Laver Trees in the Generalized Baire Space

We prove that any suitable generalization of Laver forcing to the space $\kappa^\kappa$, for uncountable regular $\kappa$, necessarily adds a Cohen $\kappa$-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 $\kappa^{<\kappa}=\kappa$, then every $<\kappa$-distributive tree forcing on $\kappa^\kappa$ adding a dominating $\kappa$-real which is the image of the generic under a continuous function in the ground model, adds a Cohen $\kappa$-real. This is a contribution to the study of generalized Baire spaces and answers a question from arXiv:1611.08140