Boundedly finite conjugacy classes of tensors

TL;DR

This paper investigates when bounded conjugacy classes of tensors in the non-abelian tensor square context imply strong finiteness properties. By analyzing the group $\nu(G)$ and its tensor subgroup $T_{\otimes}(G)$, the authors prove that if every tensor has at most $n$ conjugates in $\nu(G)$, then the second derived subgroup $\nu(G)''$ is finite with $n$-bounded order, extending DS-type results to tensor frameworks. They also derive a corollary giving a BFC-condition under the additional hypothesis that $G'$ is finitely generated, and provide two counterexamples demonstrating the sharpness of the hypotheses. The work highlights how bounded tensor conjugacy controls higher derived structure and contributes to understanding when a group is a BFC-group based on tensor data.

Abstract

Let $n$ be a positive integer and let $G$ be a group. We denote by $ν(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. Set $T_{\otimes}(G) = \{g \otimes h \mid g,h \in G\}$. We prove that if the size of the conjugacy class $\left |x^{ν(G)} \right| \leq n$ for every $x \in T_{\otimes}(G)$, then the second derived subgroup $ν(G)''$ is finite with $n$-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.

Theorems & Definitions (6)

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