A non-commutative Reidemeister-Turaev torsion of homology cylinders

We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the $K_1$-group of the $I$-adic completion of the group ring $\mathbb{Q}{\pi }_1{\mathrm{\Sigma }}_{g,1}$, and prove that its reduction to $\widehat{\mathbb{Q}{\pi }_1{\mathrm{\Sigma }}_{g,1}}\mathrm{/}{\widehat{I}}^{d+1}$ is a finite-type invariant of degree $d$. We also show that the $1$-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion.