Solvable groups with polynomial Dehn functions
HTML articles powered by AMS MathViewer
- by G. N. Arzhantseva and D. V. Osin
- Trans. Amer. Math. Soc. 354 (2002), 3329-3348
- DOI: https://doi.org/10.1090/S0002-9947-02-02985-9
- Published electronically: April 3, 2002
- PDF | Request permission
Abstract:
Given a finitely presented group $H$, finitely generated subgroup $B$ of $H$, and a monomorphism $\psi :B\to H$, we obtain an upper bound of the Dehn function of the corresponding HNN-extension $G=\langle H, t\; |\; t^{-1}Bt=\psi (B)\rangle$ in terms of the Dehn function of $H$ and the distortion of $B$ in $G$. Using such a bound, we construct first examples of non-polycyclic solvable groups with polynomial Dehn functions. The constructed groups are metabelian and contain the solvable Baumslag-Solitar groups. In particular, this answers a question posed by Birget, Ol’shanskii, Rips, and Sapir.References
- Juan M. Alonso, Inégalités isopérimétriques et quasi-isométries, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 12, 761–764 (French, with English summary). MR 1082628
- Gilbert Baumslag, On finitely presented metabelian groups, Bull. Amer. Math. Soc. 78 (1972), 279. MR 291260, DOI 10.1090/S0002-9904-1972-12958-6
- Aldo A. Bernasconi, On HNN-extensions and the complexity of the word problem for one-relator groups, Ph.D. thesis, University of Utah, June 1994; available on http://www.math.utah.edu/$\sim$gersten/Papers/bernasconi-thesis.ps.gz.
- M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. MR 1152226
- Robert Bieri and Ralph Strebel, Almost finitely presented soluble groups, Comment. Math. Helv. 53 (1978), no. 2, 258–278. MR 498863, DOI 10.1007/BF02566077
- J. C. Birget, A. Yu. Ol’shanskii, E. Rips, and M. V. Sapir, Isoperimetric functions of groups and computational complexity of the word problem, (1998), preprint.
- Noel Brady, Branched coverings of cubical complexes and subgroups of hyperbolic groups, J. London Math. Soc. (2) 60 (1999), no. 2, 461–480. MR 1724853, DOI 10.1112/S0024610799007644
- Stephen G. Brick, On Dehn functions and products of groups, Trans. Amer. Math. Soc. 335 (1993), no. 1, 369–384. MR 1102884, DOI 10.1090/S0002-9947-1993-1102884-1
- M. R. Bridson and S. M. Gersten, The optimal isoperimetric inequality for torus bundles over the circle, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 1–23. MR 1380947, DOI 10.1093/qmath/47.1.1
- Martin R. Bridson, Fractional isoperimetric inequalities and subgroup distortion, J. Amer. Math. Soc. 12 (1999), no. 4, 1103–1118. MR 1678924, DOI 10.1090/S0894-0347-99-00308-2
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Cornelia Druţu, Remplissage dans des réseaux de $\mathbf Q$-rang 1 et dans des groupes résolubles, Pacific J. Math. 185 (1998), no. 2, 269–305 (French, with English summary). MR 1659046, DOI 10.2140/pjm.1998.185.269
- C. Drutu, Filling in lattices and solvable groups, (2000), preprint.
- Benson Farb, The extrinsic geometry of subgroups and the generalized word problem, Proc. London Math. Soc. (3) 68 (1994), no. 3, 577–593. MR 1262309, DOI 10.1112/plms/s3-68.3.577
- Steve M. Gersten, Isoperimetric and isodiametric functions of finite presentations, Geometric group theory, Vol. 1 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 181, Cambridge Univ. Press, Cambridge, 1993, pp. 79–96. MR 1238517, DOI 10.1017/CBO9780511661860.008
- S. M. Gersten, Dehn functions and $l_1$-norms of finite presentations, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 195–224. MR 1230635, DOI 10.1007/978-1-4613-9730-4_{9}
- S.M. Gersten, Some remarks on subgroup of hyperbolic groups, (1999), preprint.
