A. Yu. Ol'shanskii

Isoperimetric functions of groups and computational complexity of the word problem

We prove that the word problem of a finitely generated group G is in NP (solvable in polynomial time by a nondeterministic Turing machine) if and only if this group is a subgroup of a finitely presented group H with polynomial isoperimetric function. The embedding can be chosen in such a way that G has bounded distortion in H. This completes the work started in [6] and [25].

Large groups and their periodic quotients

We first give a short group theoretic proof of the following result of Lackenby. If G is a large group, H is a finite index subgroup of G admitting an epimorphism onto a non-cyclic free group, and g 1 ,..., g k are elements of H, then the quotient of G by the normal subgroup generated by g n 1 ,..., g n k is large for all but finitely many n ∈ Z. In the second part of this note we use similar methods to show that for every infinite sequence of primes (p 1 , p 2 ,...), there exists an infinite finitely generated periodic group Q with descending normal series Q = Q 0 >Q 1 >..., such that ∩ i Q i = {1} and Q i-1 /Q i is either trivial or abelian of exponent p i .

The class of groups all of whose subgroups with lesser number of generators are free is generic

It is shown that, in a certain statistical sense, in almost every group withm generators andn relations (withm andn chosen), any subgroup generated by less thanm 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 groups.

The number of generators and orders of Abelian subgroups of finite p-groups

Let f(F) be the smallest function such that every finite p-group, all of whose Abelian subgroups are generated by at most n elements (all of whose Abelian subgroups have orders at most pn, has at most f(n) generators (has order not exceeding pF(n)). It is established that the functions f and F have quadratic order of growth.

Subnormal subgroups in free groups, their growth and cogrowth

In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (3) gives sharp estimates from below for the e.g.r. of subnormal subgroups in free groups; and (4) finds cogrowth for the subnormal closures of free generators in F.

Imbedding of countable periodic groups into simple 2-generated periodic groups

We prove a theorem on the isomorphic imbedding of an arbitrary countable periodic group H into a simple 2-generated periodic group G. In addition, we show that for any integers k ≥ 2 and ℓ ≥ 3 the group G contains a pair of generating elements whose orders are k and ℓ.

Imbedding of periodic groups in simple periodic groups

It is proved that each periodic group is isomorphic to a subgroup of some simple periodic group.