Algebraic degrees of stretch factors in mapping class groups

Pseudo-Anosov;Stretch factor;Dilatation;Algebraic degree;Salem number;Train track

Shin, Hyunshik ; Margalit, Dan Mathematics Etnyre, John Garoufalidis, Stavros Ulmer, Douglas Wortman, Kevin ; Margalit, Dan

University:Georgia Institute of Technology

Department:Mathematics

关键词: Pseudo-Anosov;    Stretch factor;    Dilatation;    Algebraic degree;    Salem number;    Train track;   

Others  :  https://smartech.gatech.edu/bitstream/1853/51910/1/SHIN-DISSERTATION-2014.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

Given a closed surface Sg of genus g, a mapping class f in \MCG(Sg) is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number \lambda(f). The number \lambda(f) is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur.Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on Sg with orientable foliations whose stretch factor \lambda has algebraic degree 2g. Moreover, the stretch factor \lambda is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d≤g. Our examples also give a new approach to a conjecture of Penner.