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
- V. S. Guba and M. V. Sapir, On Dehn functions of free products of groups, Proc. Amer. Math. Soc. 127 (1999), no. 7, 1885–1891. MR 1469408, DOI 10.1090/S0002-9939-99-04579-7
- Helmut Hasse, Number theory, Akademie-Verlag, Berlin, 1979. Translated from the third German edition of 1969 by Horst Günter Zimmer. MR 544018
- M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the theory of groups, Graduate Texts in Mathematics, vol. 62, Springer-Verlag, New York-Berlin, 1979. Translated from the second Russian edition by Robert G. Burns. MR 551207
- Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
- E. M. Levich, The representation of two–step solvable groups by matrices over a field of characteristic zero, Mat. Sb. 81 (1970), 352–357.
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- Charles F. Miller III and Paul E. Schupp, The geometry of Higman-Neumann-Neumann extensions, Comm. Pure Appl. Math. 26 (1973), 787–802. Collection of articles dedicated to Wilhelm Magnus. MR 344352, DOI 10.1002/cpa.3160260522
- J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR 232311
- A. Yu. Ol′shanskiĭ, Hyperbolicity of groups with subquadratic isoperimetric inequality, Internat. J. Algebra Comput. 1 (1991), no. 3, 281–289. MR 1148230, DOI 10.1142/S0218196791000183
- 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, DOI 10.1007/978-94-011-3618-1
- A. Yu. Ol′shanskii and M. V. Sapir, Embeddings of relatively free groups into finitely presented groups, Combinatorial and computational algebra (Hong Kong, 1999) Contemp. Math., vol. 264, Amer. Math. Soc., Providence, RI, 2000, pp. 23–47. MR 1800686, DOI 10.1090/conm/264/04208
- Alexander Yu. Ol′shanskii and Mark V. Sapir, Length and area functions on groups and quasi-isometric Higman embeddings, Internat. J. Algebra Comput. 11 (2001), no. 2, 137–170. MR 1829048, DOI 10.1142/S0218196701000401
- Christophe Pittet, Isoperimetric inequalities in nilpotent groups, J. London Math. Soc. (2) 55 (1997), no. 3, 588–600. MR 1452267, DOI 10.1112/S0024610797005140
- V. N. Remeslennikov, Representation of finitely generated metabelian groups by matrices. , Algebra i Logika 8 (1969), 72–75 (Russian). MR 0283102
- E. Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), no. 1, 45–47. MR 642423, DOI 10.1112/blms/14.1.45
- Joseph J. Rotman, An introduction to the theory of groups, 3rd ed., Allyn and Bacon, Inc., Boston, MA, 1984. MR 745804
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Bertram A. F. Wehrfritz, On finitely generated soluble linear groups, Math. Z. 170 (1980), no. 2, 155–167. MR 562585, DOI 10.1007/BF01214771
Bibliographic Information
- G. N. Arzhantseva
- Affiliation: Section de Mathématiques, Université de Genève, CP 240, 1211 Genève 24, Switzerland
- Email: Goulnara.Arjantseva@math.unige.ch
- D. V. Osin
- Affiliation: Department of High Algebra, MEHMAT, Moscow State University, 119899 Moscow, Russia
- MR Author ID: 649248
- Email: Denis.Osin@mtu-net.ru
- Received by editor(s): August 2, 2000
- Received by editor(s) in revised form: October 13, 2000
- Published electronically: April 3, 2002
- Additional Notes: The work has been supported by the Swiss National Science Foundation
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3329-3348
- MSC (2000): Primary 20F69, 20F06, 20F65, 20F16, 20F05
- DOI: https://doi.org/10.1090/S0002-9947-02-02985-9
- MathSciNet review: 1897402