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Laver Trees in the Generalized Baire Space

We prove that any suitable generalization of Laver forcing to the space $ κ^κ$, for uncountable regular $κ$, necessarily adds a Cohen $κ$-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if $ κ^{<κ}=κ$, then every $<κ$-distributive tree forcing on $κ^κ$ adding a dominating $κ$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $κ$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08140