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The Isomorphism Problem for Finite Extensions of Free Groups Is In PSPACE

Authors Géraud Sénizergues, Armin Weiß

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Computational complexity and cryptography
Keywords
  • virtually free groups
  • context-free groups
  • isomorphism problem
  • structure tree
  • graph of groups

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Abstract

We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group G, compute a finite graph of groups G with finite vertex groups and fundamental group G. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of context-free groups can be solved in doubly exponential space. Moreover, if, instead of a grammar, a finite extension of a free group is given as input, the construction of the graph of groups is in NP and, consequently, the isomorphism problem in PSPACE.

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Géraud Sénizergues and Armin Weiß. The Isomorphism Problem for Finite Extensions of Free Groups Is In PSPACE. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 139:1-139:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.139

Author Details

Géraud Sénizergues
  • LABRI, Bordeaux, France
Armin Weiß
  • Universität Stuttgart, FMI, Germany

References

  1. Yago Antolin. On Cayley graphs of virtually free groups. Groups - Complexity - Cryptology, 3:301-327, 2011. URL: http://dx.doi.org/10.1515/gcc.2011.012.
  2. Ron Book and Friedrich Otto. String-Rewriting Systems. Springer-Verlag, 1993. Google Scholar
  3. W. W. Boone. The Word Problem. Ann. of Math., 70(2):207-265, 1959. Google Scholar
  4. Matt Clay and Max Forester. Whitehead moves for G-trees. Bull. Lond. Math. Soc., 41(2):205-212, 2009. URL: http://dx.doi.org/10.1112/blms/bdn118.
  5. François Dahmani and Vincent Guirardel. The isomorphism problem for all hyperbolic groups. Geom. Funct. Anal., 21(2):223-300, 2011. URL: http://dx.doi.org/10.1007/s00039-011-0120-0.
  6. Max Dehn. Ueber unendliche diskontinuierliche Gruppen. Math. Ann., 71:116-144, 1911. Google Scholar
  7. Warren Dicks and Martin J. Dunwoody. Groups acting on graphs. Cambridge University Press, 1989. Google Scholar
  8. Volker Diekert and Armin Weiß. Context-Free Groups and Their Structure Trees. International Journal of Algebra and Computation, 23:611-642, 2013. URL: http://dx.doi.org/10.1142/S0218196713500124.
  9. Volker Diekert and Armin Weiß. Context-Free Groups and Bass-Serre Theory. In Juan González-Meneses, Martin Lustig, and Enric Ventura, editors, Algorithmic and Geometric Topics Around Free Groups and Automorphisms, Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser, Basel, Switzerland, 2017. URL: http://dx.doi.org/10.1007/978-3-319-60940-9.
  10. Max Forester. Deformation and rigidity of simplicial group actions on trees. Geom. Topol., 6:219-267, 2002. URL: http://dx.doi.org/10.2140/gt.2002.6.219.
  11. Vincent Guirardel and Gilbert Levitt. Deformation spaces of trees. Groups Geom. Dyn., 1(2):135-181, 2007. URL: http://dx.doi.org/10.4171/GGD/8.
  12. John E. Hopcroft and Jeffrey D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979. Google Scholar
  13. Matthias Jantzen. Confluent String Rewriting, volume 14 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988. Google Scholar
  14. A. Karrass, A. Pietrowski, and D. Solitar. Finite and infinite cyclic extensions of free groups. Journal of the Australian Mathematical Society, 16(04):458-466, 1973. URL: http://dx.doi.org/10.1017/S1446788700015445.
  15. Sava Krstic. Actions of finite groups on graphs and related automorphisms of free groups. Journal of Algebra, 124:119-138, 1989. URL: http://dx.doi.org/10.1016/0021-8693(89)90154-3.
  16. David E. Muller and Paul E. Schupp. Groups, the theory of ends, and context-free languages. J. Comput. Syst. Sci., 26:295-310, 1983. URL: http://dx.doi.org/10.1016/0022-0000(83)90003-X.
  17. David E. Muller and Paul E. Schupp. The theory of ends, pushdown automata, and second-order logic. Theor. Comput. Sci., 37(1):51-75, 1985. URL: http://dx.doi.org/10.1016/0304-3975(85)90087-8.
  18. P. S. Novikov. On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov, pages 1-143, 1955. In Russian. Google Scholar
  19. Christos H. Papadimitriou. Computational Complexity. Addison Wesley, 1994. Google Scholar
  20. Alan L. Selman. A taxonomy of complexity classes of functions. J. Comput. Syst. Sci., 48(2):357-381, 1994. URL: http://dx.doi.org/10.1016/S0022-0000(05)80009-1.
  21. Géraud Sénizergues. An effective version of Stalling’s theorem in the case of context-free groups. In Andrzej Lingas, Rolf G. Karlsson, and Svante Carlsson, editors, Proc. 20th International Colloquium Automata, Languages and Programming (ICALP 93), Lund (Sweden), volume 700 of Lecture Notes in Computer Science, pages 478-495. Springer-Verlag, 1993. Google Scholar
  22. Géraud Sénizergues. On the finite subgroups of a context-free group. In Gilbert Baumslag, David Epstein, Robert Gilman, Hamish Short, and Charles Sims, editors, Geometric and Computational Perspectives on Infinite Groups, number 25 in DIMACS series in Discrete Mathematics and Theoretical Computer Science, pages 201-212. Amer. Math. Soc., 1996. Google Scholar
  23. Géraud Sénizergues and Armin Weiß. The isomorphism problem for finite extensions of free groups is in PSPACE. CoRR, To appear soon in arXiv, 2018. URL: https://arxiv.org/abs.
  24. Jean-Pierre Serre. Trees. Springer, 1980. French original 1977. Google Scholar
  25. Carsten Thomassen and Wolfgang Woess. Vertex-transitive graphs and accessibility. J. Comb. Theory Ser. B, 58(2):248-268, 1993. URL: http://dx.doi.org/10.1006/jctb.1993.1042.
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