathbb {C} \in \mathcal {E}_{X, \varepsilon }$ due to the fact that $\mathbb {C}$ is a subspace of every non-trivial Banach space. It follows from Proposition 2.4 that if G has robust property (T) with respect to $\mathcal {E}_{X, \varepsilon }$ , then G has property $(F \mathcal {E}_{X, \varepsilon })$ . Thus, it is sufficient to prove that G has robust property with respect to $\mathcal {E}_{X, \varepsilon }$ .

Denote

$$ \begin{align*}\beta_1 = \sqrt{\frac{ \frac{1}{8n-3} - \frac{\varepsilon}{2}}{ \frac{1}{8n-3} - \varepsilon}}, \gamma = \frac{1}{8n-3} - \frac{\varepsilon}{2},\end{align*} $$

and

$$ \begin{align*}\beta_2 = \sqrt{ 1+ \frac{1}{2} \frac{1-(8n-3)\gamma}{n-1 + (3n-1)\gamma}}.\end{align*} $$

Further, denote $\beta = \min \lbrace \beta _1, \beta _2 \rbrace $ 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; that is, the functions such that for every representation $\pi $ ,

$$ \begin{align*}\pi (f_k) = R_2 T_{\pi}^k R_1,\end{align*} $$

where $T = \frac {1}{n+1} \sum _{\nu \subseteq \lbrace 0,...,n \rbrace ,\vert \nu \vert =n} P_{\nu }^{\pi }$ . 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 \lbrace 0,...,n \rbrace $ , $\vert \nu \vert =n$ that

$$ \begin{align*}\Vert P_{\nu}^{\pi} \Vert \leq \beta_2^2 < 1+ \frac{1-(8n-3)\gamma}{n-1 + (3n-1)\gamma}.\end{align*} $$

By Lemma 3.8 and the choice of $\beta _1$ , it holds for every $\nu , \nu ' \subseteq \lbrace 0,...,n \rbrace , \nu \neq \nu '$ , $\vert \nu \vert = \vert \nu ' \vert =n$ that

$$ \begin{align*}\cos (\angle (P_{\nu}^{\pi}, P_{\nu '}^{\pi})) \leq \beta_1^2 \left( \frac{1}{8n-3} - \varepsilon \right) = \gamma < \frac{1}{8n-3}.\end{align*} $$

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

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

$$ \begin{align*}\Vert T_{\pi}^k - T_{\pi}^{\infty} \Vert \leq Cr^k.\end{align*} $$

Thus, it holds for every k that

$$ \begin{align*} \Vert R_2 T_{\pi}^{\infty} R_1 - \pi (f_k) \Vert \leq \Vert R_2 \Vert \Vert R_1 \Vert C r^k \leq^{\text{Proposition } 3.4} \left( \beta \sqrt{\sum_{\sigma \in D(n)} \frac{1}{\mu (G_{\sigma} )}} \max_{\sigma \in D(n)} \sqrt{\mu (G_{\sigma} )} \right) C r^k. \end{align*} $$

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.

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 $W_l'$ and then from $W_l \setminus W_l'$ . Thus, it is clear that $\Gamma \in \mathcal {B} (k,l,\rho )$ is a quotient of $\Gamma ' \in \mathcal {B} ' (k,l,\rho )$ and by the assumption that P is monotone increasing,

$$ \begin{align*}\mathbb{P} (\Gamma \in \mathcal{B} (k,l,\rho) \text{ has } P) \geq \mathbb{P} (\Gamma ' \in \mathcal{B} ' (k,l,\rho) \text{ has } P).\\[-37pt] \end{align*} $$

Proof. By Corollary 4.4, for $\rho _1 (l) = \frac {1}{2}(2k-1)^{-(1-d)l}$ , it holds that

$$ \begin{align*}\mathbb{P} (\Gamma \in \mathcal{D} (k,l,d) \text{ has property } (F \mathcal{E}_l)) \geq \mathbb{P} (\Gamma \in \mathcal{B} (k,l,\rho_1) \text{ has property } (F \mathcal{E}_l)) - \frac{6}{(2k-1)^{\frac{l}{2}}}.\end{align*} $$

