Quasi-Isometric Bounded Generation by Q-Rank-One Subgroups

Abstract

We say that a subset X quasi-isometrically boundedly generates a finitely generated group Γ if each element γ of a finite-index subgroup of Γ can be written as a product γ = x₁x₂⋯xᵣ of a bounded number of elements of X, such that the word length of each xᵢ is bounded by a constant times the word length of γ. A. Lubotzky, S. Mozes, and M.S. Raghunathan observed in 1993 that SL(n, ℤ) is quasi-isometrically boundedly generated by the elements of its natural SL(2, ℤ) subgroups. We generalize (a slightly weakened version of) this by showing that every S-arithmetic subgroup of an isotropic, almost-simple Q-group is quasi-isometrically boundedly generated by standard ℚ-rank-1 subgroups.