Roots of Dehn twists
D. Margalit and S. Schleimer found examples of roots of the Dehn twist about a nonseparating curve in a closed orientable surface, that is, homeomorphisms such that in the mapping class group. Our main theorem gives elementary number-theoretic conditions that describe the for which an root of exists, given the genus of the surface. Among its applications, we show that must be odd, that the Margalit-Schleimer roots achieve the maximum value of among the roots for a given genus, and that for a given odd , roots exist for all genera greater than . We also describe all roots having greater than or equal to the genus.
Key words and phrases:surface, mapping class, Dehn twist, nonseparating, curve, root
2000 Mathematics Subject Classification:Primary 57M99; Secondary 57M60
A natural question about mapping class groups is whether a Dehn twist has a root. That is, given a Dehn twist about a simple closed curve in a closed orientable surface and an integer degree , does there exist an orientation-preserving homeomorphism with in the mapping class group ? It is easy to find examples of roots when is a separating curve, but for a nonseparating curve it is not immediately apparent that roots exist. Note that since all nonseparating curves are equivalent under homeomorphisms of , the question is independent of the particular curve used.
Recently some beautiful examples of such roots were found by D. Margalit and S. Schleimer . They constructed roots of degree for the Dehn twist about a nonseparating curve in the surface of genus . We will describe those examples from our viewpoint in Section 2, after stating and proving our main result Theorem 1.1 in Section 1.
Theorem 1.1 says that given and , has a root of degree if and only if there exists a collection of integers satisfying certain equations. In fact, the conjugacy classes of roots correspond to the solutions. Its proof is an exercise in the well-studied theory of group actions on surfaces. We present it using the language of orbifolds (see W. Thurston [11, Chapter 13]), rather than the classical description that uniformizes the action and then works with the lifted isometries of the hyperbolic plane.
A number of applications can be obtained from Theorem 1.1 by elementary considerations. An immediate consequence is
Suppose that has a root of degree . Then
Thus the Margalit-Schleimer roots always have the maximum degree among the roots of for a given genus.
Section 3 concerns the set of for which has a root of degree a fixed . This set always contains all but at most values, those in a set which is easy to describe, and whose maximum element is :
For odd, has a root of degree whenever . Consequently, has a root of degree for all .
For prime , has no root of degree exactly when , but when is not prime, roots of degree may occur when .
In general, it is hard to use Theorem 1.1 to work out the full set of degrees of roots of for a given genus, although our results allow easy computation of the possible prime degrees of roots. Roots of large degree are rather limited, however, and in Theorem 4.2 of Section 4 we describe the set of roots having degree . Curiously, there is a root of degree only when . For and degrees satisfying , all roots are of a restricted type that we call -roots, and the only larger degree is that of the Margalit-Schleimer roots, . The genera for which has a -root of a given degree are easily found from a prime factorization of , and the for which a given genus has a -root of degree can be calculated on a desktop computer using GAP  for genera up to 1,000,000.
As shown in Figure 1, we can combine these results to get a sense of the set of pairs for which there is a root of degree for .
Our definition of roots requires them to be orientation-preserving, but this restriction is not necessary. In Section 5, we check that can be isotopic to with orientation-reversing only when . We also observe that roots can only be conjugate by orientation-preserving homeomorphisms. Thus Theorem 1.1 is a complete classification of all roots of in the homeomorphism group, up to conjugacy.
Theorem 1.1 gives some information on roots of powers of , that is, on fractional powers of , as we discuss in Section 6. For example, has a fourth root although as we saw in Corollary 1.2, does not have a square root. Our methods can also be used to understand the roots of Dehn twists about separating curves. Of course in this case, the roots will depend on the genera of the complementary components. We expect to pursue these ideas in future work.
1. The main theorem
For us, a Dehn twist means a left-handed Dehn twist, one for which the image of an arc crossing turns to the left approaching , as seen from the outside of the oriented surface.
By a data set we mean a tuple where , , , , the and the are integers satisfying
, , each , and each divides ,
and each ,
By condition (ii), and are units , so condition (iii) requires to be odd, and conditions (iii) and (iv) require . The number is called the degree of the data set, and the positive integer defined by
is its genus. Note that is independent of the values of , , and the , and no data set has genus . Later we will check that .
We consider two data sets to be the same if they differ by interchanging and , changing or , changing a , or reordering the pairs , . With this understanding, we have our main result.
For a given and , data sets of genus and degree correspond to the conjugacy classes in of the roots of of degree .
We will first prove that every conjugacy class of roots of degree yields a data set of degree and genus .
Fix a nonseparating curve in an oriented surface of genus . Choose a closed tubular neighborhood of , and put . By isotopy we may assume that , , and .