By Proposition 4.6,

$$ \begin{align*}\mathbb{P} (\Gamma \in \mathcal{B} (k,l,\rho_1) \text{ has property } (F \mathcal{E}_l)) \geq \mathbb{P} (\Gamma ' \in \mathcal{B}' (k,l,\rho_1) \text{ has property } (F \mathcal{E}_l)) .\end{align*} $$

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

$$ \begin{align*}\mathbb{P} (\Gamma ' \in \mathcal{B}' (k,l,\rho_1) \text{ has property } P) \geq {\mathbb{P} (\Gamma \in \mathcal{M}^+ (m (l) , \rho) \text{ has property } (F \mathcal{E}_l)),}\end{align*} $$

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

Assume that l is divisible by $3$ . By Lemma 4.8, for every $\Gamma ' \in \mathcal {B} ' (k,l,\frac {1}{2} (2k-1)^{-(1-d)l})$ , there is a group $\Gamma \in \mathcal {M}^+ (m_1 , \rho _2)$ with $m_1 (l) = \vert W_{\frac {l}{3}}" \vert = k^2 (2k-1)^{\frac {l}{3}-2}$ and $\rho _2 (m _1) = \frac {1}{2} (2k-1)^{-(1-d)l}$ such that $\varphi (\Gamma )$ is a finite index subgroup in $\Gamma '$ and $\Gamma $ , $\Gamma '$ occur in the same probability.

By Proposition 2.6, if $\Gamma $ has property $(F \mathcal {E}_l)$ , then $\Gamma '$ 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

$$ \begin{align*}m = \frac{1}{2} (2k-1)^{\frac{l}{3}} \geq m_1.\end{align*} $$

That is,

$$ \begin{align*} \mathbb{P} (\Gamma ' \in \mathcal{B}' (k,l,\rho_1) \text{ has property } P) &\geq {\mathbb{P} (\Gamma \in \mathcal{M}^+ (m_1 , \rho_2 (m_1)) \text{ has property } (F \mathcal{E}_l))}\\ &\geq {\mathbb{P} (\Gamma \in \mathcal{M}^+ (m , \rho_2 (m_1)) \text{ has property } (F \mathcal{E}_l)).} \end{align*} $$

Note that

$$ \begin{align*} \rho_2 (m_1) = \frac{1}{2} (2k-1)^{-(1-d)l} =\frac{1}{2} \frac{1}{((2k-1)^{\frac{l}{3}})^{3(1-d)}} = \frac{1}{2} \frac{1}{ 2 m^{3(1-d)}} = \frac{1}{4 m^{3(1-d)}}. \end{align*} $$

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

$$ \begin{align*}\mathbb{P} (\Gamma ' \in \mathcal{B}' (k,l,\rho_1) \text{ has property } P) \geq {\mathbb{P} (\Gamma \in \mathcal{M}^+ (m , \rho (m)) \text{ has property } (F \mathcal{E}_l)),}\end{align*} $$

as needed.

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

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

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

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

Recall that

$$ \begin{align*}\sum_{v \in V} m(v) ((I-M) \phi (v)) \phi (v) = \sum_{\lbrace u,v \rbrace \in E_1 \cup ... \cup E_k} \vert \phi (u) - \phi (v) \vert^2,\end{align*} $$

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,

$$ \begin{align*}\sum_{v \in V} m(v) ((I-M_i) \phi (v)) \phi (v) = \sum_{\lbrace u,v \rbrace \in E_i} \vert \phi (u) - \phi (v) \vert^2.\end{align*} $$

Thus,

$$ \begin{align*} &\sum_{v \in V} m(v) ((I-M) \phi (v)) \phi (v) = \sum_{\lbrace u,v \rbrace \in E_1 \cup ... \cup E_k} \vert \phi (u) - \phi (v) \vert^2 = \sum_{i=1}^k \sum_{\lbrace u,v \rbrace \in E_i} \vert \phi (u) - \phi (v) \vert^2 \\& \quad =\sum_{i=1}^k \sum_{v \in V} m(v) ((I-M_i) \phi (v)) \phi (v) \geq (1- \lambda) \sum_{i=1}^k \sum_{v \in V} m_i (v)\left\vert \phi (v) - \frac{1}{m_i (V)} \sum_{u \in V} m_i (u) \phi (u) \right\vert{}^2\\& \quad =(1- \lambda) \sum_{i=1}^k \sum_{v \in V} m_i (v) \vert \phi (v) \vert^2 - (1- \lambda) \sum_{i=1}^k \sum_{v \in V} m_i (v) \left\vert \sum_{u \in V} \frac{m_i (u)}{m_i (V)} \phi (u) \right\vert{}^2 \\& \quad =.(1- \lambda) \sum_{v \in V} m (v) \vert \phi (v) \vert^2 - (1- \lambda) \sum_{i=1}^k m_i (V) \left\vert \sum_{u \in V} \frac{m_i (u)}{m_i (V)} \phi (u) \right\vert{}^2. \end{align*} $$

Thus, it is sufficient to prove that

$$ \begin{align*}\sum_{i=1}^k m_i (V) \left\vert \sum_{u \in V} \frac{m_i (u)}{m_i (V)} \phi (u) \right\vert{}^2 \leq \frac{16 \varepsilon^2}{(1- \varepsilon)^4} \sum_{v \in V} m (v) \vert \phi (v) \vert^2 .\end{align*} $$

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

$$ \begin{align*}\frac{(1-\varepsilon)d}{k (1+\varepsilon)d} \leq \frac{m_i (u)}{m(u)} \leq \frac{(1+\varepsilon)d}{k (1-\varepsilon)d}.\end{align*} $$

Thus,

$$ \begin{align*}\left( \frac{1-\varepsilon}{1+\varepsilon} \right)^2 \leq \frac{m(u)}{m_i (u)} \frac{m_i (V)}{m(V)} \leq \left( \frac{1+\varepsilon}{1-\varepsilon} \right)^2.\end{align*} $$

Therefore,

$$ \begin{align*}\left\vert 1 - \frac{m(u)}{m_i (u)} \frac{m_i (V)}{m(V)} \right\vert \leq \max \left\lbrace \left( \frac{1+\varepsilon}{1-\varepsilon} \right)^2 - 1, 1- \left( \frac{1-\varepsilon}{1+\varepsilon} \right)^2 \right\rbrace = \frac{4 \varepsilon}{(1- \varepsilon)^2}.\end{align*} $$

It follows that

$$ \begin{align*} & \sum_{i=1}^k m_i (V) \left\vert \sum_{u \in V} \frac{m_i (u)}{m_i (V)} \phi (u) \right\vert{}^2 = \sum_{i=1}^k m_i (V) \left\vert \sum_{u \in V} \left( \frac{m_i (u)}{m_i (V)} - \frac{m(u)}{m (V)} \right) \phi (u) \right\vert{}^2 \\& \quad ={\sum_{i=1}^k m_i (V) \left\vert \sum_{u \in V} \frac{m_i (u)}{m_i (V)} \left( 1 - \frac{m(u)}{m_i (u)} \frac{m_i (V)}{m(V)} \right) \phi (u) \right\vert{}^2 }\leq {\sum_{i=1}^k m_i (V) \sum_{u \in V} \frac{m_i (u)}{m_i (V)} \left\vert \left( 1 - \frac{m(u)}{m_i (u)} \frac{m_i (V)}{m(V)} \right) \phi (u) \right\vert{}^2 } \\& \quad \leq {\sum_{i=1}^k \sum_{u \in V} m_i (u) \left( \frac{4 \varepsilon}{(1- \varepsilon)^2} \right)^2 \vert \phi (u) \vert^2 }= \frac{16 \varepsilon^2}{(1- \varepsilon)^4} \sum_{u \in V} m (u) \vert \phi (u) \vert^2, \end{align*} $$

as needed.

