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Infinitely presented simple groups separated by homological finiteness properties

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OGG - Operators, Graphs, Groups

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 $FP_{\infty}$ that is not finitely presented. Moreover we show that for every $n \in \mathbb{N}$ there is a simple group of type $FP_n$ that is neither finitely presented nor of type $FP_{n+1}$. To prove these results we construct a self-similar version of the Bestvina-Brady groups, which allows us to apply the so-called R\”over—Nekrashevich construction to them. In particular, we obtain the first examples of infinitely presented groups of type $FP_2$ that live in the world of Thompson-like groups. This is joint work with Claudio Llosa Isenrich and Xiaolei Wu.

This talk is part of the Isaac Newton Institute Seminar Series series.

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