Groups with no free subsemigroups
Authors:
P. Longobardi, M. Maj and A. H. Rhemtulla
Journal:
Trans. Amer. Math. Soc. 347 (1995), 14191427
MSC:
Primary 20F16; Secondary 20F60
MathSciNet review:
1277124
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Abstract: We look at groups which have no (nonabelian) free subsemigroups. It is known that a finitely generated solvable group with no free subsemigroup is nilpotentbyfinite. Conversely nilpotentbyfinite groups have no free subsemigroups. Torsionfree residually finite groups with no free subsemigroups can have very complicated structure, but with some extra condition on the subsemigroups of such a group one obtains satisfactory results. These results are applied to ordered groups, rightordered groups, and locally indicable groups.
 [1]
H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603614.
 [2]
R. Botto Mura and A.H. Rhemtulla, Orderable groups, Dekker, 1977.
 [3]
P.F. Conrad, Right ordered groups, Michigan Math. J. 6 (1959), 267275.
 [4]
L. Fuchs, Partially ordered algebraic systems, Pergamon Press, 1963.
 [5]
R. I. Grigorchuk, On the growth degrees of groups and torsionfree groups, Math. Sb. 126 (1985), 194214; English transl. Math. USSRSb. 54 (1986), 347352.
 [6]
R. I. Grigorchuk and A. Machi, An intermediate growth automorphism group of the real line, preprint.
 [7]
N. Gupta and S. Sidki, Some infinite groups, Algebra i Logika 22 (1983), 584589.
 [8]
E. Hecke, Lectures on the theory of algebraic numbers, Translated by George U. Brauer and Jay R. Goldman, SpringerVerlag, 1981.
 [9]
Y. K. Kim and A. H. Rhemtulla, Weak maximality condition and polycyclic groups.
 [10]
P. H. Kropholler, Amenability and right orderable groups, Bull. London Math. Soc. 25 (1993), 347352.
 [11]
J. Milnor, Growth of finitely generated solvable groups, J. Differential Geom. 2 (1968), 447449.
 [12]
J. M. Rosenblatt, Invariant measures and growth conditions, Trans. Amer. Math. Soc. 197 (1974), 3353.
 [13]
A. Shalev, Combinatorial conditions in residually finite groups, II, J. Algebra 157 (1993), 5162.
 [14]
S. Wagon, The BanachTarski paradox, Cambridge University Press, 1985.
 [15]
J. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geom. 2 (1968), 421446.
 [16]
E. I. Zelmanov, On some problems of group theory and lie algebras, Math. USSRSb. 66 (1990), 159168.
 [1]
 H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603614.
 [2]
 R. Botto Mura and A.H. Rhemtulla, Orderable groups, Dekker, 1977.
 [3]
 P.F. Conrad, Right ordered groups, Michigan Math. J. 6 (1959), 267275.
 [4]
 L. Fuchs, Partially ordered algebraic systems, Pergamon Press, 1963.
 [5]
 R. I. Grigorchuk, On the growth degrees of groups and torsionfree groups, Math. Sb. 126 (1985), 194214; English transl. Math. USSRSb. 54 (1986), 347352.
 [6]
 R. I. Grigorchuk and A. Machi, An intermediate growth automorphism group of the real line, preprint.
 [7]
 N. Gupta and S. Sidki, Some infinite groups, Algebra i Logika 22 (1983), 584589.
 [8]
 E. Hecke, Lectures on the theory of algebraic numbers, Translated by George U. Brauer and Jay R. Goldman, SpringerVerlag, 1981.
 [9]
 Y. K. Kim and A. H. Rhemtulla, Weak maximality condition and polycyclic groups.
 [10]
 P. H. Kropholler, Amenability and right orderable groups, Bull. London Math. Soc. 25 (1993), 347352.
 [11]
 J. Milnor, Growth of finitely generated solvable groups, J. Differential Geom. 2 (1968), 447449.
 [12]
 J. M. Rosenblatt, Invariant measures and growth conditions, Trans. Amer. Math. Soc. 197 (1974), 3353.
 [13]
 A. Shalev, Combinatorial conditions in residually finite groups, II, J. Algebra 157 (1993), 5162.
 [14]
 S. Wagon, The BanachTarski paradox, Cambridge University Press, 1985.
 [15]
 J. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geom. 2 (1968), 421446.
 [16]
 E. I. Zelmanov, On some problems of group theory and lie algebras, Math. USSRSb. 66 (1990), 159168.
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DOI:
http://dx.doi.org/10.1090/S00029947199512771245
PII:
S 00029947(1995)12771245
Article copyright:
© Copyright 1995
American Mathematical Society
