Torsion-free $H$-structures on almost Abelian solvmanifolds

In this article, we provide a general set-up for arbitrary linear Lie groups $H\le \mathrm{G}\mathrm{L}(n,\mathbb{R})$ which allows to characterise the almost Abelian Lie algebras admitting a torsion-free $H$-structure. In more concrete terms, using that an $n$-dimensional almost Abelian Lie algebra $\mathfrak{g}={\mathfrak{g}}_f$ is fully determined by an endomorphism $f$ of ${\mathbb{R}}^{n-1}$, we give a description of the subspace ${\mathcal{F}}_{\mathfrak{h}}$ of all $f\in \mathrm{E}\mathrm{n}\mathrm{d}({\mathbb{R}}^{n-1})$ for which ${\mathfrak{g}}_f$ admits a ``special’’ torsion-free $H$-structure in terms of the image of a certain linear map. For large classes of linear Lie groups $H$, we are able to explicitly compute ${\mathcal{F}}_{\mathfrak{h}}$ and so give characterisations of the almost Abelian Lie algebras admitting a torsion-free $H$-structure. Our results reprove all the known characterisations of the almost Abelian Lie algebras admitting a torsion-free $H$-structure for different single linear Lie groups $H$ and extends them to big classes of linear Lie groups $H$. For example, we are able to provide characterisations in the case $n=2m$, $H\le \mathrm{G}\mathrm{L}(m,\mathbb{C})$ and $H$ either being a complex Lie group or being totally real, or in the case that $H$ preserves a pseudo-Riemannian metric. In many cases, we show that the space ${\mathcal{F}}_{\mathfrak{h}}$ coincides with what we call the \emph{characteristic subalgebra} ${\tilde{\mathfrak{k}}}_{\mathfrak{h}}$ associated to $\mathfrak{h}$, and that then the torsion-free condition is equivalent to the left-invariant flatness condition. In particular, we prove this to be the case if $H$ is a complex linear Lie group or if $\mathfrak{h}$ does not contain any elements of rank one or two and is either metric or totally real.