The surname of "Kyōjurō Rengoku", the hero who failed, the
Flame Hashira.
The 9th form of
Flame breathing is not to be mistaken with "fire" or "
sun" breathing, the strongest and most dangerous technique for both the user and enemies due to leaving the user of this 9th form completely open after the strike, and the enemy due to it having all the force of the user.
The name "Rengoku" for the 9th form is the
Japanese name for the 9th form, while the English name being called purgatory, ironically or coincidentally the early translations for the manga also used the name "purgatory" for the last and strongest move, the 9th form.