On an algorithm to decide whether a free group is a free factor of another

Silva, Pedro V.; Weil, Pascal

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 2, pp. 395-414

We revisit the problem of deciding whether a finitely generated subgroup $H$ is a free factor of a given free group $F$. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of $H$ and exponential in the rank of $F$. We show that the latter dependency can be made exponential in the rank difference rank$\left(F\right)$ - rank$\left(H\right)$, which often makes a significant change.

Classification : 20E05, 05C25
Keywords: combinatorial group theory, free groups, free factors, inverse automata, algorithms

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