The holonomy of the Obata connection on Joyce hypercomplex manifolds

We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except $\mathrm{S}\mathrm{U}(2n+1)$, we show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $\mathrm{S}\mathrm{U}(2n+1)$ is more subtle: for every $n>1$, we show that there exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in $\mathrm{G}\mathrm{L}(n(n+1),\mathbb{H})$. On the other hand, Soldatenkov showed that $\mathrm{S}\mathrm{U}(3)$ has Obata holonomy equal to $\mathrm{G}\mathrm{L}(2,\mathbb{H})$ \cite{Sol}, and we present here a new example on $\mathrm{S}\mathrm{U}(5)$ with holonomy equal to $\mathrm{G}\mathrm{L}(6,\mathbb{H})$. Finally, we investigate Joyce hypercomplex manifolds whose restricted holonomy lie in $\mathrm{S}\mathrm{L}(n,\mathbb{H})$, yielding new compact examples of twisted Calabi-Yau manifolds.