"what evidence exists that some of the asteroids were geologically active in the distant past"

The geological activity of a celestial body like a planet, satellite or asteroid can be determined by looking at the surface features like the presence of a valley like terrain or plains or basins created by the lava flow. The presence of heavy elements like uranium, lead, iron and gold on the surface also indicate that the body once spewed out lava since these are found within the internal core of the body. Ceres, an asteroid has these elements present on its surface which hints at the presence of an active geological process in the past.