The surname of "KyÅjurÅ 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.
"Flame breathing, nineth form! RENGOKU!"