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Unformatted text preview: POLYNOMIAL-TIME WORD PROBLEMS SAUL SCHLEIMER Abstract. We find polynomial-time solutions to the word prob- lem for free-by-cyclic groups, the word problem for automorphism groups of free groups, and the membership problem for the han- dlebody subgroup of the mapping class group. All of these re- sults follow from observing that automorphisms of the free group strongly resemble straight line programs, which are widely stud- ied in the theory of compressed data structures. In an effort to be self-contained we give a detailed exposition of the necessary results from computer science. 1. Introduction Automorphisms of the free group are closely connected to two tech- niques in computer science: string matching and compression . The relevance of the first is obvious. The second is less clear. So, consider the fact that an automorphism of complexity n can produce, by acting on a generator, a word of size at most exp( n ). Now, there are only exp( n ) such automorphisms while there are exp(exp( n )) words avail- able as output. Thus most words in the free group cannot be obtained in this way. Those which can are highly regular and thus susceptible to compression. Compression techniques have already made an appearance in algo- rithmic topology. Word equations play a starring role in the work of Schaefer, Sedgwick, and ˇ Stefankoviˇ c [ 25 ]. One of the problems they consider, connectedness of normal curves and surfaces, is also addressed by the orbit-counting techniques of Agol, Hass, and Thurston [ 1 ]. Both results rely, directly or indirectly, on Plandowski’s Algorithm [ 24 ] (The- orem 8.1 below). The structure of the paper is as follows: Section 2 reviews straight line programs and also a slight variant, composition systems . Such programs are called compressed words . Section 3 is an exposition of Lohrey’s Theorem [ 18 ]: Date : March 6, 2007. This work is in the public domain. 1 2 SAUL SCHLEIMER Theorem 3.5 . The word problem for compressed words in the free group is solvable in polynomial time. We use this theorem to answer a variety of questions; in each case the compression technique “accelerates” an obvious exponential-time algorithm. Theorem 4.1 . For any automorphism Φ ∈ Aut( F m ) the word problem for the free-by-cyclic group G Φ = F m o Φ Z is polynomial time. This problem is already known to be in NP : Bridson and Groves [ 7 ] show that G Φ has a quadratic isoperimetric inequality. A generalization gives: Theorem 5.2 . The word problem for Aut( F m ) is polynomial time. This solves problem (C1) on the list maintained by Baumslag, Myas- nikov, and Shpilrain [ 3 ]. In Section 6 we discuss membership problems: deciding whether or not a given word belongs to a subgroup. In partic- ular, if V is a handlebody and S = ∂V then Broaddus asks if there is a polynomial-time algorithm to decide whether or not a homeomorphism of S extends over V . We prove: Theorem 6.4 . The membership problem for MCG ( V ) in MCG ( S ) is polynomial time.polynomial time....
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