The conjugacy problem for HNN extensions with infinite cyclic associated groups

Horadam, K. J.; Farr, G. E.

doi:10.1090/S0002-9939-1994-1185267-4

The conjugacy problem for HNN extensions with infinite cyclic associated groups
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by K. J. Horadam and G. E. Farr PDF

Proc. Amer. Math. Soc. 120 (1994), 1009-1015 Request permission

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