Representation of tree permutations by words

Abstract

The problem of solving equations in groups can be stated as follows: given a group G G and a free group F = F ( x 1 , x 2 , … ) F = F\left ( {{x_1},{x_2}, \ldots } \right ) , for which pairs ( w , g ) \left ( {w,g} \right ) with w = w ( x 1 , x 2 , … ) ∈ F , g ∈ G w = w\left ( {{x_1},{x_2}, \ldots } \right ) \in F,g \in G , is it possible to find elements g i ∈ G {g_i} \in G such that w ( g 1 , g 2 , … ) = g w\left ( {{g_1},{g_2}, \ldots } \right ) = g ? We investigate the corresponding question of solving equations in the group A ( Ω ) A\left ( \Omega \right ) of all automorphisms of a transitive tree Ω \Omega . If the tree has isomorphic cones at a branch point, then certain equations of the form x n = g {x^n} = g cannot be solved (Theorem 2.3). If the tree is sufficiently transitive, we find large classes of equations w = g w = g which can be solved (Theorems 2.13, 2.16).

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…Some such classes were discussed in the literature: infinite symmetric groups [86] (surjectivity of the commutator word was established by Ore [102]), groups of automorphisms of trees [92] and of random graphs [29].…”