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Application of the most robust method of measuring black hole masses,spatially resolved kinematics of gas and stars, is presently limited to nearbygalaxies. The Atacama Large Millimeter/sub-millimeter Array (ALMA) and thirtymeter class telescopes (the Thirty Meter Telescope, the Giant MagellanTelescope, and the European Extremely Large Telescope) with milli-arcsecondresolution are expected to extend such measurements to larger distances. Here,we study the possibility of exploiting the angular magnification provided bystrong gravitational lensing to measure black hole masses at high redshifts (z~1-6), using resolved gas kinematics with these instruments. We show that in~15% and ~20% of strongly lensed galaxies, the inner 25 and 50 pc could beresolved, allowing the mass of ~$10^8 M_{\odot}$ black holes to be dynamicallymeasured with ALMA, if moderately bright molecular gas is present at thesesmall radii. Given the large number of strong lenses discovered in currentmillimeter surveys and future optical surveys, this fraction could constitute astatistically significant population for studying the evolution of theM-$\sigma$ relation at high redshifts.

Prospects for Measuring the Mass of Black Holes at High Redshifts with Resolved Kinematics Using Gravitational Lensing