Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T09:22:59.488Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  20 December 2007

Pedro V. Silva
Affiliation:
Centro de Matemática, Faculdade de Ciências, Universidade do Porto, R. Campo Alegre 687, 4169-007 Porto, Portugal; [email protected]
Pascal Weil
Affiliation:
LaBRI, CNRS, 351 cours de la Libération, 33405 Talence Cedex, France; [email protected]
Get access

Abstract

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(F) - rank(H), which often makes a significant change.

Type
Research Article
Copyright
© EDP Sciences, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J. Almeida, Finite semigroups and universal algebra. World Scientific Publishing, Singapore (1994).
Anshel, I., Anshel, M., Fisher, B. and Goldfeld, D., New key agreement protocols in braid group cryptography, in CT-RSA 2001. Lect. Notes Comput. Sci. 2020 (2001) 115.
S. Cleary and J. Taback, Metric properties of the lamplighter group as an automata group. Contemp. Math. 372, AMS (2005)
Dehornoy, P., Braid-based cryptography. Contemp. Math. 360 (2004) 533. CrossRef
Gersten, S., Whitehead's, On algorithm. Bull. Amer. Math. Soc. 10 (1984) 281284. CrossRef
Henckell, K., Margolis, S.W., Pin, J.-E. and Rhodes, J., Ash's type II theorem, profinite topology and Malcev products. Int. J. Algebr. Comput. 1 (1991) 411436. CrossRef
Kambites, M., Silva, P.V. and Steinberg, B., The spectra of lamplighter groups and Cayley machines. Geom. Dedicata 120 (2006) 193227. CrossRef
Kapovich, I. and Miasnikov, A.G., Stallings foldings and subgroups of free groups. J. Algebra 248 (2002) 608668. CrossRef
Kapovich, I., Schupp, P. and Shpilrain, V., Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups. Pacific J. Math. 223 (2006) 113140. CrossRef
B. Khan, The structure of automorphic conjugacy in the free group of rank two, in Proc. Special Session on Interactions between Logic, Group Theory and Computer Science. Contemp. Math. 349 (2004).
Lee, D., A tighter bound for the number of words of minimum length in an automorphic orbit. J. Algebra 305 (2006) 10931101. CrossRef
R. Lyndon and P. Schupp, Combinatorial group theory. Springer (1977, reprinted 2001).
Margolis, S. and Meakin, J., Free inverse monoids and graph immersions. Int. J. Algebr. Comput. 3 (1993) 79100. CrossRef
Margolis, S., Sapir, M. and Weil, P., Closed subgroups in pro-V topologies and the extension problem for inverse automata. Int. J. Algebr. Comput. 11 (2001) 405445. CrossRef
Miasnikov, A.G. and Shpilrain, V., Automorphic orbits in free groups. J. Algebra 269 (2003) 1827. CrossRef
A.G. Miasnikov, E. Ventura and P. Weil, Algebraic extensions in free groups, in Geometric Group theory, Geneva and Barcelona conferences, G.N. Arzhantseva, L. Bartholdi, J. Burillo and E. Ventura, eds. Trends Math. (2007) 225-253.
D. Perrin, Automata, in Handbook of Theoretical Computer Science, Vol. B, J. Leeuwen ed. Elsevier, 1990.
J.-E. Pin, Variétés de langages formels, Masson, Paris (1984); English translation: Varieties of formal languages. Plenum, New-York (1986).
J. Rotman, An introduction to the theory of groups. 4th edition, Springer (1995).
J.-P. Serre, Arbres, amalgames, SL 2 , Astérisque 46, Soc. Math. France (1977). English translation: Trees, Springer Monographs in Mathematics, Springer (2003).
V.M. Sidelnikov, M.A. Cherepnev and V.Y. Yaschenko, Systems of open distribution of keys on the basis of non-commutative semigroups. Ross. Acd. Nauk Dokl. 332-5 (1993). English translation: Russian Acad. Sci. Dokl. Math. 48-2 (1994) 383–386.
Silva, P.V. and Steinberg, B., On a class of automata groups generalizing lamplighter groups. Intern. J. Algebra Comput. 15 (2005) 12131234. CrossRef
Stallings, J., Topology of finite graphs. Invent. Math. 71 (1983) 551565. CrossRef
J. Stephen, Applications of automata theory to presentations of monoids and inverse monoids. Ph.D. Dissertation, University of Nebraska (1987).
N. Touikan. A fast algorithm for Stallings? Folding process. J. Algebra Comput. 16 (2006) 1031–1046.
Ventura, E., On fixed subgroups of maximal rank. Comm. Algebra 25 (1997) 33613375. CrossRef