Nonstandard Cayley Automatic Representations for Fundamental Groups of Torus Bundles over the Circle

We construct a new family of Cayley automatic representations of semidirect products [Formula: see text] for which none of the projections of the normal subgroup [Formula: see text] onto each of its cyclic components is finite automaton recognizable. For [Formula: see text] we describe a family of matrices from [Formula: see text] corresponding to these representations. We are motivated by a problem of characterization of all possible Cayley automatic representations of these groups.