  Supermassive black holes at the center of each galaxy may be surrounded by
dark matter. Such dark matter admits a spike structure and vanishes at a
certain distance from the black hole. This dark matter will impact the
spacetime near the black hole and the related ringing gravitational waves can
show distinguished features of the black hole without dark matter. In the
present work, we focus on the quasi-normal modes of the axial gravitational
perturbation who dominate the ringdown process of the perturbed black holes
surrounded by dark matter spikes. The relativistic modification results in less
impact on QNMs. And the relative ringing frequency difference between the black
holes with and without dark matter can be as large as $10^{-2}$. These features
can be used in future gravitational wave detection about extremal mass ratio
inspiral systems to probe the existence of dark matter around supermassive
black holes.
