بسم الله الرحمن الرحيم
In the above mentioned case three talaqs took place on the wife and their nikah came to an end completely. Now it is not lawful to marry her without a Shariah halalah, rather it is haram.
قال فى العالمكيرى وان كان الطلاق ثلاثا فى الحرة …..لم تحل له حتى تنكح زوجاً غيره نكاحاً صحيحاً ويدخل بها ام يطلقها او يموث عنها كذا فى الهداية (هندیه ج 1 ص 473)
Three talaqs occurs as three in any case whether given in single meeting or in different meetings three talaqs. The husband cannot marry her without a Shariah halalah. It is mentioned in the holy Quran and Hadith in clearly. The Sahabah have a consensus opinion about it as well the tabiyeen, tabe-tabiyeen, the four mujtahid Imams and all the Muhaddithin have the same opinion. The person who says that only one talaq took place he is a deviated person. You should study the book عمدة الاثاث فى الطلقات الثلاث or read Fatawa Rahimia, Ahsanul Fatawa in Urdu. You would find detailed and complete fatwa there. You may also find the detailed fatwa in the book named “Chand Aham Asari Masail”, a compilation of Fatawas issued by Darul Ifta and published by Maktaba Darul Uloom Deoband.
Allah knows Best!
Darul Uloom Deoband