On use of iterates of endomorphisms for constructing groups with
specific properties

R.I. Grigorchuk; M.J. Mamaghani

We produce a new approach for constructing groups with unusual properties such for example as to have intermediate growth or to be amenable. It is based on the use of iterations of endomorphism of a group. The particular case of the construction is when one starts the limit procedure with non-hopfian group. The example of Baumslag-Solitar group is considered carefully.

1. 19. . S. Andradakis, E. Raptis, and D. Varsos Hopficity of certain HNN-extensions// Comm. in Alg. V.20 N.5–1992 p.1511–1533

2. M. Day Amenable semigroups II, Ill.// J. Math. V.1–1957 p. 509–544

3. D. Gildenhuyse Classification of Solvable Groups of Cohomological dimension Two// Math. Z. V.166–1979 p.21–25

4. R.I. Grigorchuk On the Burnside problem about periodic groups// Funkts. Anal. Prilozhen. V.14 N.1–1980 p.53–54

5. R.I. Grigorchuk On Milnor's problem on group growth // Dokl. AN SSSR V. 271 N.1–1983 p.31–33

6. R.I. Grigorchuk Degrees of growth of finitely generated groups and theory of invariant means// Izv. Akad. Nauk SSSR, Ser. Mat. V.48 N.3–1984 p.417–481

7. R.I. Grigorchuk On rate of growth of $p$-groups and torsion-free groups// Math. Sbornik V.126 N.2--1986 p.194--214

8. R.I. Grigorchuk On the problem of M. Day on non-elementary amenable groups in the class of finitely presented groups // Mat. Zametki V.61 N.6–1996

9. M. Gromov Groups of polynomial growth and expanding maps // Publ. Math. IHES V.13–1981 p.53–73

10. N. Gupta, S. Sidki On the Burnside problem for periodic groups// Math. Z. V.182–1983 p.385–388

11. H. Kesten Symmetric random walks on groups// Trans. Amer. Math. Soc. V.92 N.2–1959 p.336–354

12. A.G. Kurosh The theory of groups, Second ed., Chelsea pub. co.,addr New-York, N.Y.–1960

13. R.C. Lyndon, P.E. Schupp Combinatorial group theory , Springer-Verlag,addr New-York, Berlin– 1977

14. I.G. Lysionok A system of defining relations for a Grigorchuk group// Math. Zametki V.38 N.4–1984 p.503–516

15. D. Meier Non-hopfian Groups// J. London Math.Soc. N. 2 V.26–1982 p.265–270

16. J. Milnor Problem 5603// Amer. Math. Monthly V.76–1968 p.686–688

17. G. Polya, G. Szego Problems and Theorems in Analysis , Springer-Verlag,addr Berlin, New-York– 1976

18. J. von Neumann Zur Algemeinen theorie des masses II // Fundam. Math. V.13–1929 p. 73–116

19. D.T. Wise A non-hopfian automatic group // J. of Alg. V. 180–1996 p. 845–847

	On use of iterates of endomorphisms for constructing groups with specific properties
Author	R.I. Grigorchuk, M.J. Mamaghani R.I. Grigorchuk,Steklov Mathematical InstituteVavilova St. 42 Moscow 117966, RUSSIAM.J. Mamaghani,Department of Mathematics and Statistics,Allameh Tabatabaii University,Ahmad Qasir St. 3, Tehran 15134 IRAN
Abstract	We produce a new approach for constructing groups with unusual properties such for example as to have intermediate growth or to be amenable. It is based on the use of iterations of endomorphism of a group. The particular case of the construction is when one starts the limit procedure with non-hopfian group. The example of Baumslag-Solitar group is considered carefully.
Keywords	group construction, intermediate growth, amenable groups, endomorphism iterations, non-hopfian group, Baumslag-Solitar group
DOI	doi:10.30970/ms.8.2.198-206
Reference	1. 19. . S. Andradakis, E. Raptis, and D. Varsos Hopficity of certain HNN-extensions// Comm. in Alg. V.20 N.5–1992 p.1511–1533 2. M. Day Amenable semigroups II, Ill.// J. Math. V.1–1957 p. 509–544 3. D. Gildenhuyse Classification of Solvable Groups of Cohomological dimension Two// Math. Z. V.166–1979 p.21–25 4. R.I. Grigorchuk On the Burnside problem about periodic groups// Funkts. Anal. Prilozhen. V.14 N.1–1980 p.53–54 5. R.I. Grigorchuk On Milnor's problem on group growth // Dokl. AN SSSR V. 271 N.1–1983 p.31–33 6. R.I. Grigorchuk Degrees of growth of finitely generated groups and theory of invariant means// Izv. Akad. Nauk SSSR, Ser. Mat. V.48 N.3–1984 p.417–481 7. R.I. Grigorchuk On rate of growth of $p$-groups and torsion-free groups// Math. Sbornik V.126 N.2--1986 p.194--214 8. R.I. Grigorchuk On the problem of M. Day on non-elementary amenable groups in the class of finitely presented groups // Mat. Zametki V.61 N.6–1996 9. M. Gromov Groups of polynomial growth and expanding maps // Publ. Math. IHES V.13–1981 p.53–73 10. N. Gupta, S. Sidki On the Burnside problem for periodic groups// Math. Z. V.182–1983 p.385–388 11. H. Kesten Symmetric random walks on groups// Trans. Amer. Math. Soc. V.92 N.2–1959 p.336–354 12. A.G. Kurosh The theory of groups, Second ed., Chelsea pub. co.,addr New-York, N.Y.–1960 13. R.C. Lyndon, P.E. Schupp Combinatorial group theory , Springer-Verlag,addr New-York, Berlin– 1977 14. I.G. Lysionok A system of defining relations for a Grigorchuk group// Math. Zametki V.38 N.4–1984 p.503–516 15. D. Meier Non-hopfian Groups// J. London Math.Soc. N. 2 V.26–1982 p.265–270 16. J. Milnor Problem 5603// Amer. Math. Monthly V.76–1968 p.686–688 17. G. Polya, G. Szego Problems and Theorems in Analysis , Springer-Verlag,addr Berlin, New-York– 1976 18. J. von Neumann Zur Algemeinen theorie des masses II // Fundam. Math. V.13–1929 p. 73–116 19. D.T. Wise A non-hopfian automatic group // J. of Alg. V. 180–1996 p. 845–847
Pages	198-206
Volume	8
Issue	2
Year	1997
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On use of iterates of endomorphisms for constructing groups with specific properties