For a finitely generated group $\Gamma$ and its finite generating subset $S$, the Kazhdan constant $\kappa(\Gamma,S)$ is defined to be $$\kappa(\Gamma,S)=\inf_{\pi,v} \max_{g\in S} \| v - \pi_g v \|,$$ where $\inf$ is taken over all unitary representations $\pi$ having no nonzero $\pi(\Gamma)$-invariant vectors and over all unit vectors $v \in H_\pi$. The group $\Gamma$ has Kazhdan property (T) iff $\kappa(\Gamma,S)>0$ for some/every finite generating subset $S$. The group $\Gamma$ is said to have the uniform Kazhdan property if $\inf_S \kappa(\Gamma,S)>0$, where $\inf$ is taken over all finite generating subsets $S$. It was asked in Lubotzky's popular book "Discrete groups, expanding graphs and invariant measures" (1994) whether every Kazhdan group has uniform Kazhdan property. Counterexamples were found by Gelander--Zuk (2002; MR1910934) and by Osin (2002; MR1958994), but it is still an open problem whether $\mathrm{SL}(3,\mathbb{Z})$ has the uniform Kazhdan property (T). I wonder specifically whether the Kazhdan constants for the following family of generating subsets are bounded away from $0$. For relatively prime pairs $(p,q)$, $$S_{p,q} = \{ I + pE_{i,j}, I + qE_{i,j} : i \neq j \}.$$
Remark: The "parent" group for $(\mathrm{SL}(3,\mathbb{Z}),S_{p,q})_{p,q}$ is $\mathrm{EL}(3,I)$, where $I:=\mathbb{Z}(X,Y)$ is the non-unital polynomial ring. Since it has the infinite abelian quotient $\mathrm{EL}(3,I/I^2) \cong (I/I^2)^{\oplus 6}$, it does not have property (T). If $I$ were unital, it would have property (T) by the Ershov--Jaikin-Zapirain theorem (2010; MR2570119). Instead, I conjecture $\mathrm{EL}(3,I)$ has Shalom's property $H_{\mathrm{T}}$ (see MR2096453), but it's another problem (which may not be relevant to the above problem).
