A Note on Pseudo-Anosov Maps with Small Growth Rate

We present an explicit sequence of pseudo-Anosov maps φk : S2k → S2k of surfaces of genus 2k whose growth rates converge to one. Introduction In this note, we present an explicit sequence φk of pseudo-Anosov of surfaces of genus 2k whose growth rates converge to one. This answers a question of Joan Birman, who had previously asked whether such growth rates are bounded away from one. Norbert A’Campo, Mladen Bestvina, and Klaus Johannson independently communicated this question to me. McMullen previously obtained a similar result using quite different techniques [McM00]. The growth of the genus is not an artifact of our construction. For a surface S of fixed genus g, the growth rates of pseudo-Anosov maps of S are clearly bounded away from one, for they are Perron-Frobenius eigenvalues of irreducible integral m ×m matrices, with m ≤ 6g − 3 [BH95]. Finding the smallest possible growth rate for each genus is an interesting problem that remains open. One curious observation, due to Norbert A’Campo, is that our sequence of pseudo-Anosov maps is a sequence of monodromies of w-slalom knots, as defined in [A’C98]. In Section 1, we review the part of the theory of train tracks [BH92, BH95] that we use in this paper. Section 2 explains the intuition that led to the result, and Section 3 contains the statement and proof of the main results (Theorem 3.2 and Corollary 3.3). The result of this paper grew out of massive computer experiments with my software package XTrain [Bri00, BS01] in the context of the REU program