What is cause of the lack of hair in people with progeria?

Lamin A/C mutation. Different mutations within the lamin a/c gene cause different genetic disorders. Lamin a/c is needed to provide stability to the cell. With the particular mutation found in progeria, skin and hair (along with other systems) get damaged. Though i'm not sure, i suspect that rapid cell division makes skin and hair more likely to get damaged sooner from this particular lamin a/c mutation.