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Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients

Part of: Group theory and generalizations Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 August 2010

René Hartung
René Hartung*
Affiliation:
Institute of Computational Mathematics, University of Braunschweig, Pockelsstraße 14, 38106 Braunschweig, Germany (email: r.hartung@tu-bs.de)

Abstract

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We describe an algorithm for computing successive quotients of the Schur multiplier M(G) of a group G given by an invariant finite L-presentation. As applications, we investigate the Schur multipliers of various self-similar groups, including the Grigorchuk super-group, the generalized Fabrykowski–Gupta groups, the Basilica group and the Brunner–Sidki–Vieira group.

MSC classification

Secondary: 20-04: Explicit machine computation and programs (not the theory of computation or programming) 20F05: Generators, relations, and presentations 20F14: Derived series, central series, and generalizations

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 260 - 271
Copyright
Copyright © London Mathematical Society 2010

References

[1] Bartholdi, L., ‘Endomorphic presentations of branch groups’, J. Algebra 268 (2003) no. 2, 419–443.CrossRefGoogle Scholar
[2] Bartholdi, L., Eick, B. and Hartung, R., ‘A nilpotent quotient algorithm for certain infinitely presented groups and its applications’, Internat. J. Algebra Comput. 18 (2008) no. 8, 1321–1344.CrossRefGoogle Scholar
[3] Bartholdi, L. and Grigorchuk, R. I., ‘On parabolic subgroups and Hecke algebras of some fractal groups’, Serdica Math. J. 28 (2002) no. 1, 47–90.Google Scholar
[4] Bartholdi, L. and Siegenthaler, O., ‘The twisted twin of the Grigorchuk group’, Internat. J. Algebra Comput. 24 (2010) no. 4, 465–488.CrossRefGoogle Scholar
[5] Bartholdi, L. and Virág, B., ‘Amenability via random walks’, Duke Math. J. 130 (2005) no. 1, 39–56.CrossRefGoogle Scholar
[6] Baumslag, G., Topics in combinatorial group theory,, Lectures in Mathematics, ETH Zürich (Birkhäuser, Basel, 1993).CrossRefGoogle Scholar
[7] Blackburn, N. and Evens, L., ‘Schur multipliers of p-groups’, J. reine angew. Math. 309 (1979) 100–113.Google Scholar
[8] Bludov, V. V., ‘On residually torsion-free nilpotent groups’, J. Group Theory 12 (2009) no. 4, 579–590.CrossRefGoogle Scholar
[9] Brunner, A. M., Sidki, S. and Vieira, A. C., ‘A just nonsolvable torsion-free group defined on the binary tree’, J. Algebra 211 (1999) no. 1, 99–114.CrossRefGoogle Scholar
[10] Eick, B. and Nickel, W., ‘Computing the Schur multiplicator and the nonabelian tensor square of a polycyclic group’, J. Algebra 320 (2008) no. 2, 927–944.CrossRefGoogle Scholar
[11] Fabrykowski, J. and Gupta, N., ‘On groups with sub-exponential growth functions’, J. Indian Math. Soc. (N.S.) 49 (1987) no. 3–4, 249–256, 1985.Google Scholar
[12] The GAP group, ‘GAP—groups, algorithms and programming, version 4.4.10’, 2007.Google Scholar
[13] Gordon, C. M., ‘Some embedding theorems and undecidability questions for groups’, Combinatorial and geometric group theory (Edinburgh, 1993),, London Mathematical Society Lecture Note Series 204 (Cambridge University Press, Cambridge, 1995) 105–110.Google Scholar
[14] Grigorchuk, R. I., ‘On Burnside’s problem on periodic groups’, Funktsional. Anal. i Prilozhen. 14 (1980) no. 1, 53–54.CrossRefGoogle Scholar
[15] Grigorchuk, R. I., ‘On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata’, Groups St Andrews 1997 in Bath, I,, London Mathematical Society Lecture Note Series 260 (Cambridge University Press, Cambridge, 1999) 290–317.CrossRefGoogle Scholar
[16] Grigorchuk, R. I., ‘Just infinite branch groups’, New horizons in pro-p groups,, Progress in Mathematics 184 (Birkhäuser, Boston, MA, 2000) 121–179.CrossRefGoogle Scholar
[17] Grigorchuk, R. I. and Żuk, A., ‘On a torsion-free weakly branch group defined by a three state automaton’, Internat. J. Algebra Comput. 12 (2002) no. 1–2, 223–246. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000).CrossRefGoogle Scholar
[18] Grigorchuk, R. I. and Żuk, A., ‘Spectral properties of a torsion-free weakly branch group defined by a three state automaton’, Computational and statistical group theory: AMS special sessions on geometric group theory (Las Vegas, NV, 2001) and computational group theory, Hoboken, NJ, 2001,, Contemporary Mathematics 298 (American Mathematical Society, Providence, RI, 2002) 57–82.CrossRefGoogle Scholar
[19] Hartung, R., ‘NQL—nilpotent quotients of L-presented groups’, a refereed GAP 4 package, 2009; see [12].Google Scholar
[20] Holt, D. F., ‘The calculation of the Schur multiplier of a permutation group’, Computational group theory, Durham, 1982, (Academic Press, London, 1984) 307–319.Google Scholar
[21] Lysënok, I. G., ‘A set of defining relations for the Grigorchuk group’, Mat. Zametki 38 (1985) no. 4, 503–516.Google Scholar
[22] Mikhaĭlov, R. V., ‘Residual nilpotence and residual solvability of groups’, Mat. Sb. 196 (2005) no. 11, 109–126.Google Scholar
[23] Nickel, W., ‘Computing nilpotent quotients of finitely presented groups’, Geometric and computational perspectives on infinite groups: Proceedings of a joint DIMACS/Geometry Center workshop, Minneapolis, MN and New Brunswick, NJ, 1994,, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 25 (American Mathematical Society, Providence, RI, 1996) 175–191.Google Scholar
[24] Robinson, D. J. S., A course in the theory of groups,, 2nd edn, Graduate Texts in Mathematics 80 (Springer, New York, 1996).CrossRefGoogle Scholar
[25] Sergiescu, V., ‘A quick introduction to Burnside’s problem’, Group theory from a geometrical viewpoint, Trieste, 1990, (World Scientific Publishers, River Edge, NJ, 1991) 622–629.Google Scholar