A note on finiteness conditions for the non-abelian tensor square of groups

TL;DR

This paper investigates conditions under which the non-abelian tensor square and related groups are finite or locally finite, focusing on finitely generated and locally residually finite groups with finite simple tensors.

Contribution

It establishes finiteness and local finiteness conditions for the non-abelian tensor square and the group bla(G) based on the finiteness of simple tensors in specific group classes.

Findings

If G is finitely generated with finitely many simple tensors, then G igotimes G and bla(G) are finite.

For locally residually finite G with finite simple tensors in all finitely generated subgroups, bla(G) is locally finite.

Abstract

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is a finitely generated group in which the set of all simple tensors $T_{\otimes}(G)$ is finite, then the non-abelian tensor square $G \otimes G$ and the group $\nu(G)$ are finite. Moreover, we show that if $G$ is a locally residually finite group in which the set of simple tensors $T_{\otimes}(H)$ is finite for every proper finitely generated subgroup $H$ of $G$, then the group $\nu(G)$ is locally finite.