Bounded automorphisms and quasi-isometries of finitely generated groups

Abstract

Let Γ be a finitely generated infinite group. Denote by K (Γ) the FC-centre of Γ, i.e. the subgroup of all elements of Γ having only finitely many conjugates in Γ. Let QI(Γ) denote the group of quasi-isometries of Γ with respect to a word metric. We prove that the natural homomorphism θ Γ: Aut(Γ) → QI(Γ) is a monomorphism only if K (Γ) equals the centre Z (Γ of Γ. The converse holds if K (Γ) = Z (Γ) is torsion-free. When K (Γ) is finite we show that is a monomorphism where Γ= Γ/K (Γ). We apply this criterion to a number of classes of groups arising in combinatorial and geometric group theory.