The Groups of Isometries of Metric Spaces over Vector Groups

In this paper, we consider the groups of isometries of metric spaces arising from finitely generated additive abelian groups. Let A be a finitely generated additive abelian group. Let R={1,ϱ} where ϱ is a reflection at the origin and T={ta:A→A,ta(x)=x+a,a∈A}. We show that (1) for any finitely generated additive abelian group A and finite generating set S with 0∉S and −S=S, the maximum subgroup of IsomX(A,S) is RT; (2) D⊴RT if and only if D≤T or D=RT′ where T′={h2:h∈T}; (3) for the vector groups over integers with finite generating set S={u∈Zn:|u|=1}, IsomX(Zn,S)=On(Z)Zn. The paper also includes a few intermediate technical results.