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RESEARCH STATEMENT 1. Introduction the Mapping Class RESEARCH STATEMENT KASHYAP RAJEEVSARATHY 1. Introduction The mapping class group Mod(S) of an orientable surface S is de- fined as the group of isotopy classes of orientation-preserving self- diffeomorphisms on S. The mapping class group has been one of the central objects in the field of 2-dimensional geometric topology. It has been widely studied since it also plays an important role in several other fields including Teichm¨ullertheory and algebraic geometry, where it is called the modular group. Max Dehn in the 1920s tried to understand the mapping class groups of surfaces by addressing such questions as the existence of a finite set of generators. In the process, he introduced a basic element of Mod(S) called a Dehn twist. Regarding S1 as R=Z, a Dehn twist of an annulus S1 × I is a homeo- morphism h : S1 × I −! S1 × I defined by h(x; s) = (x + s; s). A Dehn twist tC about a simple closed curve C on a surface S is defined to be the map h on an annular neighborhood S1 × I of C and the identity elsewhere. The effect of a Dehn twist on an arc transverse to the curve C is shown in the figure below. C Figure 1. A Dehn twist A natural question concerning mapping class groups is whether a Dehn twist can have a root. It is easy to find examples of roots for Dehn twists about separating curves, that is curves whose complement is a union of two disjoint subsurfaces. As a simple example, we can obtain a square root of tC by rotating one of the subsurfaces on either side of C by an angle π producing a half-twist near C. However, in the case of nonseparating curves, the existence of a root is not obvious. 1 2 KASHYAP RAJEEVSARATHY Recently, D. Margalit and S. Schleimer [2] showed the existence of such roots by constructing roots of degree 2g + 1 in the surface of genus g + 1 ≥ 2. The natural questions were whether there exist roots of other degrees, and whether we could classify them. These questions motivated me to look deeper into this subject. My first research work (on nonseparating curves) [3], a collaborative effort with my thesis adviser Dr. Darryl McCullough, was a direct outcome of this pursuit, and my research has centered around completely un- derstanding and classifying the roots of Dehn twists about separating and nonseparating curves. The machinery developed in my research uses some elementary number theory in addition to techniques in clas- sical topology. I will now summarize the key ideas and results of my research. 2. Research Summary 2.1. Nonseparating curves. My first collaborative research paper ti- tled \Roots of Dehn twists" [3] explored the case of Dehn twists about nonseparating curves. In this paper, we derived necessary and suffi- cient conditions for the existence of a root of tC in Sg+1 (the closed orientable surface of genus g + 1) of degree n up to conjugacy. The 0 geometric idea behind the construction is to obtain a Cn action h that has two distinguished fixed points so that for two of the fixed points, the rotation angles add up to 2π=n. We then remove disk neighbor- hoods around these two fixed points and attach an annulus extending 0 n h to a homeomorphism h on Sg+1. Thus h ' tC for a Dehn twist tC about a nonseparating curve C in Sg+1. It can be shown that h(C) is isotopic to C, and hence we may assume that h preserves C up to isotopy. Making extensive of Thurston's theory of orbifolds (see W. Thurston[6, Chapter 13]), we obtain algebraic conditions that describe this geomet- ric construction of roots. The main theorem says that the Dehn twist has a root of degree n if and only if there exists a collection of integers satisfying some simple identities mod n. This collection of integers will be called a data set of degree n. A data set is basically a tuple that encodes the essential information required to describe the Cn action on Sg. We now state the main result in the paper. Theorem 2.1 (Main Theorem). For a given n > 1 and g ≥ 0, data sets of genus g and degree n correspond to the conjugacy classes in Mod(Sg+1) of the roots of tC of degree n. RESEARCH STATEMENT 3 A number of applications can be obtained from the main theorem by elementary considerations. An immediate consequence is the following corollary. Corollary 2.2. Suppose that tC in Sg+1 has a root of degree n. Then (a) n is odd. (b) n ≤ 2g + 1. This shows that the Margalit-Schleimer roots always have the maxi- mum degree among the roots of tC for a given genus. Among the other applications of the main result, is the following corollary. Corollary 2.3. A Dehn twist tC in Sg+1 has a root of degree n when- ever g + 1 > (n − 2)(n − 1)=2. 