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O. Bogopolski, A. Martino, and E. Ventura (2009)

Orbit decidability and the conjugacy problem for some extensions of groups

Transactions of the American Mathematical Society, 362(04):2003–2036..

Given a short exact sequence of groups with certain conditions, , we prove that has solvable conjugacy problem if and only if the corresponding action subgroup is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form , , , and with virtually solvable action group . Also, we give an easy way of constructing groups of the form and with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and we give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in is given.

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