Following the topic of the book Canonical Ramsey Theory on Polish Spaces by V. Kanovei, M. Sabok, and J. Zapletal we study Borel equivalences on Laver trees. We prove that equivalence relations Borel reducible to an equivalence relation on 2ω given by some FσP-ideal on ω can be canonized to the full equivalence relation or to the identity relation.
This has several consequences, e.g., Silver type dichotomy for the Laver ideal and equivalences Borel reducible to equivalence relations given by FσP-ideals.
