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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The conjugacy problem for HNN extensions with infinite cyclic associated groups


Authors: K. J. Horadam and G. E. Farr
Journal: Proc. Amer. Math. Soc. 120 (1994), 1009-1015
MSC: Primary 20F10; Secondary 20E06, 20M18
MathSciNet review: 1185267
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Abstract: Under suitable recursive conditions, the conjugacy problem for HNN extensions of the form $ \langle G,{t_i},i \in I:{x^{{k_i}}} = t_i^{ - 1}{x^{{l_i}}}{t_i},\;{k_i},\;{l_i} \geqslant 1,\;i \in I\rangle $ is solvable if and only if the inverse subsemigroup generated by $ \{ ({k_i},{l_i}),\;i \in I\} $ has solvable extended word problem in the semigroup

$\displaystyle S = (\mathbb{N} \times \mathbb{N},\;(a,b)(c,d) = (ac/\gcd (b,c),\;bd/\gcd (b,c))).$

Furthermore, $ S$ is isomorphic to the direct sum of countably many copies of the bicyclic semigroup, which has a central place in the theory of inverse semigroups.

This new approach to the conjugacy problem is used to determine several classes of HNN extensions with infinitely many stable letters and solvable conjugacy problem.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1185267-4
PII: S 0002-9939(1994)1185267-4
Keywords: Conjugacy problem, HNN extension, extended word problem, inverse semigroup, bicyclic semigroup
Article copyright: © Copyright 1994 American Mathematical Society