ELECTRONIC RESEARCH ANNOUNCEMENTS doi:10.3934/era.2011.18.97 IN MATHEMATICAL SCIENCES Volume 18, Pages 97{111 (August 31, 2011) S 1935-9179 AIMS (2011) SIMPLE LOOPS ON 2-BRIDGE SPHERES IN 2-BRIDGE LINK COMPLEMENTS DONGHI LEE AND MAKOTO SAKUMA (Communicated by Walter Neumann) Abstract. The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when are they homotopic in the link complement? We also announce applications of these results to character varieties and McShane's identity. 1. Introduction Let K be a knot or a link in S3 and S a punctured sphere in the complement S3 − K obtained from a bridge sphere of K. Then the following natural question arises. Question 1.1. (1) Which essential simple loops on S are null-homotopic in S3−K? (2) For two distinct essential simple loops on S, when are they homotopic in S3 − K? A refined version of the first question for 2-bridge spheres of 2-bridge links was proposed in the second author's joint work with Ohtsuki and Riley [20, Ques- tion 9.1(2)], in relation with epimorphisms between 2-bridge links. It may be re- garded as a special variation of a question raised by Minsky [9, Question 5.4] on essential simple loops on Heegaard surfaces of 3-manifolds. The purpose of this note is to announce a complete answer to Question 1.1 for 2- bridge spheres of 2-bridge links, established by the series of papers [11, 12, 13, 14], and to explain its application to the study of character varieties and McShane's identity [15]. The key tool for solving the question is small cancellation theory, applied to two-generator and one-relator presentations of 2-bridge link groups. We note that Received by the editors April 19, 2011 and, in revised form, June 24, 2011. 2000 Mathematics Subject Classification. Primary: 57M25; Secondary: 20F06 . Key words and phrases. 2-bridge link groups, character varieties, McShane's identity, small cancellation theory. The authors would like to thank Norbert A'Campo, Hirotaka Akiyoshi, Brian Bowditch, Danny Calegari, Max Forester, Koji Fujiwara, Yair Minsky, Toshihiro Nakanishi, Caroline Series and Ser Peow Tan for stimulating conversations. The first author was supported by a 2-Year Research Grant of Pusan National University. The second author was supported by JSPS Grants-in-Aid 22340013 and 21654011. c 2011 American Institute of Mathematical Sciences 97 98 DONGHI LEE AND MAKOTO SAKUMA Weinbaum [32] and Appel and Schupp [5] have used small cancellation theory to prove that the word and conjugacy problems for prime alternating link groups are solvable (see also [10] and references in it). Moreover, it was shown by Sela [24] and Pr´eaux[21] that the word and conjugacy problems are solvable for any link group. A characteristic feature of our work is that it gives complete answers to special (but also natural) word and conjugacy problems for the link groups of 2-bridge links, which form a special (but also important) family of prime alternating links. (See [1,4] for the role of 2-bridge links in Kleinian group theory.) This note is organized as follows. In Sections2,3 and4, we describe the main results and applications to character varieties and McShane's identity. The remain- ing sections are devoted to explanation of the idea of the proof of the main results. In Section5, we describe the two-generator and one-relator presentation of the 2- bridge link group, to which small cancellation theory is applied, and give a natural decomposition of the relator, which plays a key role in the proof. In Section6, we introduce a certain finite sequence associated with the relator and state its key properties. In Section7, we recall small cancellation theory and present a charac- terization of the \pieces" of the symmetrized subset arising from the relator. In Sections8 and9, we describe outlines of the proofs of the main results. 