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

$$ \begin{align*}\sum_{v \in V} m_1 (v) (M_1 \phi (v) ) \phi (v) \leq \left( \lambda + 4 \varepsilon \right) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2\end{align*} $$

or, equivalently, that

$$ \begin{align*}\sum_{v \in V} m_1 (v) ((I-M_1) \phi (v) ) \phi (v) \geq \left( 1-\lambda - 4 \varepsilon \right) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2.\end{align*} $$

Observe that

$$ \begin{align*} \sum_{v \in V} m(v) ((I-M) \phi (v)) \phi (v)& = \sum_{v \in V} m_1 (v) ((I-M_1) \phi (v)) \phi (v) + \sum_{v \in V} m_2 (v) ((I-M_2) \phi (v)) \phi (v) \\ &\leq \sum_{v \in V} m_1 (v) ((I-M_1) \phi (v)) \phi (v) + 2 \sum_{v \in V} m_2 (v) \vert \phi (v) \vert^2 \\ &\leq \sum_{v \in V} m_1 (v) ((I-M_1) \phi (v)) \phi (v) + 2 \varepsilon \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2. \end{align*} $$

Thus,

$$ \begin{align*}\sum_{v \in V} m_1 (v) ((I-M_1) \phi (v)) \phi (v) \geq \sum_{v \in V} m(v) ((I-M) \phi (v)) \phi (v) - 2 \varepsilon \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2,\end{align*} $$

and we are left to prove that

$$ \begin{align*}\sum_{v \in V} m(v) ((I-M) \phi (v)) \phi (v) \geq (1- \lambda - 2\varepsilon) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2.\end{align*} $$

Note that for every $u \in V$ , $\frac {1}{1+ \varepsilon } \leq \frac {m_1 (u)}{m (u)} \leq 1$ and

$$ \begin{align*}\frac{1}{1+ \varepsilon} \leq \frac{m (V)}{m(u)} \frac{m_1 (u)}{m_1 (V)} \leq 1+ \varepsilon.\end{align*} $$

Thus,

$$ \begin{align*} \sum_{v \in V} m(v) ((I-M) \phi (v)) \phi (v) &\geq (1- \lambda) \sum_{v \in V} m(v) \vert \phi (v) \vert^2 - (1- \lambda) m (V) \left\vert \sum_{u \in V} \frac{m (u)}{m(V)} \phi (u) \right\vert{}^2\\& \geq (1- \lambda) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2 - m (V) \left\vert \sum_{u \in V} \frac{m (u)}{m(V)} \phi (u) \right\vert{}^2\\& =(1- \lambda) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2 - m (V) \left\vert \sum_{u \in V} \frac{m (u)}{m(V)} \left(1 - \frac{m (V)}{m(u)} \frac{m_1 (u)}{m_1 (V)} \right) \phi (u) \right\vert{}^2\\& \leq(1- \lambda) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2 - \varepsilon^2 \sum_{u \in V} m (u) \vert \phi (u) \vert^2 \\&\leq (1- \lambda) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2 - \varepsilon^2 (1+\varepsilon) \sum_{u \in V} m_1 (u) \vert \phi (u) \vert^2 \\&\leq(1- \lambda) \sum_{v \in V} m_1 (v) \vert \phi (v) \vert^2 - 2\varepsilon \sum_{u \in V} m_1 (u) \vert \phi (u) \vert^2, \end{align*} $$

as needed.

