Cogrowth and Essentiality in Groups and Algebras

The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects non-trivially every non-trivial subgroup. The main result of this paper is that every function f : N ∪ {0}−→N which is strictly increasing, but at most exponential, is equivalent to a cogrowth function of an essential subgroup of infinite index of the free group of rank two. This class of functions properly contains the class of growth functions of groups. The notions of growth and cogrowth of right ideals in algebras are introduced. We show that when the algebra is without zero divisors then every right ideal, whose cogrowth is less than that of the algebra, is essential. 1 Growth, Cogrowth and Essentiality in Groups 1.1 Growth and Cogrowth of Subgroups A growth function ΓS(n) on a set S with a length function l on it is defined by ΓS(n) := card{s ∈ S | l(s) ≤ n}, (1) assuming that ΓS(n) is finite for each n. A preorder is given on the growth functions by Γ1(n) Γ2(n) ⇐⇒ ∃C [Γ1(n) ≤ Γ2(Cn) ]. (2) The notion of growth when applied to finitely generated groups (see [6] for an overview) has been investigated mainly after Milnor’s paper ([12]). A geometric interpretation can be given, for example, when computing the growth function Supported by the Minerva Fellowship