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

TL;DR

This paper establishes lower bounds on the minimal representation dimension of direct products of non-abelian groups and demonstrates that branch groups cannot be boundedly generated, advancing understanding of their algebraic structure.

Contribution

It proves new lower bounds for representation dimensions of direct products and shows that branch groups lack bounded generation, answering longstanding questions.

Findings

Minimal representation dimension of direct product ≥ n+1

If groups are non-solvable, lower bound improves to 2n

Branch groups are not boundedly generated

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.