More discrete copies of Z in βN

In a previous paper we established that if q is any minimal idempotent in βN, then for all except possibly one , q+p+q generates an infinite discrete group. Responding to a question of Wis Comfort, we extend this result in two directions. We show on the one hand that for a minimal idempotent q, there is at most one prime r for which there exists such that the group generated by q+p+q is not both infinite and discrete. On the other hand, we show that for any p∈βN, if p∈cℓ(nN) for infinitely many n∈N, then there is some minimal idempotent q such that the group generated by q+p+q is infinite and discrete. We also show that if G is a countable discrete group and if p is a right cancelable element of G∗, then there is an idempotent q∈G∗ such that q⋅p⋅q generates a discrete copy of Z in G∗.We do not know whether there exist any minimal idempotent q and any p with p∈cℓ(nN) for infinitely many n∈N such that the group generated by q+p+q is not discrete. We show that if such a “bad” q exists, then there are many of them.