Back in 1983, Gupta and Said (Math Z.) were constructing finitely generated infinite p-groups as groups of automorphisms of p-regular trees. There is a later treatment of their work by Gilbert Baumslag, but I don't have an immediate reference for his exposition. I think that Baumslag mostly looks at the special case where p=3. This work might be irrelevant to Dominik, but it might be of interest.

All trees are planar.

Back in 1983, N. Gupta and S. Sidki (Math Z.) were constructing finitely generated infinite p-groups as groups of automorphisms of p-regular trees. There is a later treatment of their work by Gilbert Baumslag, but I don't have an immediate reference for his exposition. I think that Baumslag mostly looks at the special case where p=3. This work might be irrelevant to Dominik, but it might be of interest.

All trees are planar.