To Infinity and Beyond

TL;DR

This paper proves that groups generated by bireversible Mealy automata with an element of infinite order exhibit exponential growth, ruling out the possibility of such automata generating infinite virtually nilpotent groups.

Contribution

It establishes a link between infinite order elements in these groups and exponential growth, providing new constraints on the types of groups generated by bireversible Mealy automata.

Findings

Groups with infinite order elements have exponential growth

No infinite virtually nilpotent group can be generated by a bireversible Mealy automaton

Growth behavior is directly linked to the automaton's properties

Abstract

We prove that if a group generated by a bireversible Mealy automaton contains an element of infinite order, its growth blows up and is necessarily exponential. As a direct consequence, no infinite virtually nilpotent group can be generated by a bireversible Mealy automaton.