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On stallings' unique factorisation groups

Published online by Cambridge University Press:  17 April 2009

Donald I. Cartwright  and
Bernhard Krön
Donald I. Cartwright
Affiliation:
School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia, e-mail: donaldc@maths.usyd.edu.au, bernhard@maths.usyd.edu.au
Bernhard Krön
Affiliation:
School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia, e-mail: donaldc@maths.usyd.edu.au, bernhard@maths.usyd.edu.au
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Let Γ be a group and Σ a symmetric generating set for Γ. In 1966, Stallings called Γ a unique factorisation group if each group element may be written in a unique way as a product a1…am, where ai ∈ Σ for each i and aiai+1 ∉ Σ ∪ {1} for each i < m. In this paper we give a complete combinatorial proof of a theorem, not explicitly stated by Stallings in 1966, characterising all such pairs (Γ, Σ). We also characterise the unique factorisation pairs by a certain tree-like property of their Cayley graphs.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 1 , February 2006 , pp. 27 - 36
Copyright
Copyright © Australian Mathematical Society 2006

References

Referenes

[1]Avenhaus, J., Madlener, K. and Otto, F., ‘Groups presented by finite two-monadic Church-Rosser Thue systems’, Trans. Amer. Math. Soc. 297 (1986), 427–443.CrossRefGoogle Scholar
[2]D'Agostino, G., ‘Cayley graphs of virtually free groups’, Internat. J. Algebra Comput. 3 (1993), 189–199.CrossRefGoogle Scholar
[3]Haring-Smith, R.H., ‘Groups and simple languages’, Trans. Amer. Math. Soc. 279 (1983), 337–356.CrossRefGoogle Scholar
[4]Mairesse, J., ‘Random walks on groups and monoids with a Markovian harmonic measure’, (preprint, 2004) Available at http://www.liafa.jussieu.fr/~mairesse/Article/ and http://www.liafa.jussieu.fr/web9/rapportrech/description_en.php?idrapportrech=666.Google Scholar
[5]Mairesse, J. and Mathéus, F., ‘Random walks on groups with a tree-like Cayley graph’, in Mathematics and computer science. III. Algorithms, trees, combinatorics and probabilities pages 445–460, Trends Math. (Birkhäuser Verlag, Basel, 2004).Google Scholar
[6]Shapiro, M., ‘Pascal's triangles in Abelian and hyperbolic groups’, J. Austral. Math. Soc. Ser. A 63 (1997), 281–288.CrossRefGoogle Scholar
[7]Stallings, J., ‘A remark about the description of free products of groups’, Proc. Cambridge Philos. Soc. 62 (1966), 129–134.CrossRefGoogle Scholar