Minimal Penner dilatations on nonorientable surfaces

Abstract

For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner’s construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points, in contrast to the case of orientable surfaces where there is only one accumulation point. One of our key techniques is representing pseudo-Anosov dilatations as roots of Alexander polynomials of fibered links and comparing dilatations using the skein relation for Alexander polynomials.

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.…”

Minimal pseudo-Anosov stretch factors on nonoriented surfaces

“…We expect the limits to be different for other genus sequences. For example, in a forthcoming paper [LS18b] we show that if δ P (N g ) denotes the minimal stretch factor among pseudo-Anosov mapping classes on N g obtained from Penner's construction, then the sequence δ P (N g ) has exactly two accumulation points as g → ∞. One accumulation point, (1 + √ 2) 2 , is the limit for the sequence restricted to even g. The other accumulation point, strictly greater than (1 + √ 2) 2 , is the limit for the sequence restricted to odd g. We expect this dichotomy to be indicative how the sequence (δ + (N g )) g behaves for odd and even genus sequences, respectively, since so far no pseudo-Anosov mapping class of a nonorientable surface is known to not have a power arising from Penner's construction (compare with Question 1.10 below).…”