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Representation of tree permutations by words

Author: John A. Maroli
Journal: Proc. Amer. Math. Soc. 110 (1990), 859-869
MSC: Primary 06A06; Secondary 06F15, 20B27
MathSciNet review: 1037214
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Abstract: 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).

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