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

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

$$ \begin{align*}\mathbb{P} (m(v) \leq (1- \delta) n \rho) \leq \exp (-\frac{\delta^2}{3} n \rho (n)).\end{align*} $$

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

$$ \begin{align*}\mathbb{P} \left(\vert m(v) - n \rho (n) \vert \geq \frac{n \rho (n)}{n^{\frac{1-\alpha}{3}}} \right) \leq 2 \exp \left(-\frac{n \rho (n)}{3 n^{\frac{2(1-\alpha)}{3}}} \right) \leq 2 \exp \left( -\frac{C n^{\frac{1-\alpha}{3}}}{3} \right).\end{align*} $$

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

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 appears four times or more is bounded by $m^4 \rho (m)^4$ . By union bound, the probability that some edge appears four 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 $2m^{3K+1} \rho (m)^{2K} \leq 2 C^{2K} m^{3K+1-2K \alpha }$ , and this tends to $0$ since $K> \frac {1}{2 \alpha - 3}$ .

Let $L_i '$ 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 $L_i '$ differs than the degree of v in $L_i$ by at most $2 K$ (which is constant and does not depend on m).

Each of the $L_i '$ is a random bipartite Erdös-Rényi graph with

$$ \begin{align*}\rho ' (m) = 1- (1-\rho (m))^m \geq \frac{C}{m^{\alpha -1}}\end{align*} $$

for every large m.

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

$$ \begin{align*}\vert w_i '(v) - m \rho ' (m) \vert \leq \frac{m \rho ' (m)}{m^{\frac{2-\alpha}{3}}},\end{align*} $$

where $w_i '$ denotes the degree of v in $L_i '$ . Thus, by Lemma 4.11, it holds a.a.s. that for every large m, the second largest eigenvalue of the random walk operator on $L_1 ' \cup L_2 ' \cup L_3 '$ is bounded from above by

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

Note that by the discussion above, $L_1 \cup L_2 \cup L_3$ is a union of $L_1 ' \cup L_2 ' \cup L_3 '$ and the graph H of the deleted edges. Also note that a.a.s. it holds that every vertex v, H has at most $6 K = 6 \lfloor \frac {1}{2 \alpha - 3} \rfloor + 6$ edges connected to it. Therefore, if we denote $w'$ to be the degree of a vertex is $ L_1 ' \cup L_2 ' \cup L_3 '$ and $w"$ to be the degree of a vertex in H, it holds a.a.s. for every large m that

$$ \begin{align*}\frac{w" (v)}{w' (v)} \leq \frac{6K}{3 ( m \rho ' (m) - \frac{m \rho ' (m)}{m^{\frac{2-\alpha}{3}}})} \leq \frac{4K}{C m^{2 - \alpha}}.\end{align*} $$

Thus, from Lemma 4.12, it follows that a.a.s. the second largest eigenvalue of the random walk operator on $L_1 \cup L_2 \cup L_3 $ is bounded from above by

$$ \begin{align*}\frac{9}{\sqrt{C} m^{1- \frac{\alpha}{2}}} + \frac{16K}{C m^{2 - \alpha}} < \frac{10}{\sqrt{C} m^{1- \frac{\alpha}{2}}}\end{align*} $$

as needed.

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 operator 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 $v \in X(0)$ ,

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

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) $$ \begin{align} \left( \frac{10}{\sqrt{C} m^{1- \frac{\alpha}{2}}} \right)^{\theta_0} = \frac{1}{26}, \end{align} $$

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

$$ \begin{align*}\theta_0 = \frac{\log (26)}{(1-\frac{\alpha}{2}) \log (m) + \frac{1}{2} \log (C) - \log (10)}\end{align*} $$

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

$$ \begin{align*}\frac{2}{p} \leq \frac{\log (26)}{(1-\frac{\alpha}{2}) \log (m) + \frac{1}{2} \log (C) - \log (10)},\end{align*} $$

it follows that a.a.s. $\mathcal {M}^+ (m, \rho )$ has property $(F L^p)$ . The condition $\frac {2}{p} \geq \theta _0$ is equivalent to

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

as needed.

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.