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The Grigorchuk group is finitely generated and has subexponential non-polynomial growth but I'm not aware of a finite presentation. Does a finite presentation imply that the group is polynomial or exponential as well?

saolof
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  • Grigorchuk himself credits I. Lysenok with a presentation of his group, but one of the relations is parametrized by an integer k so the presentation is not finite. He also mentions that the Grigorchuk group is not finitely presentable. – saolof Oct 06 '20 at 04:25

1 Answers1

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There are no known examples of such groups. Grigorchuk group is infinitely presented and so are all other known infinite finitely generated groups of intermediate growth (there are many examples: Gupta-Sidki, Erschler, and others).

markvs
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