Infinitely presented simple groups separated by homological finiteness properties

Abstract

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $\hat{I}“$ 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 ${\mathrm{FP}}_{\infty }$ that is not finitely presented. Moreover we show that for every $n\in \mathbb{N}$ there is a simple group of type ${\mathrm{FP}}_n$ that is neither finitely presented nor of type ${\mathrm{FP}}_{n+1}$. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky.

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