This paper describes a family of pseudo-Anosov braids with small dilatation. The smallest dilatations occurring for braids with 3,4 and 5 strands appear in this family. A pseudo-Anosov braid with 2g + 1 strands determines a hyperelliptic mapping class with the same dilatation on a genus-g surface. Penner showed that logarithms of least dilatations of pseudo-Anosov maps on a genus-g surface grow asymptotically with the genus like 1/g, and gave explicit examples of mapping classes with dilatations bounded above by log 11/g. Bauer later improved this bound to log 6/g. The braids in this paper give rise to mapping classes with dilatations bounded above by log(2 + √ 3)/g. They show that least dilatations for hyperelliptic mapping classes have the same asymptotic behavior as for general mapping classes on genus-g surfaces. 37E30, 57M50 IntroductionIn this paper, we study a family of generalizations of these examples to arbitrary numbers of strands. Let B(D, s) denote the braid group on D with s strands, where D denotes a 2-dimensional closed disk. First consider the braids β m,n in B(D, m + n + 1) given byMatsuoka's example [22] appears as β 1,1 , and Ko, Los and Song's example [18] as β 2,1 . For any m, n ≥ 1, β m,n is pseudo-Anosov (Theorem 3.9). The dilatations of β m,m coincide with those found by Brinkmann [7] (see also Section 4.2), who also shows that the dilatations arising in this family can be made arbitrarily close to 1.It turns out that one may find smaller dilatations by passing a strand of β m,n once around the remaining strands. As a particular example, we consider the braids σ m,n defined by taking the rightmost-strand of β m,n and passing it counter-clockwise once around the remaining strands. Figure 1 gives an illustration of β m,n and σ m,n . The braid σ 1,3 is conjugate to Ham and Song's braid σ 1 σ 2 σ 3 σ 4 σ 1 σ 2 . For |m − n| ≤ 1, we show that σ m,n is periodic or reducible. Otherwise σ m,n is pseudo-Anosov with dilatation strictly less than the dilatation of β m,n (Theorem 3.11, Corollary 3.32). The dilatations of σ g−1,g+1 (g ≥ 2) satisfy the inequalityLet M s g denote the set of mapping classes (or isotopy classes) of homeomorphisms on the closed orientable genus-g surface F g set-wise preserving s points. We denote M 0 g by M g . For any subset Γ ⊂ M s g , define λ(Γ) to be the least dilatation among pseudo-Anosov elements of Γ, and let δ(Γ) be the logarithm of λ(Γ). g . An element of M g is called hyperelliptic if it commutes with an involution ι on F g such that the quotient of F g by ι is S 2 . Let M g,hyp ⊂ M g denote the subset of hyperelliptic elements of M g . Any pseudo-Anosov braid on 2g + 1 strands determines a hyperelliptic element of M g with the same dilatation (Proposition 2.10). Thus, (1) implies:This improves the upper bounds on δ(M g ) found by Penner This paper is organized as follows. Section 2 reviews basic terminology and results on mapping class groups. In Section 3, we determine the Thurston-Nielsen types of β m,n and σ m,n by finding efficient graph maps for their m...
We denote by δg (resp. δ + g ), the minimal dilatation for pseudo-Anosovs (resp. pseudo-Anosovs with orientable invariant foliations) on a closed surface of genus g. This paper concerns the pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the Whitehead sister link exterior W by Dehn filling two cusps, where the fillings are on the boundary slopes of fibers of W . We give upper bounds of δg for g ≡ 0, 1, 5, 6, 7, 9 (mod 10), δ + g for g ≡ 1, 5, 7, 9 (mod 10). Our bounds improve the previous one given by Hironaka. We note that the monodromies of fibrations on W were also studied by Aaber and Dunfield independently.
We consider a surface bundle over the circle, the so called magic manifold M . We determine homology classes whose minimal representatives are genus 0 fiber surfaces for M , and describe their monodromies by braids. Among those classes whose representatives have n punctures for each n, we decide which one realizes the minimal entropy. We show that for each n ≥ 9 (resp. n = 3, 4, 5, 7, 8), there exists a pseudo-Anosov homeomorphism Φ n : D n → D n with the smallest known entropy (resp. the smallest entropy) which occurs as the monodromy on an n-punctured disk fiber for the Dehn filling of M . A pseudo-Anosov homeomorphism Φ 6 : D 6 → D 6 with the smallest entropy occurs as the monodromy on a 6-punctured disk fiber for M .(Proposition 4.14). If gcd(m − 1, p) = 1, then the mapping torus T(Γ(T m,p )) is homeomorphic to M magic (Corollary 3.28). Otherwise Γ(T m,p ) is reducible. We setThe dilatation λ(b) ≈ 1.72208 equals the largest real root of f (5,3,2) (t) = t 6 − t 5 − 2t 3 − t + 1.(3b-ii) Γ(T 8k+7,2k+1 ) in case n = 8(k + 1) for k ≥ 1. The dilatation λ(T 8k+7,2k+1 ) equals the largest real root of f (4k+5,4k+1,0) (t) = t 8k+6 − 2(t 4k+1 + t 4k+5 ) + 1.
We discuss a comparison of the entropy of pseudo-Anosov maps and the volume of their mapping tori. Recent study of the Weil-Petersson geometry of Teichmüller space tells us that the entropy and volume admit linear inequalities for both directions under some bounded geometry condition. Based on experiments, we present various observations on the relation between minimal entropies and volumes, and on bounding constants for the entropy over the volume from below. We also provide explicit bounding constants for a punctured torus case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.