Strong lensing as a giant telescope to localize the host galaxy of gravitational wave event

Standard siren cosmology of gravitational wave (GW) merger events relies on the identification of host galaxies and their redshifts. But this can be highly challenging due to numerous candidates of galaxies in the GW localization area. We point out that the number of candidates can be reduced by orders of magnitude for strongly lensed GW events, due to extra observational constraints. For the next-generation GW detectors like Einstein Telescope (ET), we estimate that this number is usually significantly less than one, as long as the GW localization uncertainty is better than similar to 10 deg(2). This implies that the unique identification of the host galaxy of lensed GW event detected by ET and Cosmic Explorer (CE) is possible. This provides us a promising opportunity to measure the redshift of the GW event and facilitate the standard siren cosmology. We also discuss its potential applications in understanding the evolution process and environment of the GW event.