A Recipe for Short-word Pseudo-Anosovs, and Group Growth.

[摘要] The dissertation solves the short-word pseudo-Anosov problem posed by Fujiwara.Given any generating set of any pseudo-Anosov-containing subgroup of the mapping class group of a surface, we construct a pseudo-Anosov with word length bounded by a constant depending only on the surface. More generally, in any subgroup G we find an element f with the property that the minimal subsurface supporting a power of f is as large as possible for elements of G; the same constant bounds the word length of f. Along the way we find examples of all-pseudo-Anosov free subgroups quasi-isometrically embedded in the curve complex.Combined with a theorem of Fujiwara, the solution to the short-word problem yields, as a corollary, a new proof of the ;;strong;; Tits alternative for the mapping class group, which we also describe.

[发布机构] University of Michigan

[效力级别] Pseudo-Anosov

[关键词] Mapping Class Group;Pseudo-Anosov;Curve Complex;Mathematics;Science;Mathematics