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Mathematical Notes, Voi. 59, No. 4, 1996 The Class of Groups All of Whose Subgroups with Lesser Number

Summary: Mathematical Notes, Voi. 59, No. 4, 1996
The Class of Groups All of Whose Subgroups with Lesser Number
of Generators are Free is Generic
G. N. Arzhantseva and A. Yu. OFshanskii UDC 512.5
ABSTRACT. It is shown that, in a certain statistical sense, in almost every group with rn generators and n
relations (with m and n chosen), any subgroup generated by less than m elements (which need not belong to
the system of generators of the whole group) is free. In particular, this solves Problem 11.75 from the Kourov
Notebook. In the proof we introduce a new assumption on the defining relations stated in terms of finite marked
As is well known, in the fundamental group of an oriented dosed compact surface of genus g, all
subgroups of infinite index are free. In particular, this means that in the group
S, -- (al,...,a,, bx,... ,b, [ [ax,blJ...[a,,b,] = 1)
with 2g generators, any subgroup with lesser mlmber of generators (not necessarily from the original
set {al,..., a~, bl,..., bg}) is free. In this connection the second author posed the problem of finding
conditions on the defining relations under which the subgroups of a 6nitely defined group with a "small"
n,lmber of generators are free. Such conditions were found by V. S. Guba in the paper [1], where subgroups
with two generators were considered. Besides the condition of small cancellation, there was a condition
on "long 2-subwords" of defining words.
To study groups with free k-generated subgroups with k > 2, we had to find an appropriate gene-


Source: Arzhantseva, Goulnara N. - Section de Mathématiques, Université de Genève


Collections: Mathematics