Nonstandard Cayley automatic representations of fundamental groups

Abstract

We construct a new family of Cayley automatic representations of semidirect products ${\mathbb{Z}}^n{⋊}_A\mathbb{Z}$ 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.

14 pages, a paper for Language and Automata Theory and Applications 2020 (extended version)

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