Q-tensor square of finitely generated nilpotent groups, q >= 0

TL;DR

This paper extends structural results of the non-abelian tensor square to the q-tensor square for finitely generated nilpotent groups, providing explicit computations and bounds for these groups and their tensor squares.

Contribution

It generalizes known results to all q ≥ 0, computes q-tensor squares for free nilpotent groups of class 2, and verifies bounds on generators for these groups.

Findings

Explicit formulas for q-tensor squares of free nilpotent groups of class 2.

Generalization of bounds on minimal generators to all q ≥ 0.

Verification that bounds are achieved for q > 1.

Abstract

The authors extend to the $q-$tensor square $G \otimes^q G$ of a group $G$, $q$ a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square $G \otimes G$ ($q = 0$). The results are applied to the computation of $G \otimes^q G$ for finitely generated nilpotent groups $G$, specially for free nilpotent groups of finite rank. They also generalize to all $q \geq 0$ results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square $G \otimes G$ when $G$ is a $n-$generator nilpotent group of class 2. The paper ends with the computation of the $q-$tensor squares of the free $n-$generator nilpotent group of class 2, $n \geq 2$, for all $q \geq 0.$ This shows that the above mentioned upper bound is also achieved for these groups when $q > 1.$