Infinitely presented simple groups separated by homological finiteness properties

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $\mathrm{\Gamma }$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type ${\mathrm{F}\mathrm{P}}_{\mathrm{\infty }}$ that is not finitely presented. Moreover we show that for every $n\in \mathbb{N}$ there is a simple group of type ${\mathrm{F}\mathrm{P}}_n$ that is neither finitely presented nor of type ${\mathrm{F}\mathrm{P}}_{n+1}$. Since our simple groups arise as Röver–Nekrashevych groups, this answers a question of Zaremsky.