Does a triple divorce occur if a wife speaks to her sisters after her husband swore "by triple divorce, if you speak to them," knowing that the conversation took place with the husband's permission and his intention was that the oath was temporary for her commitment, and is expiation incumbent upon him?

The oath is based on the intention of the swearer. If you intended to prevent your wife from speaking to her sisters because they were the cause of conflict, then there is no breaking of the oath if she speaks to her mother. If your intention was for the prohibition to be temporary until your wife complies with your demands and harmony returns between her and her sisters, and the conflict between you ceases, then if she does not speak to them until after her compliance and reconciliation, there is no breaking of the oath. However, if she spoke to them before that, then according to the majority of scholars, a triple divorce has occurred, and she becomes unlawful to you until she marries another husband. Shaykh al-Islam Ibn Taymiyyah held the view that an expiation for an oath is due if divorce was not intended, and one divorce if it was intended. The preponderant view is that of the majority of scholars.