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\Sigma _{g,1}, and prove that its reduction to \widehat{\mathbb{Q}\pi _1\Sigma _{g,1}}/\hat{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.