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arXiv · 2510.01952

Infinitely presented simple groups separated by homological finiteness properties

Claudio Llosa Isenrich·Eduard Schesler·Xiaolei Wu
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Abstract

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $\Gamma$ 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 $\mathrm{FP}_{\infty}$ 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 R\"over--Nekrashevych groups, this answers a question of Zaremsky.

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Claudio Llosa Isenrich, Eduard Schesler, Xiaolei Wu. 2025-10-02. Infinitely presented simple groups separated by homological finiteness properties. https://arxiv.org/abs/2510.01952

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