Asymptotic Geometry, Bounded Generation and Subgroups of Mapping Class Groups

A group G is boundedly generated (by cyclic groups) if there is a finite ordered set (gi[special characters omitted], such that G = {[special characters omitted]}. Let (gi[special characters omitted] be a finite sequence of elements in [special characters omitted](S). For any word [special characters omitted] without obvious cancellation, we can estimate its stable length with [special characters omitted]|ni|. We demonstrate that if a subgroup of [special characters omitted](S) has exponential growth, it cannot be boundedly generated. Combining this with the Tits alternative for mapping class groups, we derive that a subgroup of [special characters omitted](S) is boundedly generated if and only if it is virtually abelian. Suppose further that each gi is non-elliptic, and that any gi and gj are not commensurable up to conjugacy if i ≠ j. We apply the same estimate to show that there exists m > 0 such that normal subgroup ⟨⟨[special characters omitted]|1 ≤ i ≤ k⟩⟩ is an infinitely generated right-angled Artin group provided n is sufficiently large. We use a different method to show that if G < [special characters omitted](S) has exponential growth, G cannot be boundedly generated by cyclic subgroups and/or curve stabilizers. That is G ≠ Γ1···Γ k if each Γi is either cyclic or a curve stabilizer