Linearly Mismatched Free-by-Cyclic Groups are Asynchronously Automatic

We call the family of free-by-cyclic groups defined by $G = \left< a, t, b_1, b_2, \ldots b_k \mid at = ta, b_1^{-1}tb_1 = a^{n_1}t, \ldots b_k^{-1}tb_k = a^{n_k}t \right>$ for $n_1, n_2, \ldots n_k \in \mathbb Z$ linearly mismatched since the automorphisms used to define the HNN extensions grow linearly at different rates. Using techniques from Elder's thesis, namely words with a parallel stable letter structure, we prove that linearly mismatched free-by-cyclic groups are asynchronously automatic, and thus they have a solvable word problem.