We construct a new family of Cayley automatic representations of semidirect products $\mathbb{ Z}^n \rtimes_A \mathbb{ Z}$ for which none of the projections of the normal subgroup $\mathbb{ Z}^n$ onto each of its cyclic components is finite automaton recognizable. For $n=2$ we describe a family of matrices from $\mathrm{ GL}(2,\mathbb{ Z})$ corresponding to these representations. We are motivated by a problem of characterization of all possible Cayley automatic representations of these groups.
Categories: math.GR cs.FL