More discrete copies of in

In a previous paper we established that if q is any minimal idempotent in β N , then for all except possibly one p ∈ c ℓ { 2 n : n ∈ N } ∖ N , 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 p ∈ c ℓ { r n : n ∈ N } ∖ N 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 ∈ β N , if p ∈ c ℓ ( n N ) 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 ∗ . not know whether there exist any minimal idempotent q and any p with p ∈ c ℓ ( n N ) 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.