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We realize lamplighter groups , with a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize as a bireversible automaton group if and only if the 2-Sylow subgroup of has no multiplicity one summands in its expression as a direct sum of cyclic groups of order a power of 2.
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Rachel Skipper, Benjamin Steinberg, Lamplighter groups, bireversible automata, and rational series over finite rings. Groups Geom. Dyn. 14 (2020), no. 2, pp. 567–589DOI 10.4171/GGD/555