A recipe for short-word pseudo-Anosovs (Q2845591)

In the article under review, the author affirmatively answers a question of Fujiwara, showing that if \(S\) is an orientable hyperbolic surface, there exists a \(K=K(S)\) such that if \(G\) is any subgroup of the mapping class group of \(S\) which contains a pseudo-Anosov element and which is finitely generated by a set \(\Sigma\), then \(G\) contains a pseudo-Ansov element of \(\Sigma\)--length at most \(K\).NEWLINENEWLINEThe author also shows the existence of a constant \(Q=Q(S)\) such that if \(a\) and \(b\) are reducible mapping classes which are ``sufficiently different'' in some technical sense, then for every \(n,m\geq Q\), the group \(G=\langle a^n,b^m\rangle\) is free of rank two and consists entirely of pseudo-Anosov elements aside from the elements which are conjugate to powers of \(a^n\) and \(b^m\). Furthermore, all finitely generated purely pseudo-Anosov subgroups of \(G\) are convex cocompact.NEWLINENEWLINEThe proof relies on the hyperbolicity of the curve complex of \(S\).