Abstract
The computational complexity of the word problem in HNN-extension of groups is studied. HNN-extension is a fundamental construction in combinatorial group theory. It is shown that the word problem for an ascending HNN-extension of a group H is logspace reducible to the so-called compressed word problem for H. The main result of the paper states that the word problem for an HNN-extension of a hyperbolic group H with cyclic associated subgroups can be solved in polynomial time. This result can be easily extended to fundamental groups of graphs of groups with hyperbolic vertex groups and cyclic edge groups.
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Notes
- 1.
The concept of undistorted subgroups is defined for arbitrary finitely generated subgroups but we will need it only for the cyclic case.
References
Artin, E.: Theorie der Zöpfe. Abh. Math. Semin. Univ. Hambg. 4(1), 47–72 (1925)
Avenhaus, J., Madlener, K.: The Nielsen reduction and P-complete problems in free groups. Theoret. Comput. Sci. 32(1–2), 61–76 (1984)
Bieri, R., Strebel, R.: Almost finitely presented soluble groups. Commentarii Mathematici Helvetici 53, 258–278 (1978)
Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005). https://doi.org/10.1007/3-540-27596-7
Boone, W.W.: The word problem. Ann. Math. Second Series 70, 207–265 (1959)
Britton, J.L.: The word problem. Ann. Math. 77(1), 16–32 (1963)
Charney, R.: An introduction to right-angled Artin groups. Geom. Dedicata. 125, 141–158 (2007). https://doi.org/10.1007/s10711-007-9148-6
Dehn, M.: Über unendliche diskontinuierliche Gruppen. Math. Ann. 71, 116–144 (1911)
Dehn, M.: Transformation der Kurven auf zweiseitigen Flächen. Math. Ann. 72, 413–421 (1912)
Diekert, V., Kausch, J.: Logspace computations in graph products. J. Symb. Comput. 75, 94–109 (2016)
Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Bartlett (1992)
Epstein, D.B.A., Holt, D.F.: The linearity of the conjugacy problem in word-hyperbolic groups. Internat. J. Algebra Comput. 16(2), 287–306 (2006)
Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, Heidelberg (1987). https://doi.org/10.1007/978-1-4613-9586-7_3
Hagenah, C.: Gleichungen mit regulären Randbedingungen über freien Gruppen. Ph.D. thesis, University of Stuttgart (2000)
Haubold, N., Lohrey, M.: Compressed word problems in HNN-extensions and amalgamated products. Theory Comput. Syst. 49(2), 283–305 (2011). https://doi.org/10.1007/s00224-010-9295-2
Holt, D.: Word-hyperbolic groups have real-time word problem. Internat. J. Algebra Comput. 10, 221–228 (2000)
Holt, D.F., Lohrey, M., Schleimer, S.: Compressed decision problems in hyperbolic groups. In: 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, Berlin, Germany, 13–16 March 2019, LIPIcs, vol. 126, pp. 37:1–37:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019). http://www.dagstuhl.de/dagpub/978-3-95977-100-9
Lipton, R.J., Zalcstein, Y.: Word problems solvable in logspace. J. ACM 24(3), 522–526 (1977)
Lohrey, M.: Decidability and complexity in automatic monoids. Int. J. Found. Comput. Sci. 16(4), 707–722 (2005)
Lohrey, M.: The Compressed Word Problem for Groups. Springer Briefs in Mathematics, Springer, Heidelberg (2014). https://doi.org/10.1007/978-1-4939-0748-9
Lohrey, M.: Complexity of word problems for HNN-extensions. CoRR abs/2107.01630 (2021). https://arxiv.org/abs/2107.01630
Magnus, W.: Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106(1), 295–307 (1932). https://doi.org/10.1007/BF01455888
Mattes, C., Weiß, A.: Parallel algorithms for power circuits and the word problem of the Baumslag group. CoRR abs/2102.09921 (2021). https://arxiv.org/abs/2102.09921
Minasyan, A.: On products of quasiconvex subgroups in hyperbolic groups. Int. J. Algebra Comput. 14(2), 173–195 (2004)
Myasnikov, A., Nikolaev, A.: Verbal subgroups of hyperbolic groups have infinite width. J. Lond. Math. Soc. 90(2), 573–591 (2014)
Myasnikov, A., Nikolaev, A., Ushakov, A.: Knapsack problems in groups. Math. Comput. 84, 987–1016 (2015)
Myasnikov, A., Ushakov, A., Won, D.W.: The word problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable. J. Algebra 345(1), 324–342 (2011)
Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Am. Math. Soc. Transl. II. Ser. 9, 1–122 (1958)
Rabin, M.O.: Computable algebra, general theory and theory of computable fields. Trans. Am. Math. Soc. 95, 341–360 (1960)
Rips, E.: Subgroups of small cancellation groups. Bull. Lond. Math. Soc. 14, 45–47 (1982)
Simon, H.U.: Word problems for groups and contextfree recognition. In: Proceedings of Fundamentals of Computation Theory, FCT 1979, pp. 417–422. Akademie-Verlag (1979)
Stillwell, J.: Classical Topology and Combinatorial Group Theory, 2nd edn. Springer, Heidelberg (1995). https://doi.org/10.1007/978-1-4612-4372-4
Waack, S.: The parallel complexity of some constructions in combinatorial group theory. J. Inf. Process. Cybern. EIK 26, 265–281 (1990)
Wehrfritz, B.A.F.: On finitely generated soluble linear groups. Math. Z. 170, 155–167 (1980)
Weiß, A.: On the complexity of conjugacy in amalgamated products and HNN extensions. Ph.D. thesis, University of Stuttgart (2015)
Weiß, A.: A logspace solution to the word and conjugacy problem of generalized Baumslag-Solitar groups. In: Algebra and Computer Science. Contemporary Mathematics, vol. 677. American Mathematical Society (2016)
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This work is supported by the DFG project LO748/12-1.
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Lohrey, M. (2021). Complexity of Word Problems for HNN-Extensions. In: Bampis, E., Pagourtzis, A. (eds) Fundamentals of Computation Theory. FCT 2021. Lecture Notes in Computer Science(), vol 12867. Springer, Cham. https://doi.org/10.1007/978-3-030-86593-1_26
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