Linearly Mismatched Free-by-Cyclic Groups are Asynchronously Automatic

We call the family of free-by-cyclic groups defined by $G=<a,t,b_1,b_2,\dots b_k\mid at=ta,b_1^{-1}t{b}_1=a^{n_1}t,\dots b_k^{-1}t{b}_k=a^{n_k}t>$ for $n_1,n_2,\dots n_k\in \mathbb{Z}$ 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.