Suppose that is a root of of degree . We have , which implies that is isotopic to . Changing by isotopy, we may assume that preserves and takes to . Put .
Since and both preserve , there is an isotopy from to preserving and hence one taking to at each time. That is, is isotopic to . By the Nielsen-Kerckhoff theorem, is isotopic to a homeomorphism whose power is . (The Nielsen-Kerckhoff Theorem was proven in general by S. Kerckhoff [5, 6]. The cyclic case we need here was given by J. Nielsen in , although W. Fenchel [1, 2] gave the first complete proof. See the introduction in H. Zieschang’s monograph .) So we may change by isotopy so that .
We cannot have for any with . For a minimal such would have to divide , and then would be isotopic either to the identity or to some power of , forcing with either or greater than . So defines an effective action of the cyclic group of order on . Filling in the two boundary circles of with disks and extending to a homeomorphism by coning, we obtain a -action on the closed orientable surface of genus , where .
Later, we will show that cannot interchange the sides of . For now, assume that it does not. Under this assumption, fixes the center points and of the two disks of . The orientation of determines one for and hence for , so we may speak of directed angles of rotation about and (and any other fixed points of ). The rotation angle of at is for some with . As illustrated in Figure 2, the rotation angle at must be , in order that be a single Dehn twist.
Now, let be the quotient orbifold for the action of on , and let be the genus of the underlying -manifold . Figure 3 shows , with cone points and of order (the images of the points and of ) and possibly other cone points , of some orders , . The figure also shows some of the generators , , and of . Along with similar generators going around the other and standard generators and , in the “surface part” of , we have a presentation
From orbifold covering space theory, the orbifold covering map corresponds to an exact sequence
Here, is the group of covering transformations, generated by , and is obtained by lifting path representatives of elements of — these do not pass through the cone points so the lifts are uniquely determined. To find , we note first that the loop lifts as shown in Figure 4, so maps to where has rotation angle about . Since acts with rotation angle , we have so . Similarly at , the rotation angle of is . Since , the left-hand twisting angle along in Figure 2 is . This requires , giving . Multiplying by produces condition (iii) of a data set.
For , the preimage of consists of points cyclically permuted by . Each of the points has stabilizer generated by . The rotation angle of must be the same at all points of the orbit, since its action at one point is conjugate by a power of to its action at each other point. So the rotation angle at each point is of the form , where , and as before, lifting shows that where .
Finally, we have , since is abelian, so
giving condition (iv) of a data set.
The fact that the genus of the data set equals follows from the multiplicativity of the orbifold Euler characteristic for the orbifold covering :
Thus leads to a data set of degree and genus .
Suppose now that interchanges the sides of . Its degree must be even, and we will write it as . The points and are now interchanged by , while is a root of of order that does not interchange the sides. In particular, must be odd. We assume for now that .
Let and be the disks centered at and , for which . The actions of at and are conjugate, by , so there exists an equivariant homeomorphism from to , where the latter has the action and (the minus sign is not necessary, but is natural for our construction). We think of and as corresponding to and respectively.
Since is twice and twice , we may further assume that the action of on in these coordinates is either and , or and .
Figure 5 illustrates the effect of on for the first action, in which . The indicated angles are . If we extend to by sending to followed by a simple left-hand twist, as in Figure 5, then the twisting angle is , and consequently . Other extensions to will differ from this by full twists, giving for some integer . In any case, cannot equal . For the second action, in which , the amount of twisting on is still plus some number of full twists, so again .
Finally, suppose that . Then in the previous construction, is the identity on , and is either and , or and . In either case, any extension of to has some number of full twists, so is some even power of .
At this point, we have shown how every root of produces a data set. If the original roots are conjugate in , then their restrictions to are conjugate and isotopic to conjugate homeomorphisms of order , and their extensions to are conjugate by a homeomorphism preserving . Therefore their orbifold quotients and are homeomorphic by an orientation-preserving orbifold homeomorphism preserving taking the distinguished cone points to the distinguished cone points of , and compatible with the representations of the orbifold fundamental groups to . It follows that our procedure produces equivalent data sets.
Given a data set, we can reverse the argument to produce the root . We construct the corresponding orbifold and representation . Any finite subgroup of is conjugate to a subgroup of one of the cyclic subgroups generated by , , or a , so condition (ii) ensures that the kernel of is torsionfree. Therefore the orbifold covering corresponding to the kernel is a manifold, and calculation of its Euler characteristic shows that has genus . Removing disks around the fixed points and corresponding to the cone points and produces the surface , and attaching an annulus produces the surface of genus . Condition (iii) ensures that the rotation angles work correctly to allow an extension of to an with a single Dehn twist.
