Some Results on Non-Club Isomorphic Aronszajn Trees

Abstract

Let λ be a regular cardinal satisfying λ<λ = λ and ♦(Sλ + λ ). Then there exists a family of 2λ + many completely club rigid special λ+-Aronszajn trees which are pairwise far. In this article we will be concerned with building Aronszajn trees which are not club isomorphic and have strong rigidity properties. This topic goes back to Gaifman-Specker [4], who proved that if λ is a regular cardinal satisfying λ = λ, then there exists a family of 2 + many normal λ-complete λ-Aronszajn trees which are pairwise non-isomorphic. Abraham [1] and Todorcevic [7] constructed in ZFC ω1-Aronszajn trees which are rigid, that is, have no automorphisms other than the identity. Later the focus shifted from isomorphisms between trees to club isomorphisms. Abraham-Shelah [2] proved that under PFA, any two normal ω1Aronszajn trees are club isomorphic. Krueger [6] provided a generalization of this result to higher cardinals. Abraham-Shelah [2] also showed that the weak diamond principle on ω1 implies the existence of a family of 2 ω1 many normal club rigid ω1-Aronszajn trees which are pairwise not club embeddable into each other. Building off of this work, we will use the diamond principle to construct a family of pairwise non-club isomorphic Aronszajn trees. Specifically, assume that λ is a regular cardinal satisfying λ = λ and the diamond principle ♦(S + λ ) holds, where S + λ := {α < λ : cf(α) = λ}. Then there exists a family {Tα : α < 2 +} of normal λ-complete special λ-Aronszajn trees such that for each α < 2 + , the only club embedding from a downwards closed normal subtree of Tα into Tα is the identity, and for all α < β < 2 + , Tα and Tβ do not contain club isomorphic downwards closed normal subtrees. We also discuss some related results, such as obtaining a large family of Suslin trees with similar properties and generalizing the Abraham-Shelah result on weak diamond to higher cardinals.