On torsionfree groups with finite regular file bases
Author:
Alexey Muranov
Journal:
Trans. Amer. Math. Soc. 359 (2007), 36093645
MSC (2000):
Primary 20F05; Secondary 20F06
Published electronically:
March 7, 2007
MathSciNet review:
2302509
Fulltext PDF Free Access
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Abstract: The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsionfree group has a finite system of generators , ..., such that every element of has a unique presentation in the form where , is it true that is virtually polycyclic? The answer is ``not always.'' A counterexample is constructed in this paper as a group presented by generators and defining relations.
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 [ChCoH]
 Chiswell, Ian M.; Collins, Donald J.; Huebschmann, Johannes, Aspherical group presentations, Math Z. 178 (1) (1981), 136. MR 0627092 (83a:20046)
 [H1]
 Huebschmann, Johannes, Cohomology theory of aspherical groups and of small cancellation groups, J. Pure Appl. Algebra 14 (2) (1979), 137143. MR 0524183 (80e:20064)
 [H2]
 , The homotopy type of a combinatorially aspherical presentation, Math Z. 173 (2) (1980), 163169. MR 0583383 (81m:57003)
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 Itô, Noboru, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955), 400401 (German). MR 0071426 (17:125b)
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 The Kourovka Notebook, unsolved problems in group theory, 13th augmented edition, Russian Acad. of Sci. Siber. Div., Inst. Math., Novosibirsk, 1995 transl. from Russian. MR 1392713 (97d:20001)
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 Lennox, John C.; Roseblade, James E., Soluble products of polycyclic groups, Math. Z. 170 (2) (1980), 153154. MR 0562584 (81a:20047)
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 Lyndon, Roger C.; Schupp, Paul E., Combinatorial group theory, reprint of the 1977 edition, SpringerVerlag, Berlin, 2001. MR 1812024 (2001i:20064)
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 Minasyan, Ashot, On products of quasiconvex subgroups in hyperbolic groups, Internat. J. Algebra Comput. 14 (2) (2004), 173195. MR 2058319 (2005a:20063)
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 Muranov, Alexey Yu., Diagrams with selection and method for constructing boundedly generated and boundedly simple groups, Comm. Algebra 33 (4) (2005), 12171258. MR 2136699 (2006b:20044)
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Additional Information
Alexey Muranov
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240–0001
Address at time of publication:
Institut Camille Jordan, Université Lyon 1, 43 blvd du 11 novembre 1918, 69622 Villeurbanne cedex, France
Email:
muranov@math.univlyon1.fr
DOI:
http://dx.doi.org/10.1090/S0002994707042560
PII:
S 00029947(07)042560
Keywords:
Group presentation,
van Kampen's lemma,
diagram with selection,
Sdiagram,
virtually polycyclic group,
boundedly generated group,
file basis
Received by editor(s):
March 4, 2005
Published electronically:
March 7, 2007
Additional Notes:
This work was supported in part by the NSF grant DMS 0245600 of Alexander Ol’shanskiy and Mark Sapir.
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
