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There currently exists no algebraic algorithm for computing twisted conjugacyclasses in free groups. We propose a new technique for deciding twistedconjugacy relations using nilpotent quotients. Our technique is generalizationof the common abelianization method, but admits significantly greater rates ofsuccess. We present experimental results demonstrating the efficacy of thetechnique, and detail how it can be applied the related settings of surfacegroups and doubly twisted conjugacy.

Computing twisted conjugacy classes in free groups using nilpotent quotients