2000
DOI: 10.1016/s0166-8641(99)00063-2
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On complexity of the word problem in braid groups and mapping class groups

Abstract: We prove that the word problem in the mapping class group of the once-punctured surface of genus g has complexity O(|w| 2 g) for |w| ≥ log(g) where |w| is the length of the word in a (standard) set of generators. The corresponding bound in the case of the closed surface is O(|w| 2 g 2 ). We also carry out the same methods for the braid groups, and show that this gives a bound which improves the best known bound in this case; namely, the complexity of the word problem in the n−braid group is O(|w| 2 n), for |w|… Show more

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Cited by 8 publications
(7 citation statements)
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“…Finally, recall Nielsen's Theorem (see [19, page 175]): the mapping class group MCG(S) is isomorphic to the outer automorphism group of π 1 (S). So, similar to the proof of Theorem 6.1, well-tempered paths give: Theorem A.8 ([21], [11]). The problem in MCG(S) is solvable in polynomial time.…”
Section: Appendix a On Surfacesmentioning
confidence: 66%
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“…Finally, recall Nielsen's Theorem (see [19, page 175]): the mapping class group MCG(S) is isomorphic to the outer automorphism group of π 1 (S). So, similar to the proof of Theorem 6.1, well-tempered paths give: Theorem A.8 ([21], [11]). The problem in MCG(S) is solvable in polynomial time.…”
Section: Appendix a On Surfacesmentioning
confidence: 66%
“…Choosing a basepoint on the boundary makes B n act on π 1 (D m ) ∼ =F m and so embeds B n into Aut(F m ). A simple corollary to Theorem 5.2 is the well-known: [15], [8], [4], [11]). The word problem for B m is polynomial time.…”
Section: The Automorphism Group Of a Free Groupmentioning
confidence: 98%
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“…Although we use little of this vast theory, it is quite connected to our problem: If G 1 and G 2 are cellularly embedded on S, a homeomorphism of S that maps G 1 to G 2 represents a unique element of the mapping class group, and our problem amounts to testing whether this element is the identity. Hence it is closely related to the word problem in mapping class groups, which can be solved in quadratic time [21,31]. However, these algorithms take as input the collection of Dehn twists corresponding to a mapping class, which is an input incomparable to ours.…”
Section: Related Workmentioning
confidence: 99%
“…Here we mention some of them. The complexity of different algorithms varies, but to our knowledge the best known solution is of complexity O(l 2 n), where l is the length of the longer braid word, and n is the number of strings in the corresponding braid group [9].…”
Section: Two Known Algorithms For the Word Problemmentioning
confidence: 99%