On torsion-free groups with finite regular file bases

Author:
Alexey Muranov

Journal:
Trans. Amer. Math. Soc. **359** (2007), 3609-3645

MSC (2000):
Primary 20F05; Secondary 20F06

DOI:
https://doi.org/10.1090/S0002-9947-07-04256-0

Published electronically:
March 7, 2007

MathSciNet review:
2302509

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Abstract | References | Similar Articles | Additional Information

Abstract: The following question was asked by V. V. Bludov in The Kourovka Notebook in 1995: If a torsion-free 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.

**[ABC+]**J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short,*Notes on word hyperbolic groups*, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 3–63. Edited by Short. MR**1170363****[Bl]**V. V. Bludov,*Thread bases in groups*, Algebra i Logika**34**(1995), no. 3, 247–261, 363 (Russian, with Russian summary); English transl., Algebra and Logic**34**(1995), no. 3, 131–139. MR**1364464**, https://doi.org/10.1007/BF02341869**[Br]**Kenneth S. Brown,*Cohomology of groups*, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR**1324339****[ChCoH]**Chiswell, Ian M.; Collins, Donald J.; Huebschmann, Johannes,*Aspherical group presentations*, Math Z.**178**(1) (1981), 1-36. MR**0627092 (83a:20046)****[H1]**Huebschmann, Johannes,*Cohomology theory of aspherical groups and of small cancellation groups*, J. Pure Appl. Algebra**14**(2) (1979), 137-143. MR**0524183 (80e:20064)****[H2]**-,*The homotopy type of a combinatorially aspherical presentation*, Math Z.**173**(2) (1980), 163-169. MR**0583383 (81m:57003)****[I]**Noboru Itô,*Über das Produkt von zwei abelschen Gruppen*, Math. Z.**62**(1955), 400–401 (German). MR**0071426**, https://doi.org/10.1007/BF01180647**[kn]**V. D. Mazurov and E. I. Khukhro (eds.),*Unsolved problems in group theory. The Kourovka notebook*, Thirteenth augmented edition, Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 1995. MR**1392713****[LeR]**Lennox, John C.; Roseblade, James E.,*Soluble products of polycyclic groups*, Math. Z.**170**(2) (1980), 153-154. MR**0562584 (81a:20047)****[LyS]**Roger C. Lyndon and Paul E. Schupp,*Combinatorial group theory*, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR**1812024****[Mi]**Ashot Minasyan,*On products of quasiconvex subgroups in hyperbolic groups*, Internat. J. Algebra Comput.**14**(2004), no. 2, 173–195. MR**2058319**, https://doi.org/10.1142/S0218196704001712**[Mu]**Alexey Muranov,*Diagrams with selection and method for constructing boundedly generated and boundedly simple groups*, Comm. Algebra**33**(2005), no. 4, 1217–1258. MR**2136699**, https://doi.org/10.1081/AGB-200053951**[O1]**A. Yu. Ol′shanskiĭ,*Geometry of defining relations in groups*, Mathematics and its Applications (Soviet Series), vol. 70, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturin. MR**1191619****[O2]**A. Yu. Ol′shanskiĭ,*On residualing homomorphisms and 𝐺-subgroups of hyperbolic groups*, Internat. J. Algebra Comput.**3**(1993), no. 4, 365–409. MR**1250244**, https://doi.org/10.1142/S0218196793000251

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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.univ-lyon1.fr

DOI:
https://doi.org/10.1090/S0002-9947-07-04256-0

Keywords:
Group presentation,
van Kampen's lemma,
diagram with selection,
S-diagram,
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.