The exponent of the non-abelian tensor square and related constructions of $p$-groups

Let $G$ be a finite $p$-group. In this paper we obtain bounds for the exponent of the non-abelian tensor square $G \otimes G$ and of $\nu(G)$, which is a certain extension of $G \otimes G$ by $G \times G$. In particular, we bound $\exp(\nu(G))$ in terms of $\exp(\nu(G/N))$ and $\exp(N)$ when $G$ admits some specific normal subgroup $N$. We also establish bounds for $\exp(G \otimes G)$ in terms of $\exp(G)$ and either the nilpotency class or the coclass of the group $G$, improving some existing bounds.