It remains to show that the resulting root of is determined up to conjugacy. Our data sets encode the fixed-point data of the periodic transformation , and it was proven by J. Nielsen  that this data determines up to conjugacy. We require in addition that the conjugating homeomophism preserve .
Suppose that and are roots obtained by applying our procedure to a data set . That is, we use the data set to define orbifolds and and homomorphisms and , then take the corresponding covers and and so on. Each of and has genus and cone points of corresponding orders, including the two distinguished order- cone points, which give elements and and and and and . We have , where the rotation angles of and at are , and similarly for the other generators coming from cone points.
We claim that the generators and of may be selected so that for all . Suppose this is not initially the case. There is an orbifold homeomorphism of whose effect on the abelianization of is to send to and to fix the other generators; it is the end map of an isotopy that slides the cone point around a loop that represents . Since is a generator of , we may repeat this homeomorphism some number of times until for the new , . Repeating this process on the other and , we obtain a new set of generators that verify the claim.
Performing a similar process, we may assume that for all . Now, we take an orientation-preserving orbifold homeomorphism such that and so on. It satisfies , so lifts to a homeomorphism such that . If we select with a bit of care, carries to , and we can extend to a homeomorphism of conjugating to . ∎
Theorem 1.1 tells us that always has a cube root when , corresponding to the data sets with the selected to achieve condition (iv). Also, if has a root of degree , then replacing by in a corresponding data set produces a root of degree for .
Of more interest is the following:
Suppose that has a root of degree . Then
Part (a) is simply the fact that data sets must have odd degree. For (b), suppose for contradiction that . From the formula for , we have so , , and . Putting , condition (iv) gives , contradicting condition (iii) since and divides . ∎
2. The Margalit-Schleimer roots
Here we will describe the examples of Margalit and Schleimer from our viewpoint. They construct the surface by identifying opposite faces of a -gon. It center point is , and the two points that come from identifying vertices are and . Pictures centered at , , and are shown in Figure 6 for the case of ; in general becomes , and so on. Let be the homeomorphism of obtained by rotating through a (counterclockwise) angle of at and . It carries to , so it rotates through an angle of at . Let be , which rotates through at and and through at . Modulo , is if is even and if is odd, so is approximately a quarter turn at , counterclockwise if is even and clockwise if not. The examples are then obtained by the construction in Theorem 1.1.
The inverse of is , so , while the inverse of is . So the data set resulting from the Margalit-Schleimer construction is .
We call a root of a Margalit-Schleimer root if it has degree . Using Theorem 1.1, is is easy to find all the Margalit-Schleimer roots. We need only find the such that and are both relatively prime to , then put and , and . A GAP function to list such roots is provided in the software at . For example, we find that has three Margalit-Schleimer roots, , , and , and has .
3. Genus sets
The genus set of , , is the set of such that has a root of degree . Corollary 1.2 tells us that is empty for even . For odd , we can gain considerable information about the genus set. For the rest of this section, will be assumed odd, and will denote .
A data set with all is called a primary data set, and the corresponding root of is called a primary root. Primary data sets exist for all , since we may take , and the selected from so that .
We now examine the genera of primary data sets. A quick example will make it much easier to follow the notation. For , so that , we position the genera according to their values :
The genus of a primary data set is . For , we obtain the values for , which for are the values in the first column other than . For , is always , and we obtain all values greater than . Similarly, gives the values in the third column greater than , and gives those in the last column beyond . Higher values of give no new values for . So the primary data sets for give all values of except the values indicated in the table.
In general, the genera obtained from data sets of degree having are , and are exactly those with . No new genera are obtained when . So the genera not obtained are those in the “triangular” set defined by
Since has elements, has elements. The maximum element in is the maximum element in , which is .
Since the primary data sets produce roots for every genus other than those in , we have
For odd, contains all that are not in . Consequently, has a root of degree whenever .
When is prime, all data sets are primary. So we have
For prime, equals the set of not in . In particular, does not have a root of degree .
For example, has a cube root for all , and a fifth root exactly when is not , , , or . For that are not prime, determination of is more complicated, as elements in often arise from non-primary data sets. For example, , but a ninth root for arises from the data set , for which condition (iv) is satisfied since .
We note that contains about half of the values with . Therefore in Figure 1, the pairs corresponding to primary roots would be about half of the pairs with odd in the region above .
4. The root set and -roots
The root set is the set of such that has a root of degree (although degree set would be a more accurate name). Corollary 3.2 allows us to effectively compute the primes in . From Corollary 3.1, contains whenever . In Theorem 4.2, we will determine all in that satisfy . First, we must introduce -roots.
