Analytic torsion for fibred boundary metrics and conic degeneration

We study the renormalized analytic torsion of complete manifolds with fibred boundary metrics, also referred to as $\phi$-metrics. We establish invariance of the torsion under suitable deformations of the metric, and establish a gluing formula. As an application, we relate the analytic torsions for complete $\phi$- and incomplete wedge-metrics. As a simple consequence we recover a result by Sher and Guillarmou about analytic torsion under conic degeneration.