Overlap-Dirac fermions with a small hopping parameter

We consider overlap-Dirac fermions at non-zero bare coupling and for a smal l hopping parameter, or, equivalently, large \M\ with M the domain-wall hei ght. We prove the existence of a phase at large positive M where the abelia n axial group U-A(1) is a symmetry, and the corresponding pseudo-scalar is an exact Goldstone boson. We also provide a conjecture for the phase diagra m of asymptotically free gauge theories with overlap-Dirac fermions. In par ticular, we suggest that, for large gauge coupling, the massive-fermion pha se at negative M possibly extends to all M < 4.