An algebraic approach to communication complexity

Let M be a finite monoid: define C-(k)(M) to be the maximum number of bits that need to be exchanged in the k-party communication game to decide membership in any language recognized by M. We prove the following: a) If M is a group then, for any k, C-(k)(M) = O(1) if M is nilpotent of class k - 1 and C-(k)(M) = -(n) otherwise. b) If M is aperiodic, then C-(2)(M) = O(1) if M is commutative, C-(2)(M) = O(log n) if M belongs to the variety DA but is not commutative and C-(2)(M) = -(n) otherwise. We also show that when M is in DA, C-(k)(M) = O(1) for some k and conjecture that this algebraic condition is also necessary.