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Lamplighter groups and automata

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Abstract

This paper investigates presentations of lamplighter groups using computational models from automata theory. The present work shows that if G can be presented such that the full group operation is recognised by a transducer, then the same is true for the lamplighter group \(G \wr {{\mathbb {Z}}}\) of G. Furthermore, Cayley presentations, where only multiplications with constants are recognised by transducers, are used to study generalised lamplighter groups of the form \(G \wr {{\mathbb {Z}}}^d\) and \(G \wr F_d\), where \(F_d\) is the free group over d generators. Additionally, \({{\mathbb {Z}}}_k \wr {{\mathbb {Z}}}^2\) and \({{\mathbb {Z}}}_k \wr {F_d}\) are shown to be Cayley tree automatic.

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Acknowledgements

Sanjay Jain and Frank Stephan have been supported in part by the Singapore Ministry of Education Academic Research Fund Tier 2 grants MOE2016-T2-1-019/R146-000-234-112 and MOE2019-T2-2-121/R146-000-304-112. In addition Sanjay Jain was also supported in part by NUS grant C252-000-087-001 and E252-00-0021-01. The paper was developed during the final year project of Tien Dat Tran [56] and Birzhan Moldagaliyev’s PhD thesis [47]. The authors would like to thank Dmitry Berdinsky and Bakhadyr Khoussainov for correspondence.

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Authors and Affiliations

  1. School of Computing, National University of Singapore, Singapore, 117417, Republic of Singapore

    Sanjay Jain, Frank Stephan & Tien Dat Tran

  2. Suleyman Demirel University, 1/1 Abylai Khan Street, 040900, Kaskelen, Almaty, Kazakhstan

    Birzhan Moldagaliyev

  3. Department of Mathematics, National University of Singapore, Singapore, 119076, Republic of Singapore

    Frank Stephan

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  1. Sanjay Jain
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  2. Birzhan Moldagaliyev
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  4. Tien Dat Tran
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Correspondence to Frank Stephan.

Additional information

This paper is dedicated to Klaus-Jörn Lange to honour his seventieth birthday; Klaus-Jörn Lange is a coauthor of Frank Stephan. Klaus-Jörn Lange’s work spans over four decades and centres at the notion of formal languages. He started off with investigating ET0L systems [40] and he made significant contributions to formal languages and computational complexity [30, 41, 43, 44]. Among these four, the work [43] stands out by showing that reversible space classes coincide with the corresponding deterministic space classes of the same bound. The connections between formal languages and computational complexity are also manifested in leaf languages [17] and this is where our ways crossed for the first time. His working area also touches group and semigroup theory [39, 42] which is the topic of the present work. Klaus-Jörn Lange has dedicated most of his life to science and we wish him health and strength to be further able to follow his passion in the golden years. Frank Stephan also still remembers the visits to Tübingen, in particular in the time when the common coauthor Bernd Borchert was working there with Klaus-Jörn Lange.

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Jain, S., Moldagaliyev, B., Stephan, F. et al. Lamplighter groups and automata. Acta Informatica 59, 451–478 (2022). https://doi.org/10.1007/s00236-022-00423-3

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