Non-amenable finitely presented torsion-by-cyclic groups
Authors:
A. Yu. Ol’shanskii and M. V. Sapir
Journal:
Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 63-71
MSC (2000):
Primary 20F05, 43A07
DOI:
https://doi.org/10.1090/S1079-6762-01-00095-6
Published electronically:
July 3, 2001
MathSciNet review:
1852901
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We construct a finitely presented non-amenable group without free non-cyclic subgroups thus providing a finitely presented counterexample to von Neumann’s problem. Our group is an extension of a group of finite exponent $n\gg 1$ by a cyclic group, so it satisfies the identity $[x,y]^n=1$.
- S. I. Adyan, Random walks on free periodic groups, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1139–1149, 1343 (Russian). MR 682486
- S. I. Adjan, Periodic products of groups, Trudy Mat. Inst. Steklov. 142 (1976), 3–21, 268 (Russian). Number theory, mathematical analysis and their applications. MR 532668
BT S. Banach, A. Tarski. Sur la décomposition des ensembles de points en parties respectivement congruentes. Fund. Math 6 (1924), 244–277.
BORS J. C. Birget, A. Yu. Ol’shanskii, E. Rips, M. V. Sapir. Isoperimetric functions of groups and computational complexity of the word problem, 1998 (submitted to Annals of Mathematics), preprint available at http://www.math.vanderbilt.edu/$\sim$msapir/publications.html.
- Matthew G. Brin and Craig C. Squier, Groups of piecewise linear homeomorphisms of the real line, Invent. Math. 79 (1985), no. 3, 485–498. MR 782231, DOI https://doi.org/10.1007/BF01388519
- J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438
- A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
- Joel M. Cohen, Cogrowth and amenability of discrete groups, J. Funct. Anal. 48 (1982), no. 3, 301–309. MR 678175, DOI https://doi.org/10.1016/0022-1236%2882%2990090-8
day Mahlon M. Day. Amenable semigroups. Illinois J. Math. 1 (1957), 509–544.
- Open problems in infinite-dimensional topology, Topology Proc. 4 (1979), no. 1, 287–338 (1980). MR 583711
- Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
- R. I. Grigorchuk, Symmetrical random walks on discrete groups, Multicomponent random systems, Adv. Probab. Related Topics, vol. 6, Dekker, New York, 1980, pp. 285–325. MR 599539
- R. I. Grigorchuk, An example of a finitely presented amenable group that does not belong to the class EG, Mat. Sb. 189 (1998), no. 1, 79–100 (Russian, with Russian summary); English transl., Sb. Math. 189 (1998), no. 1-2, 75–95. MR 1616436, DOI https://doi.org/10.1070/SM1998v189n01ABEH000293
Haus F. Hausdorff. Grundzüge der Mengenlehre. Leipzig, 1914.
- S. V. Ivanov and A. Yu. Ol′shanskiĭ, Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091–2138. MR 1327257, DOI https://doi.org/10.1090/S0002-9947-96-01510-3
- Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI https://doi.org/10.7146/math.scand.a-10568
- O. G. Kharlampovich and M. V. Sapir, Algorithmic problems in varieties, Internat. J. Algebra Comput. 5 (1995), no. 4-5, 379–602. MR 1361261, DOI https://doi.org/10.1142/S0218196795000227
- V. D. Mazurov, Yu. I. Merzlyakov, and V. A. Churkin (eds.), Kourovskaya tetrad′, 8th ed., Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1982 (Russian). Nereshennye voprosy teorii grupp. [Unsolved problems in group theory]; Including problems from the 8th All-Union symposium on group theory held at Sumy, 1982. MR 725169
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. MR 0577064
vN J. von Neumann. Zur allgemeinen Theorie des Masses. Fund. Math. 13 (1929), 73–116.
AN P. S. Novikov and S. I. Adian. Infinite periodic groups. I, II, III, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 212–244, 251–524, 709–731. ; ;
- A. Ju. Ol′šanskiĭ, An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1328–1393 (Russian). MR 567039
- A. Ju. Ol′šanskiĭ, An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 2, 309–321, 479 (Russian). MR 571100
- A. Ju. Ol′šanskiĭ, On the question of the existence of an invariant mean on a group, Uspekhi Mat. Nauk 35 (1980), no. 4(214), 199–200 (Russian). MR 586204
- A. Yu. Ol′shanskiĭ, Geometriya opredelyayushchikh sootnosheniĭ v gruppakh, Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1024791
- A. Yu. Ol′shanskiĭ, ${\rm SQ}$-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 8, 1199–1211. MR 1357360, DOI https://doi.org/10.1070/SM1995v186n08ABEH000063
OS A. Yu. Ol’shanskii, M. V. Sapir. Embeddings of relatively free groups into finitely presented groups. Contemporary Mathematics, 264 (2000), 23–47.
talk A. Yu. Ol’shanskii and M. V. Sapir. Length and area functions on groups and quasi-isometric Higman embeddings. To appear, IJAC, 2000.
OSamen A. Yu. Ol’shanskii and M. V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. (Submitted).
SBR M. V. Sapir, J. C. Birget, E. Rips. Isoperimetric and isodiametric functions of groups, 1997, submitted to Annals of Mathematics, preprint available at http://www.math.vanderbilt.edu/$\sim$msapir/publications.html.
- Wilhelm Specht, Zur Theorie der messbaren Gruppen, Math. Z. 74 (1960), 325–366 (German). MR 140641, DOI https://doi.org/10.1007/BF01180494
- J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI https://doi.org/10.1016/0021-8693%2872%2990058-0
Adian S. I. Adian. Random walks on free periodic groups. Izv. Akad. Nauk SSSR, Ser. Mat. 46 (1982), 1139–1149.
