Algebraic fibrations of certain hyperbolic 4-manifolds

Algebraically fibering group is an algebraic generalization of the fibered 3-manifold group in higher dimensions. Let $M(\mathcal{P})$ and $M(\mathcal{E})$ be the cusped and compact hyperbolic real moment-angled manifolds associated to the hyperbolic right-angled 24-cell $\mathcal{P}$ and the hyperbolic right-angled 120-cell $\mathcal{E}$, respectively. Jankiewicz-Norin-Wise showed in [13] that $\pi_1(M(\mathcal{P}))$ and $\pi_1(M(\mathcal{E}))$ are algebraic fibered. Namely, there are two exact sequences $$1\rightarrow H_{\mathcal{P}}\rightarrow \pi_1(M(\mathcal{P}))\xrightarrow{\phi_{\mathcal{P}}} \mathbb{Z}\rightarrow 1,$$ $$1\rightarrow H_{\mathcal{E}}\rightarrow \pi_1(M(\mathcal{E}))\xrightarrow{\phi_{\mathcal{E}}} \mathbb{Z}\rightarrow 1,$$ where $H_{\mathcal{P}}$ and $H_{\mathcal{E}}$ are finitely generated. In this paper, we furtherly show that the groups $H_{\mathcal{P}}$ and $H_{\mathcal{E}}$ are not $FP_2$. In particular, those fiber-kernel groups are finitely generated, but not finitely presented.