The twisted conjugacy problem for finitely generated free groups

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• 作者：Seung Won Kim ^{kimsw@ks.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：20E05 ; 20E45 ; 20F10 ; 55M20
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：220
• 期：4
• 页码：1281-1293
• 全文大小：346 K

文摘

Let G   be a finitely generated free group and let φ∈End(G)$\phi \in \mathrm{End}(G)$ be an endomorphism of G. In this paper we prove that the twisted conjugacy problem for φ is algorithmically solvable in the special case of φ having remnant. This case covers a significant set of endomorphisms. It is proved in Wagner (1999) [14] that almost all endomorphisms of G have remnant in a sense that can be made precise in terms of probability.

For φ∈End(G)$\phi \in \mathrm{End}(G)$ having remnant, we provide an upper bound on the length of elements z∈G$z\in G$ that need to be checked to solve the twisted conjugate problem for φ so that the algorithm is simple to use for a computer search. Our new algorithm improves on existing algorithms which can only handle homomorphisms with remnant words of length at least 2.