Representation of Tree Permutations by Words
John A. Maroli · Proceedings of the American Mathematical Society · 1990
The problem of solving equations in groups can be stated as follows: given a group $G$ and a free group $F = F\left ( {{x_1},{x_2}, \ldots } \right )$, for which pairs $\left ( {w,g} \right )$ with $w = w\left ( {{x_1},{x_2}, \ldots } \right ) \in F,g \in G$, is it possible to find elements ${g_i} \in G$ such that $w\left ( {{g_1},{g_2}, \ldots } \right ) = g$? We investigate the corresponding question of solving equations in the group $A\left ( \Omega \right )$ of all automorphisms of a transitive tree $\Omega$. If the tree has isomorphic cones at a branch point, then certain equations of the form ${x^n} = g$ cannot be solved (Theorem 2.3). If the tree is sufficiently transitive, we find large classes of equations $w = g$ which can be solved (Theorems 2.13, 2.16).