Schwarzschild black hole surrounded by a cloud of strings in the background of perfect fluid dark matter*

Abstract This manuscript investigates a Schwarzschild black hole surrounded by perfect fluid dark matter embedded in a cloud of strings. The effects of its surroundings on thermodynamics, timelike and null geodesics, shadows, and quasinormal modes are analyzed. It is demonstrated that changes in spacetime, induced by the surrounding environment, significantly influence the stability, thermal phases, energy dynamics, particle trajectories, and observable features of the black hole's shadow, as well as its oscillation frequency and decay rate.