Computational complexity and the conjugacy problem

The conjugacy problem for a finitely generated group \(G\) is the two-variable problem of deciding for an arbitrary pair \((u,v)\) of elements of \(G\), whether or not \(u\) is conjugate to \(v\) in \(G\). We construct examples of finitely generated, computably presented groups such that for every element \(u_0\) of \(G\), the problem of deciding if an arbitrary element is conjugate to \(u_0\) is decidable in quadratic time but the worst-case complexity of the global conjugacy problem is arbitrary: it can be any c.e. Turing degree , can exactly mirror the Time Hierarchy Theorem, or can be \(\mathcal{NP}\)-complete. Our groups also have the property that the conjugacy problem is generically linear time: that is, there is a linear time partial algorithm for the conjugacy problem whose domain has density \(1\), so hard instances are very rare. We also consider the complexity relationship of the "half-conjugacy" problem to the conjugacy problem. In the last section we discuss the extreme opposite situation: groups with algorithmically finite conjugation.