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BEING CAYLEY AUTOMATIC IS CLOSED UNDER TAKING WREATH PRODUCT WITH VIRTUALLY CYCLIC GROUPS

Part of: Theory of computing Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 April 2021

DMITRY BERDINSKY Open the ORCID record for DMITRY BERDINSKY [Opens in a new window] ,
MURRAY ELDER Open the ORCID record for MURRAY ELDER [Opens in a new window]  and
JENNIFER TABACK Open the ORCID record for JENNIFER TABACK [Opens in a new window]
DMITRY BERDINSKY
Affiliation:
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok10400, Thailand and Centre of Excellence in Mathematics, Commission on Higher Education, Bangkok 10400, Thailand e-mail: berdinsky@gmail.com
MURRAY ELDER*
Affiliation:
School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, NSW2007, Australia
JENNIFER TABACK
Affiliation:
Department of Mathematics, Bowdoin College, 8600 College Station, Brunswick, ME04011, USA e-mail: jtaback@bowdoin.edu
*
e-mail: murray.elder@uts.edu.au
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Abstract

We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

Keywords

Cayley automatic grouprestricted wreath productvirtually infinite cyclic grouplamplighter group

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20E22: Extensions, wreath products, and other compositions 68Q45: Formal languages and automata
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 3 , December 2021 , pp. 464 - 474
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author was supported by Australian Research Council grant DP160100486. The third author acknowledges support from Simons Foundation grant 31736 to Bowdoin College.

References

Berdinsky, D. and Khoussainov, B., ‘On automatic transitive graphs’, in: Developments in Language Theory, Lecture Notes in Computer Science, 8633 (Springer, Cham, 2014), 1–12.Google Scholar
Berdinsky, D. and Khoussainov, B., ‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science 27(2) (2016), 147–159.10.1142/S0129054116400049CrossRefGoogle Scholar
Bérubé, S., Palnitkar, T. and Taback, J.. ‘Higher rank lamplighter groups are graph automatic’, J. Algebra 496 (2018), 315–343.CrossRefGoogle Scholar
Cleary, S. and Taback, J.. ‘Dead end words in lamplighter groups and other wreath products’, Quarterly Journal of Mathematics 56(2) (2005), 165–178.10.1093/qmath/hah030CrossRefGoogle Scholar
Elder, M. and Taback, J., ‘ ${\mathcal C}$ -graph automatic groups’, J. Algebra 413 (2014), 289–319.CrossRefGoogle Scholar
Epstein, D. B. A., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P., Word Processing in Groups (Jones and Barlett Publishers, Boston, MA, 1992).CrossRefGoogle Scholar
Kharlampovich, O., Khoussainov, B. and Miasnikov, A., ‘From automatic structures to automatic groups’, Groups Geom. Dyn. 8(1) (2014), 157–198.CrossRefGoogle Scholar