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 defined 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 Teichmüllertheory 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 homeomorphism 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.

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 sufficient 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 ′ 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 ′ 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 geometric 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 topological 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, ℓi)-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 ∪ {0}) × N, and we write fractions as ℓ//n so as not to confuse, for example, 1//2 with 2//4. Fix a nonseparating curve C in G.A fractional power of tC of exponent ℓ//n is a root of degree RESEARCH STATEMENT 5

ℓ ≥ n of tC for some ℓ 1. In particular, a root of tC of degree n is just a fractional power of exponent 1//n. As in the root case, a fractional power of tC must preserve C up to isotopy. We then say that h is side-exchanging (SE) if it interchanges the two sides of C, and side-preserving (SP) if not. Side-preserving case. Theorem 2.1 gives some information about side-preserving fractional powers of tC . Side-preserving fractional powers can be described by tuples called SP data sets that look exactly like the data sets in [3]. The only diﬀerence in the construction is that the rotation angles at centers of the disks are of the form 2πk/n and 2π(ℓ − k)/n, and the twisting on the annulus N is through an angle 2πℓ/n rather than 2π/n. We now state the main result. Theorem 3.1 (Main theorem). For a given n > 1, 1 ≤ ℓ ≤ n, and g ≥ 0, SP data sets of genus g and exponent ℓ//n correspond to the conjugacy classes in Mod(Sg+1) of the side-preserving fractional powers of tC of exponent ℓ//n. Consequently, tC has a side-preserving fractional power of exponent ℓ//n if and only if there exists a SP data set of genus g and exponent ℓ//n. We state some of the applications of the main result in this case.

Proposition 3.2. Let h be a side-preserving fractional power of tC of ℓ exponent ℓ//n with gcd(ℓ, n) = 1. Then h = h0 for some root h0 of tC of degree n. Other applications of the main theorem are the following corollaries. Corollary 3.3. Suppose that h is a side-preserving fractional power of tC of exponent ℓ//n. Then (a) n is odd if ℓ is odd. (b) n ≤ 2g + 1 if gcd(ℓ, n) = 1. Corollary 3.4. Suppose that h is a side-preserving fractional power of tC of exponent ℓ//n. Then n ≤ 4g. Side-exchanging case. Suppose that h is a side-exchanging fractional power of tC . Then h must have exponent of the form ℓ//2n. This is because any odd integer power of h exchanges the sides. So, we use h to define a cyclic action of order 2n on the closed surface G. Since h2 fixes the sides of C its action at centers of the coned disks are conjugate by h. This means that h2 will have the same turning angle at the centers. This setup is encoded by a somewhat different kind of data set, called an SE data set. We now state the main result in this case. 6 KASHYAP RAJEEVSARATHY

Theorem 3.5 (Main Theorem). For a given n ≥ 1, 1 ≤ ℓ ≤ n, and g ≥ 0, the side-exchanging data sets of genus g and exponent ℓ//2n correspond to the conjugacy classes in Mod(Sg+1) of the side- exchanging fractional powers of tC of exponent ℓ//2n. Consequently, tC has a side-exchanging fractional power of exponent ℓ//2n if and only if there exists a SE data set of genus g and exponent ℓ//2n.

4. Future research directions Fractional powers. There is a lot more to be analyzed and examined both in the SP and SE cases. Another unexplored topic in this direction is the existence and classification of fractional powers of Dehn twists about separating curves. Multicurves. A multicurve is a collection of disjoint nonisotopic curves on a surface. The question is - Can the theory developed so far for a single curve be extended consistently to a system of curves? There are a lot of issues connected with answering this question. One such issue is that such a collection of curves can have curves of different types (i.e. both separating and nonseparating). So this requires a very careful examination of the various possibilities. Nonorientable surfaces. Topologically, a surface is said to be nonorientable if it has an imbedded Möbiusband. It is natural to try to develop an analogous theory for nonorientable surfaces. We can expect such a theory to have some significant algebraic differences. This is because the boundary of a nonorientable surface, unlike an orientable surface, is not a product a commutators. I hope to explore this case in the near future. Bounded surfaces. It might be possible to adapt our theory on Dehn twists to the case of bounded surfaces. Our roots have representatives which have finite order on most of the surface, so by just removing invariant disks around fixed points, or a non-invariant disk and all its translates, we produce many examples of roots of a Dehn twist on bounded surfaces. In adapting our general theory, the orbifold funda- mental group would lack the relation involving a product of commutators, which would allow a great deal more flexibility for homomorphisms to Cn. Nonetheless, we expect a useful theory to emerge. Roots of other elements in Mod(S). The existence of roots of Dehn twists raises additional natural questions about the root structure in Mod(S). For example, our results show that a Dehn twist will typically have many different degrees of roots, and many different roots of the RESEARCH STATEMENT 7 same degree. C. Bonatti and L. Paris [1] have shown that an element of a pure subgroup G of Mod(S) can have at most one root of degree m in G. Naturally, one would like to examine the existence and uniqueness of roots for other elements in Mod(S).

Roots of elements in Out(Fn). Another direction is the study of the root structure in Out(Fn) for the free group Fn of rank n. D. Margalit and S. Schleimer [2] note that their roots of Dehn twists in Mod(S), adapted to once-punctured surfaces, provide examples of “geometric” roots of Nielsen transformations in Out(Fn). We should be able to adapt our general theory of roots of Dehn twists to develop a systematic theory of such roots. Other directions. My research to date has centered around under- standing the fractional powers of Dehn twists, but I have broader interests in other areas of low-dimensional and geometric topology. My interests in 3-dimensional topology include areas such as Heegaard split- tings of 3-manifolds and knot complements in S3, and my interests in 2- dimensional topology include Teichm¨ullerspace, mapping class groups and the complex of curves. In the future, I wish to explore these other areas either through my own research or by seeking opportunities to collaborate with other researchers in these ﬁelds. References

[1] Christian Bonatti and Luis Paris. Roots in the mapping class groups. Proc. Lond. Math. Soc. (3), 98(2):471–503, 2009. [2] Dan Margalit and Saul Schleimer. Dehn twists have roots. Geom. Topol., 13(3):1495–1497, 2009. [3] Darryl McCullough and Kashyap Rajeevsarathy. Roots of dehn twists. Geome- triae Dedicata, 151:397–409, 2011. [4] K. Rajeevsarathy. Roots of Dehn twists about separating curves. ArXiv e-prints, arXiv:1104.0968v1 [math.GT], April 2011. [5] Kashyap Rajeevsarathy. Fractional powers of Dehn twists. in preparation. [6] William P. Thurston. The Geometry and Topology of Three-Manifolds. notes available at: http://www.msri.org/communications/books/gt3m/PDF.