PSL(2, C) is the projective special linear group of 2x2 complex matrices, modulo scalar matrices, and is a fundamental object in complex analysis and geometry, often appearing as the automorphism group of the Riemann sphere. It plays a crucial role in the theory of Möbius transformations, hyperbolic geometry, and isomorphic to the Lorentz group SO(3,1), making it essential in both mathematics and theoretical physics.