Adian2 S. I. Adian. Periodic products of groups. Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov. 142 (1976), 3–21, 268.
BT S. Banach, A. Tarski. Sur la décomposition des ensembles de points en parties respectivement congruentes. Fund. Math 6 (1924), 244–277.
BORS J. C. Birget, A. Yu. Ol’shanskii, E. Rips, M. V. Sapir. Isoperimetric functions of groups and computational complexity of the word problem, 1998 (submitted to Annals of Mathematics), preprint available at http://www.math.vanderbilt.edu/$\sim$msapir/publications.html.
BS M. G. Brin and C. C. Squier. Groups of piecewise linear homeomorphisms of the real line. Invent. Math. 79 (1985), 485–498.
CFP J. W. Cannon, W. J. Floyd and W. R. Parry. Introductory notes on Richard Thompson’s groups. L’Enseignement Mathématique (2) 42 (1996), 215–256.
Cliff-Prest A.H. Clifford and G.B. Preston, The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7. American Mathematical Society, Providence, R.I. 1961.
Cohen J. M. Cohen. Cogrowth and amenability of discrete groups. J. Funct. Anal. 48 (1982), no. 3, 301–309.
day Mahlon M. Day. Amenable semigroups. Illinois J. Math. 1 (1957), 509–544.
Geogh Open problems in infinite-dimensional topology. Edited by Ross Geoghegan. The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979). Topology Proc. 4 (1979), no. 1, 287–338 (1980).
Greenleaf F. P. Greenleaf. Invariant means on topological groups and their applications. Van Nostrand Reinhold, New York, 1969.
Grigorchuk R. I. Grigorchuk. Symmetrical random walks on discrete groups. Multicomponent random systems. Adv. Probab. Related Topics, 6, Dekker, New York, 1980, 285–325.
Grig R. I. Grigorchuk. An example of a finitely presented amenable group that does not belong to the class EG. Mat. Sb. 189 (1) (1998), 79–100.
Haus F. Hausdorff. Grundzüge der Mengenlehre. Leipzig, 1914.
OlIvanov S. V. Ivanov and A. Y. Ol’shanskii. Hyperbolic groups and their quotients of bounded exponents. Trans. Amer. Math. Soc. 348 (1996), no. 6, 2091–2138.
Kesten2 Harry Kesten. Full Banach mean values on countable groups. Math. Scand. 7 (1959), 146–156.
KhSap O. G. Kharlampovich and M. V. Sapir. Algorithmic problems in varieties. Internat. J. Algebra Comput. 5 (1995), no. 4-5, 379–602.
Kour Kourovka Notebook. Unsolved Problems in Group Theory. 8th edition, Novosibirsk, 1982.
LS Roger Lyndon and Paul Schupp. Combinatorial group theory. Springer-Verlag, 1977.
vN J. von Neumann. Zur allgemeinen Theorie des Masses. Fund. Math. 13 (1929), 73–116.
AN P. S. Novikov and S. I. Adian. Infinite periodic groups. I, II, III, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 212–244, 251–524, 709–731. ; ;
OlTarA. Yu. Ol’shanskii. An infinite simple torsion-free Noetherian group. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1328–1393.
OlTar1 A. Yu. Ol’shanskii. An infinite group with subgroups of prime order. Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 2, 309–321.
OlAmen A. Yu. Ol’shanskii. On the question of the existence of an invariant mean on a group. (Russian) Uspekhi Mat. Nauk 35 (1980), no. 4 (214), 199–200.
book A. Yu. Ol’shanskii. The geometry of defining relations in groups, Nauka, Moscow, 1989.
Ol95 A. Yu. Ol’shanskii. The SQ-universality of hyperbolic groups, Mat. Sb. 186 (1995), no. 8, 119–132.
OS A. Yu. Ol’shanskii, M. V. Sapir. Embeddings of relatively free groups into finitely presented groups. Contemporary Mathematics, 264 (2000), 23–47.
talk A. Yu. Ol’shanskii and M. V. Sapir. Length and area functions on groups and quasi-isometric Higman embeddings. To appear, IJAC, 2000.
OSamen A. Yu. Ol’shanskii and M. V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. (Submitted).
SBR M. V. Sapir, J. C. Birget, E. Rips. Isoperimetric and isodiametric functions of groups, 1997, submitted to Annals of Mathematics, preprint available at http://www.math.vanderbilt.edu/$\sim$msapir/publications.html.
Specht W. Specht. Zur Theorie der messbaren Gruppen. Math. Z. 74 (1960), 325–366.
Tits J. Tits. Free subgroups in linear groups. J. Algebra 20 (1972), 250–270.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (2000):
20F05,
43A07
Retrieve articles in all journals
with MSC (2000):
20F05,
43A07
Additional Information
A. Yu. Ol’shanskii
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, and Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia
MR Author ID:
196218
Email:
olsh@math.vanderbilt.edu, olshan@shabol.math.msu.su
M. V. Sapir
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville, TN 37240
MR Author ID:
189574
Email:
msapir@math.vanderbilt.edu
Keywords:
Amenable group,
Burnside groups,
free subgroups
Received by editor(s):
January 9, 2001
Published electronically:
July 3, 2001
Additional Notes:
Both authors were supported in part by the NSF grant DMS 0072307. In addition, the research of the first author was supported in part by the Russian fund for fundamental research 99-01-00894, and the research of the second author was supported in part by the NSF grant DMS 9978802.
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2001
American Mathematical Society
