Is it permissible for a husband to abandon his wife and treat her like a "servant" and deprive her of intimacy as revenge for her betrayal, while continuing to provide her with only housing and sustenance, and is he sinful for not being just in spending the night with her because of her actions?

A woman may waive some of her rights to maintenance or cohabitation so that her husband does not divorce her, and in such a case, there is no sin upon the husband for not equally dividing his time between his wives. However, the wife has the right to retract her waiver and demand her rights at a later time.

We advise you, if your wife has repented and has not resumed the correspondences, to keep her and cohabit with her honorably, fulfill her rights, and forgive her, for repentance erases what came before it, and the one who repents from sin is like one who has no sin.

However, if her repentance is not apparent to you, then you should not keep her in that state, and you should separate from her through divorce or khul‘, especially since her remaining under your marital bond while abandoning cohabitation with her may be a temptation for her and an aid for Satan against her.