Elasticities of Krull monoids with infinite cyclic class group

Let H be a Krull monoid with infinite cyclic class group G that we identify with Z. Let GP ⊆ G denote the set of classes containing prime divisors. It was shown that ρ(H), the elasticity of H, is finite if and only if GP is bounded above or below. By a result of Lambert, it is easy to show that ρ(H) ≤ min{− inf(GP ), sup(GP )}. In this paper, we focus on H with large elasticity. When GP is infinite but bounded above or below, we give necessary and sufficient conditions for that ρ(H) > min{− inf(GP ), sup(GP )} − 3/2. When GP is a finite set, we give a better upper bound on ρ(H), which is sharp in some sense.