Minimal dilatation in Penner’s construction

Abstract

For all orientable closed surfaces, we determine the minimal dilatation among mapping classes arising from Penner’s construction. We also discuss generalisations to surfaces with punctures.

Minimal Penner dilatations on nonorientable surfaces

“…A remarkable difference between the case of orientable and nonorientable surfaces is that in the case of orientable closed surfaces S g , the limit of the sequence of minimal dilatations δ P (S g ) arising from Penner's construction exists, whereas in the nonorientable case, the limit of the sequence δ P (N g ) does not. An orientable double cover argument implies that any accumulation point of δ P (N g ) must be at least 3 + 2 √ 2, which is the limit in the orientable case by work of the first author [10]. In fact, the limit in the orientable case is the same as for the even genus subsequence in the nonorientable case.…”

“…They will be used later on. Lemma 2.4 hints at why searching the minimal dilatation among pseudo-Anosov mapping classes arising from Penner's construction is more complicated on nonorientable surfaces than on orientable ones: the intersection graph of the curves used in the construction always contains at least one cycle, while for the minimising examples on closed orientable surfaces, it is a path [10].…”

“…We will often use the following lemma to obtain lower bounds for the dilatation of mapping classes arising from Penner's construction. It was implicitly used already in the case of orientable surfaces by the first author [10].…”

Pseudo-Anosov maps with small stretch factors on punctured surfaces

“…After Penner, there has been many works aiming to make the constants A 1 , A 2 more precise [3,16,31,15], to find the exact value of l g,n for small values of g and n [38,13,6,19,20], or to find the asymptotic of least stretch factor when restricted to certain subgroups or subsets of the mapping class group [37,7,4,14,1,25]. See also [30,22,26,24,23,44] for other related research.…”