can they still write and send mail if they are in the hole . and do they receive mail in the hole

Yes. Inmates in solitary confinement for disciplinary or administrative reasons are allowed to have mail like the rest of the institution. Mail is considered sacred by the prison administration, they do everything in their power to ensure the inmates remain in contact with their families and loved ones. There is a caveat, if the inmate is on a terror list or has committed a terror act, the courts have restricted outside contact in some cases (see Boston bomber convicted terrorist).