A Remark on the Bers Type of Some Self-Maps of Riemann Surfaces with Two Specified Points

A Reducibility Problem for Monodromy of Some Surface Bundles

“…Before writing the current version of the present paper, we showed that the statement of the theorem is true for n = 2. This was announced at talks at the Fukuoka Conference in 1999 (see Imayoshi-Ito-Yamamoto [6]) and at Ichinoseki 2000; a detailed treatment of this result appears in Imayoshi-Ito-Yamamoto [7]. It should be noted that these studies distinguished topologically the four types (elliptic, parabolic, hyperbolic, pseudohyperbolic) of Bers [3], rather than just the three types (reducible, pseudo-Anosov, of finite order) of Thurston [2].…”

Least dilatation of pure surface braids

“…The second important component of the proof of Theorem 6.1 is Theorem 2.4 below. The statement and proof of Theorem 2.4 in the case of n = 2 are due to Imayoshi-Ito-Yamamoto [21] with a weaker upper bound on the quasiconformal dilatation. The proof of Imayoshi-Ito-Yamamoto holds in the case of n > 2 punctures without any modification so we will omit the full argument and will instead provide a sketch of proof, namely the construction of F t and a justification of our improved upper bound on K t = K(F t ).…”

Pseudo-Anosov mapping classes and their representations by products of two Dehn twists

“…Here we recall the main theorem of Imayoshi, Ito and Yamamoto [8]. Denote M = S × S, a = {a, a ′ }, and ∆ = {(x, y) ∈ M : x = y}.…”

“…Then s 1 = F (a, t) and s 2 = F (a ′ , t), where 1 ≤ t ≤ 1, are closed curve on S, which define a pure braids [b F ] represented by b F = (s 1 , s 2 ) in the fundamental group π 1 (M − ∆, a). By Theorem 1.3 and the main theorem of [8], we obtain infinitely many essential pure braids [b F ] so that s 1 and s 2 are nontrivial and nonparallel.…”