In the present article, we study compact complex manifolds admitting a Hermitian metric which is strong Kähler with torsion (SKT) and Calabi–Yau with torsion (CYT) and whose Bismut torsion is parallel. We first obtain a characterization of the universal cover of such manifolds as a product of a Kähler Ricci-flat manifold with a Bismut flat one. Then, using a mapping torus construction, we provide non-Bismut flat examples. The existence of generalized Kähler structures is also investigated.

The algebraic mapping torus $M_{\Phi }$ of a group $G$ with an automorphism $\Phi$ is the HNN-extension of $G$ in which conjugation by the stable letter performs $\Phi$. We classify the Dehn functions of $M_{\Phi }$ in terms of $\Phi$ for a number of right-angled Artin groups (RAAGs) $G$, including all $3$-generator RAAGs and $F_k \times F_l$ for all $k,l \geq 2$.

Given a finitely generated free group $ {\mathbb {F} }$ of $\mathsf {rank}( {\mathbb {F} } )\geq 3$, we show that the mapping torus of $\phi$ is (strongly) relatively hyperbolic if $\phi$ is exponentially growing. As a corollary of our work, we give a new proof of Brinkmann's theorem which proves that the mapping torus of an atoroidal outer automorphism is hyperbolic. We also give a new proof of the Bridson–Groves theorem that the mapping torus of a free group automorphism satisfies the quadratic isoperimetric inequality. Our work also solves a problem posed by Minasyan and Osin: the mapping torus of an outer automorphism is not virtually acylindrically hyperbolic if and only if $\phi$ has finite order.