Let and be odd integers with . A root corresponding to a data set having , , and is called a -root. The next lemma requires an elementary number-theoretic fact for which we are unable to find a reference. To avoid interruption of the argument here, we will prove it later as Lemma 7.1.
For any odd integers , there exist -roots. Such roots satisfy the following:
, i. e. .
exactly when .
For example, a -root has (so Lemma 4.1(c) is best possible, in general), and for there are -roots when and -roots when . For even , there is always a -root given by .
Proof of Lemma 4.1.
Put and , so . Let , , and . Condition (iv) becomes . Since , we can write , and by Lemma 7.1 we may assume that . Taking and satisfies condition (iv). The genus works out to be the expressions in (b), which imply the first inequality in (c). For the second, we have
Part (d) follows because (b) gives when , and when , so . ∎
For a given one can easily compute the for which is a -root of , if we have a prime factorization of . For in (b) of Lemma 4.1, and are relatively prime divisors of . For each pair of relatively prime divisors, we write and put and giving as a -root for by Lemma 4.1(b). This gives an algorithm for computing the -roots of , again assuming that we can factor, just by checking which of the in the range allowed by Lemma 4.1(c) have among its corresponding genera. We have implemented these algorithms as a GAP script  available at . Some sample calculations include the genera having a -root of degree :
gap DERootGenera( 54573 );
[ 45476, 45477, 54571, 54572 ]
and all -roots for a given genus:
gap DERoots( 54572 );
[ 54573, 54575, 54587, 54769, 65487 ]
gap DERoots( 54573 );
The main result of this section describes all roots of large degree:
Suppose has a root of degree . Then the root is either a Margalit-Schleimer root, a -root, or the cube root of .
Since , we have
Therefore and .
Suppose first that . We cannot have , for putting , condition (iv) would say that , which is impossible since is relatively prime to and hence to . So , and is a Margalit-Schleimer root.
If , then is a -root.
Suppose that . From our expression for , we find that . Since all are odd this can only be satisfied when . Condition (iv) says that , a contradiction unless . That is, is a cube root with . In fact, this is unique, since the only data set of degree and genus is . ∎
5. There are no orientation-reversing roots
In this section, we will prove that has no orientation-reversing roots, and that roots of cannot be conjugate by orientation-reversing homeomorphisms. Consequently, Theorem 1.1 classifies all roots of in the homeomorphism group, up to conjugacy.
Let be an orientation-reversing homeomorphism of with isotopic to for some . Then .
As in the proof of Theorem 1.1, we write and change by isotopy so that restricts to a homeomorphism of finite order on for some annulus neighborhood of . On , is orientation-reversing and has finite order on .
Suppose first that preserves the components of . Then reverses orientation on each component, so is a reflection of period . It follows that has order , and for some coordinates on as , is isotopic to a homeomorphism of the form . Therefore is isotopic to the identity on , so .
Suppose now that interchanges the components of . Since is orientation-reversing and has finite order on , there are coordinates on as so that . Let be the homeomorphism of defined by . Then is isotopic relative to to for some power . Since is isotopic to relative to , and must be even, is isotopic to the identity, that is, . ∎
We note also that no two roots of can be conjugate by an orientation-reversing homeomorphism. For if and are roots and with orientation-reversing, then . But conjugation of a left-handed Dehn twist by an orientation-reversing homeomorphism produces a right-handed Dehn twist.
6. Roots of
Theorem 1.1 gives some information about the roots of powers of , that is, the fractional powers of . A tuple like a data set except that condition (iii) is replaced by the condition that produces a root of of degree . The only difference in the construction is that the rotation angles at and are of the form and , and the twisting on the annulus is through an angle rather than . Thus the data set for which yields a root of of degree . Of course we know from Corollary 1.2 that does not have a square root. The data set gives a cube root of .
There are some complications, however. If and are not relatively prime, then a root of degree of might be a power of a root of a smaller power of of lower degree, and then the action on in the proof of Theorem 1.1 will not be effective. More interesting is the fact that roots of may exchange the sides of , requiring a different kind of quotient orbifold to be analyzed.
7. An elementary lemma
Let , be relatively prime positive integers, and let be a finite set of primes. If , assume that and are not both odd. Then there exist integers and so that and neither nor is divisible by any prime in .
Choose and with , so that for all integers . We seek a so that and are nonzero for each .
For each odd , if any , then . So there is a unique such that exactly when . Similarly, if any , then such are those with for a unique . Since , there are choices of so that if , then neither nor . If , then we may assume that is even and and hence are odd, and we take equal to or