@TECHREPORT{Hironaka_mappingclasses, author = {Eriko Hironaka}, title = {Mapping classes associated to mixed-sign Coxeter graphs}, institution = {}, year = {} }

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Abstract

In this paper, we define and study properties of generalized Coxeter mapping classes on surfaces. Like the mapping classes associated to classical Coxeter graphs studied by Thurston and Leininger, the action on first homology is conjugate to the action of the Coxeter element of the associated Coxeter system considered as a reflection group, making classification and computations of invariants easier. However, unlike in the classical case, where dilatations of pseudo-Anosov examples are bounded from below by Lehmer’s number, the generalized Coxeter graphs may be used to construct pseudo-Anosov mapping classes with dilatation arbitrarily close to one. We observe that the smallest dilatation orientable pseudo-Anosov mapping classes of genus 2,3,4 and 5 found by Lanneau and Thiffeault can be realized as generalized Coxeter mapping classes, and that the smallest known accumulation point of normalized dilatations can be realized as the limit of normalized dilatations of a sequence of generalized Coxeter mapping classes. For the latter construction, we define non-classical periodic Coxeter mapping classes and use them as building blocks to define twisted mapping classes. We give sufficient conditions so that a sequence of twisted mapping classes corresponds to a convergent sequence on a fibered face. 1