Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents

Abstract

Let S S be a compact orientable surface, and M o d ( S ) \mathrm {Mod}(S) its mapping class group. Then there exists a constant M ( S ) M(S) , which depends on S S , with the following property. Suppose a , b ∈ M o d ( S ) a,b \in \mathrm {Mod}(S) are independent (i.e., [ a n , b m ] ≠ 1 [a^n,b^m]\not =1 for any n , m ≠ 0 n,m \not =0 ) pseudo-Anosov elements. Then for any n , m ≥ M n,m \ge M , the subgroup ⟨ a n , b m ⟩ \langle a^n,b^m \rangle is a free group freely generated by a n a^n and b m b^m , and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in ⟨ a n , b m ⟩ \langle a^n,b^m \rangle are pseudo-Anosov. We also show that there exists a constant N N , which depends on a , b a,b , such that ⟨ a n , b m ⟩ \langle a^n,b^m \rangle is a free group freely generated by a n a^n and b m b^m , and convex-cocompact if | n | + | m | ≥ N |n|+|m| \ge N and n m ≠ 0 nm \not =0 .