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Growth in product replacement graphs of Grigorchuk groups. (English) Zbl 1366.20029

Summary: The product replacement graph \(\Gamma_{k}(G)\) is the graph on the generating \(k\)-tuples of a group \(G\), with edges corresponding to Nielsen moves. We prove the exponential growth of product replacement graphs \(\Gamma_{k}(\mathbb G_\omega)\) of Grigorchuk groups, for \(k \geq 5\).

MSC:

20F69 Asymptotic properties of groups
20F65 Geometric group theory
20E08 Groups acting on trees
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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