May a lawyer compel a client to arbitrate a dispute over the lawyer's services?

In upper right-hand corner of the United States, yes. Bezio v. Draeger, No. 13-1910 (1st Cir. Dec. 16, 2013) (applying Maine law).

In the upper left-hand corner, no. Smith v. JEM Group, Inc., No. 11-35964 (9th Cir. Dec. 12, 2013) (applying Washington law).

Bonus:    

May lawyers in a class action case waive their right to federal court review of an arbitration award relating to a fee dispute under section 10 of the federal Arbitration Act? No. Burton v. Class Counsel, No. 11-17718 (9th Cir. Dec. 17, 2013).