Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP?

In particular, does there exist a group $G$ with the AP and a surjective group homomorphism $\varphi\colon G\to\mathrm{SL}(3,\mathbb{Z})$ such that the kernel of $\varphi$ is amenable?

**Background:**

Let $G$ be a locally compact group. We use $A(G)$ to denote the Fourier algebra of $G$, and we let $M_0A(G)$ denote the space of completely bounded multipliers on $A(G)$, equipped with completely bounded norm. Then $M_0A(G)$ has a natural weak*-topology and we say that $G$ has *the Approximation Property* (or AP, for short), if there is a net in $A(G)$ that converges to the constant function $1$ in the weak*-toplogy of $M_0A(G)$. This is Definition 1.1 in the following paper by Haagerup and Kraus:

Trans. AMS: Approximation properties for group C*-algebras and group von Neumann algebras.

**Motivation:**

An open problem in abstract harmonic analysis asks whether given a locally compact group $G$ and $p\in[1,\infty)$, the algebra $PM_p(G)$ of $p$-pseudomeasures on $G$ coincides with the algebra $CV_p(G)$ of $p$-convolvers on $G$. The answer is "yes" for $p=2$, in which case $CV_2(G)=PM_2(G)$ is the group von Neumann algebra of $G$. A positive answer to this problem for $p\neq 2$ is known for groups with the AP, as shown in the following article of Daws and Spronk:

arXiv: The approximation property implies that convolvers are pseudo-measures.

(For definitions of $PM_p(G)$ and $CV_p(G)$ we refer to the above paper.)

Now, given a locally compact group $G$ and a closed, amenable subgroup $N$ of $G$, we have by the above result that $PM_p(G)=CV_p(G)$. In general, it is known that $PM_p(G)\subset CV_p(G)$. The goal would be to show that the quotient map $Q\colon G\to G/N$ induces a surjective map $CV_p(G)\to CV_p(G/N)$. One could then try to deduce that $PM_p(G/N)=CV_p(G/N)$, thus extending the result of Daws and Spronk to groups that are quotients of groups with the AP by amenable subgroups. However, this is only interesting if this class of groups is actually larger than the groups with AP.