2.2. Separating curves. I have continued the work in [3] by exam- ining the case of a Dehn twist about a separating curve C. Although the existence of some roots is obvious in this case, there is much to be understood about their possible degrees and other behaviors. Suppose that C is a curve that separates F of genus g ≥ 2 into sub- ~ surfaces Fi of genera gi for i = 1; 2. (For simplicity, we will denote this by F = F1#C F2.) As in the case for nonseparating curves, a natural question is whether we can give necessary and sufficient conditions for the existence of a degree n root of tC . In my paper [4], we derive topo- logical and algebraic conditions for the existence of a root of degree n. Here too Thurston's orbifold theory is used to derive the algebraic conditions for the existence of a root. ~ The key idea is to define Cni actions on the subsurfaces Fi for i = 1; 2 so that have a distinguished fixed point with a rotation angle of 2π=ni around the point. The actions on either side from a compatible pair if the turning angles of the Cni -actions add up to 2π=n, where n = lcm(n1; n2). We add disks to these subsurfaces and then extend the Cni actions to these closed surfaces Fi by coning, thus obtaining a root of tC of degree n. We state the topological version of the main result in the paper. Theorem 2.4. Let F = F1#C F2 be a closed oriented surface of genus g ≥ 2. Then the conjugacy classes in Mod(F ) of roots of tC of degree n correspond to the compatible pairs ([h1]; [h2]) of equivalence classes of nestled (ni; `i)-actions hi on Fi of degree n. As in the case of nonseparating curves, we use data sets to describe ~ the Cni -actions on the subsurfaces Fi. The geometric compatibility condition can then be given by a special number-theoretic condition 4 KASHYAP RAJEEVSARATHY on data sets called the data set pair condition. The following theorem is the algebraic version of the main result. Theorem 2.5. Let F = F1#C F2 be a closed oriented surface of genus g ≥ 2. Then, data set pairs (D1;D2) of degree n and genus g, where D1 is a data set of genus g1 and D2 is a data set of genus g2, correspond to the conjugacy classes in Mod(F ) of roots of tC of degree n. We now state some applications of the Theorem 2.5. 2.3. Classifications for the closed orientable surfaces of genus 2 and 3. Let F denote the closed orientable surface of genus 2. Up to homeomorphism F has an unique curve C which separates the surface into two subsurfaces of genus 1. By the process described in the proof of the theorem above, we get Cni actions on the tori Fi for i = 1; 2 with n = lcm(n1; n2). On a torus, a cyclic actions can only be of orders 2, 3, 4 or 6. Taking the least common multiple of any two of these orders gives 12 as the only other possibility for the degree of a root of tC . We now state this application explicitly as a theorem. Theorem 2.6. Let F be the closed orientable surface of genus 2 and C a separating curve in F . Then a root of a Dehn twist tC about C can only be of degree 2, 3, 4, 6, or 12. We show that up to conjugacy in Mod(F ) there are exactly 2 roots of degrees 3, 4 and 12, one root of order 2, and three roots of order 6. Using similar considerations, we also obtain a complete classification for the closed orientable surface of genus 3. 2.4. Upper bounds on the degree of a root. Other applications of Theorem 2.5 are the derivation of the following two upper bounds on degree n of a root. Theorem 2.7. Let F = F1#C F2 be a closed oriented surface of genus g ≥ 2. Suppose that n denotes the degree of a root of the Dehn twist tC about C. (i) If g ≥ 2, then n ≤ 4g2 + 2g. ≥ ≤ 16 2 45 (ii) If g 10, then n 5 g + 12g + 4 . 3. Fractional powers of Dehn twists Currently, I am working on a paper [5], which explores the fractional powers of Dehn twists about nonseparating curves. By a fraction we mean an element of (N [ f0g) × N, and we write fractions as `==n so as not to confuse, for example, 1==2 with 2==4.
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