RELATION MODULES OF INFINITE GROUPS

Abstract

Let F_n be the free group of rank n with basis x₁, x₂, …, x_n, and let d(G) denote the minimal number of generators of the finitely generated group G. Suppose that n[ges ]d(G). There exists an exact sequence

formula here

and we may view the free abelian group R=R/R′ as a right ℤG-module by defining (rR′)^g =r^gϕ−1R′ for all g∈G, where gϕ⁻¹ is any preimage of g under ϕ, and r^gϕ−1 =(gϕ⁻¹)⁻¹r(gϕ⁻¹), the conjugate of r by gϕ⁻¹. We call R the relation module of G associated with the presentation (1), and say that R has ambient rank n. Furthermore, we call the group F_n/R′ the free abelianized extension of G associated with (1).