On representations of direct products and the bounded generation property of branch groups

Steffen Kionke; Eduard Schesler

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arXiv · 2301.01330

On representations of direct products and the bounded generation property of branch groups

Steffen Kionke·Eduard Schesler

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Abstract

We prove that the minimal representation dimension of a direct product $G$ of non-abelian groups $G_1,\ldots,G_n$ is bounded below by $n+1$ and thereby answer a question of Ab\'ert. If each $G_i$ is moreover non-solvable, then this lower bound can be improved to be $2n$. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups we show that branch groups cannot be boundedly generated.

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Steffen Kionke, Eduard Schesler. 2023-01-03. On representations of direct products and the bounded generation property of branch groups. https://arxiv.org/abs/2301.01330

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