2. Main results For a rational number r 2 Q^ := Q[f1g, let K(r) be the 2-bridge link of slope r, which is defined as the sum (S3;K(r)) = (B3; t(1)) [ (B3; t(r)) of rational tangles of slope 1 and r (see Figure1). The common boundary @(B3; t(1)) = @(B3; t(r)) 2 of the rational tangles is identified with the Conway sphere (S ; P ) := (R2; Z2)=H, where H is the group of isometries of the Euclidean plane R2 generated by the π-rotations around the points in the lattice Z2. Let S be the 4-punctured sphere S2 − P in the link complement S3 − K(r). Any essential simple loop in S, up to isotopy, is obtained as the image of a line of slope s 2 Q^ in R2 − Z2 by the covering projection onto S. The (unoriented) essential simple loop in S, so obtained, is denoted by αs. We also denote by αs the conjugacy class of an element of π1(S) 3 represented by (a suitably oriented) αs. Then the link group G(K(r)) := π1(S − K(r)) is identified with π1(S)=hhα1; αrii, where hh·ii denotes the normal closure. Let D be the Farey tessellation, whose ideal vertex set is identified with Q^ . For ^ each r 2 Q, let Γr be the group of automorphisms of D generated by reflections in the edges of D with an endpoint r. Let Γ^r be the group generated by Γr and Γ1. Then the region R, bounded by a pair of Farey edges with an endpoint 1 and a pair of Farey edges with an endpoint r, forms a fundamental domain of the action ^ 2 ^ of Γr on H (see Figure2). Let I1 and I2 be the closed intervals in R obtained as the intersection of R^ with the closure of R. Suppose that r is a rational number with 0 < r < 1. (We may always assume this except when we treat the trivial knot and the trivial 2-component link.) Write 1 r = =: [m ; m ; : : : ; m ]; 1 1 2 k m1 + m2 + .. 1 . + mk SIMPLE LOOPS ON 2-BRIDGE SPHERES IN 2-BRIDGE LINK COMPLEMENTS 99 S 2 Figure 1. (S3;K(r)) = (B3; t(1)) [ (B3; t(r)) with r = 1=3. Here (B3; t(r)) and (B3; t(1)), respectively, are the inside and the outside of the bridge sphere S2. k where k ≥ 1, (m1; : : : ; mk) 2 (Z+) , and mk ≥ 2. Then the above intervals are given by I1 = [0; r1] and I2 = [r2; 1], where ( [m1; m2; : : : ; mk−1] if k is odd, r1 = [m1; m2; : : : ; mk−1; mk − 1] if k is even, ( [m1; m2; : : : ; mk−1; mk − 1] if k is odd, r2 = [m1; m2; : : : ; mk−1] if k is even. 1/3 1/2 1 3/10 = [3, 3] 5/17 = [3, 2, 2] 2/7 = [3, 2] 0 1/4 Figure 2. A fundamental domain of Γ^r in the Farey tessellation (the shaded domain) for r = 5=17 = [3; 2; 2]. We recall the following fact, ([20, Proposition 4.6 and Corollary 4.7] and [11, Lemma 7.1]), which describes the role of Γ^r in the study of 2-bridge link groups. 0 ^ ^ Proposition 2.1. (1) If two elements s and s of Q belong to the same Γr-orbit, 3 then the unoriented loops αs and αs0 are homotopic in S − K(r). 100 DONGHI LEE AND MAKOTO SAKUMA ^ (2) For any s 2 Q, there is a unique rational number s0 2 I1 [ I2 [ f1; rg such ^ that s is contained in the Γr-orbit of s0. In particular, αs is homotopic to αs0 in 3 3 S − K(r). Thus, if s0 2 f1; rg, then αs is null-homotopic in S − K(r). Thus the following question naturally arises (see [20, Question 9.1(2)]). Question 2.2. (1) Does the converse to Proposition 2.1(2) hold? Namely, is it 3 true that αs is null-homotopic in S − K(r) if and only if s belongs to the Γ^r-orbit of 1 or r? 0 (2) For two distinct rational numbers s; s 2 I1 [ I2, when are the unoriented 3 loops αs and αs0 homotopic in S − K(r)? The following theorem, proved in [11], gives a complete answer to Question 2.2(1). 3 Theorem 2.3. The loop αs is null-homotopic in S − K(r) if and only if s belongs to the Γ^r-orbit of 1 or r. In other words, if s 2 I1 [ I2, then αs is not null- homotopic in S3 − K(r). This has the following application to the study of epimorphisms between 2-bridge link groups (see [11, Section 2] for precise meaning). Corollary 2.4. There is an upper-meridian-pair-preserving epimorphism from G(K(s)) to G(K(r)) if and only if s or s + 1 belongs to the Γ^r-orbit of r or 1. The following theorem, proved in [12, 13, 14], gives a complete answer to Ques- tion 2.2(2). Theorem 2.5. Suppose that r is a rational number such that 0 < r ≤ 1=2. For 0 3 distinct s; s 2 I1 [ I2, the unoriented loops αs and αs0 are homotopic in S − K(r) if and only if one of the following holds.