Generalized Baumslag–Solitar Groups Universally Equivalent to Tree Groups

A finitely generated group that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (GBS group). Every GBS group is the fundamental group π₁(𝔸) of a suitable labeled graph 𝔸. If 𝔸 is a labeled tree, then we call π₁(𝔸) a tree GBS group. It is known that GBS groups isomorphic to tree groups are themselves tree groups. It is shown that GBS groups universally equivalent to tree groups do not have to be tree groups. Moreover, we give a description of all GBS groups that are universally equivalent to tree groups.