The Lamplighter Group ℤ₃≀ℤ Generated by a Bireversible Automaton

Abstract

We construct a bireversible self-dual automaton with three states over an alphabet with three letters which generates the lamplighter group ℤ₃≀ℤ. In particular, this fact shows that not all groups defined by birevirsible automata are finitely presented.

Key Words:

• Automaton group
• Bireversible automaton
• Lamplighter group

2010 Mathematics Subject Classification:

• 20F65
• 20M35

ACKNOWLEDGMENT

This work was initiated while the first author was visiting Graz University of Technology, whose support and hospitality are gratefully acknowledged. The authors would also like to thank the developers of the program package AutomGrp [11] which has been used to perform many of the computations described in this paper.