[*quasi-symmetry*]{} and its quasi-symmetry class is a complete geometric invariant of the group, i.e., word-hyperbolic groups are quasi-isometric if and only their visual boundaries are quasi-symmetric [@Pau96]. Although a reducible and an irreducible automorphism can have word-hyperbolic mapping tori with homeomorphic boundaries, the conjecture asserts that the visual boundaries cannot be quasi-symmetric!

 {#app}

The following is a folk theorem that was used to construct examples of [*fully irreducible*]{} automorphisms, i.e., [*irreducible with irreducible powers (iwip)*]{}, along with examples that are infinite-order irreducible but not fully irreducible [@BH92 Example 1.4]. At the end of the appendix, we will show that the latter examples are complete/exhaustive (Corollary \[irredIWIP\]).

\[eg\] If $\phi:F \to F$ is an automorphism induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 1}$ that acts transitively on the boundary components, then $\phi$ is irreducible and it is fully irreducible if and only if $b = 1$.

Let $f : \Sigma \to \Sigma$ be the inducing pseudo-Anosov homeomorphism. Suppose $\phi$ was reducible, i.e., there exists a $\phi$-invariant proper free factor system $A_1 * \cdots * A_k$ of $F$. Thus $i_j \phi^k(A_j) \le A_j$ for some inner automorphisms $i_j$. Let $\hat \Sigma\to \Sigma$ be the cover corresponding to $A_j \le 