Roots of Dehn Twists about Multicurves
A multicurve on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist about is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism such that . We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for . Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.
Key words and phrases:surface, mapping class, Dehn twist, multicurve, root
2000 Mathematics Subject Classification:Primary 57M99; Secondary 57M60
For , let be the closed, orientable surface of genus , and let denote the mapping class group of . By a multicurve in , we mean a finite collection of disjoint non-isotopic essential simple closed curves in . Let denote the left-handed Dehn twist about an essential simple closed curve on . Since the Dehn twists about any two curves in commute, we will define the left-handed Dehn twist about to be
A root of of degree is an element such that .
When comprises a single nonseparating curve, D. Margalit and S. Schleimer  showed the existence of roots of of degree in , for . This motivated , in which D. McCullough and the first author derived necessary and sufficient conditions for the existence of a root of degree . As immediate applications of the main theorem in the paper, they showed that roots of even degree cannot exist and that . When consists of a single separating curve, the first author derived conditions  for the existence of a root of . A stable quadratic upper bound on , and complete classifications of roots for and , were derived as corollaries to the main result. In this paper, we shall derive conditions for the existence of a root of when , and since there are no such multicurves in or , we shall assume henceforth that .
In general, a root of may permute some curves in , while preserving other curves. So we define an -permuting root of to be one that induces a partition of into singletons and other subsets of size greater than one. The theory for -permuting roots, as we will see, can be obtained by generalizing the theories developed in  and , which involved the analysis of the fixed point data of finite cyclic actions.
The theory that we intend to develop for -permuting roots when can be motivated by the following example. Consider the multicurve in shown in Figure 1.
It is apparent that the rotation of by composed with for some fixed is a 5 root of in . This is a simple example of a -permuting root, which is obtained by removing invariant disks around pairs of points in two distinct orbits of the rotation of , and then attaching five 1-handles with full twists. This example indicates that a classification of such roots would require the examination of the orbit information of finite cyclic actions, in addition to their fixed point data. This is a significant departure from existing theories developed in  and .
Any subset of a multicurve will be called a submulticurve. A multicurve in is said to be pseudo-nonseparating if separates , but no proper submulticurve of separates . A multicurve that contains no pseudo-nonseparating submulticurves will be called a nonseparating multicurve, while a multicurve which is a disjoint union of pseudo-nonseparating multicurves will be called a separating multicurve. A multicurve that is neither separating nor nonseparating will be called a mixed multicurve. In Figure 2 below, the collection of curves is a mixed multicurve, while the subcollections , and form pseudo-nonseparating, separating and nonseparating multicurves, respectively.
We start by generalizing the notion of a nestled -action from  to a permuting -action. These are -actions on that have distinguished fixed points, and distinguished non-trivial orbits. In Section 3, we introduce the notion of a permuting data set, which is a generalization of a data set from . We use Thurston’s orbifold theory [11, Chapter 13] in Theorem 3.8 to establish a correspondence between permuting -actions on and permuting data sets of genus and degree . In other words, permuting data sets algebraically encode these permuting actions and contain all the relevant orbit and fixed-point information required to classify the roots that will be constructed from these actions.
Let denote the surface obtained from by deleting an annular neighbourhood of and capping. In Section 4, we prove that conjugacy classes of roots of Dehn twists about nonseparating multicurves correspond to a special subclass of permuting actions on . We use this to obtain the following bounds for the degree of such a root
Let be a nonseparating multicurve in of size , and let be an -permuting root of of degree .
If , then
Furthermore, if , then this upper bound is realizable.
If , then is odd.
If , then .
If , then .
If is a separating or a mixed multicurve, will have multiple connected components. In order to classify roots of in this case, we will require multiple actions on the various components of which can be put together and extended to a root of on . In Section 5, we will show that this extension will require compatibility of orbits across components of in addition to compatibility of fixed points as in . As an immediate consequence to this theorem, we obtain quadratic bounds for the degree of the root in terms of the genera of the individual components. Furthermore, we obtain a quadratic stable upper bound for the degree of the root as in [9, Theorem 8.14].
In Sections 6 and 7, we use our theory to obtain a complete classification of roots of on surfaces of genus 3 and 4 respectively. We conclude by proving that a root of cannot lie in the Torelli group of , and also indicate how our results can be extended to classify roots of finite products of powers of commuting Dehn twists. integrated
2. Roots and their induced partitions
In this section we shall introduce some preliminary notions, which will be used in later sections.
Let be a multicurve in , and let be a closed annular neighbourhood of .
We denote the surface by .
The closed orientable surface obtained from by capping off its boundary components is denoted by .
If is a nonseparating multicurve, then is a connected surface whose genus we denote by .
Recall that a multicurve in is said to be pseudo-nonseparating if is disconnected, but is not disconnected for any proper submulticurve .
If , we write for such a multicurve. Note that is a single separating curve.
A disjoint union of copies of is denoted by .
For integers and , we define to be the disjoint union of copies of isometrically imbedded in . In particular, , and hence we shall write for .
Given two surfaces and and a fixed , we construct a new surface with , containing a multicurve of type , in the following manner. We remove disks on and disks on each . Now connect to along a 1-handle , and choose the unique curve (upto isotopy) on . Let , then note that , so we write
for the new surface .
Similarly, given surfaces and non-negative integers , we construct a new surface with , containing a multicurve of type . Let
and , we now define
If , we simply write .
Let , and suppose that is a root of of degree in . Then, we claim that, up to isotopy, .
Suppose first that for all . Then there exists a neighbourhood of such that
However, since , it follows that
which is a contradiction.
So we assume without loss of generality that . Then we choose a neighbourhood of disjoint from for , and choose an isotopy such that
and and . Replacing by , we may assume that . Now note that for all , which allows us to proceed by induction on to conclude that, up to isotopy, .
Let be a multicurve of size in . Then for integers , an -partition of is a partition of the set into subsets such that for all ,
, , and
comprises only separating or only nonseparating curves.
Let be a multicurve in . Then for integers , a root of of is said to be -permuting if it induces an -partition of .
Let be a multicurve of size in and consider an -partition as in Definition 2.4.
We shall denote the multiset by . (From here on, we shall denote a multiset using ).
If is induced by an -permuting root of , then we shall denote it by .
3. Permuting Actions and Permuting Data Sets
In this section we shall introduce permuting -actions, which are generalizations of the nestled -actions from . We shall also introduce the notion of a permuting -data set, which is an abstract tuple involving non-negative integers that would algebraically encode a permuting -action.
For integers , an orientation-preserving -action on is called a permuting -action if
there is a set of distinguished fixed points of , and
there is a set of distinguished non-trivial orbits of .
Let be a permuting -action on .
Fix a point , and consider . By the Nielsen realisation theorem , we may change by isotopy in so that is an isometry. Hence, induces a local rotation by an angle, which we shall denote by . Note that if , then where .
Fix an orbit . If , then , and there exists a cone point in the quotient orbifold of degree . Each has stabilizer generated by and the rotation induced by around each must be the same, since its action at one point is conjugate by a power of to its action at each other point in the orbit. So the rotation angle is of the form , where and denotes the inverse of . We now associate to this orbit a pair as follows:
For any orbit , if , then we define
Consider a permuting -action on with and .
For each , define
We define the orbit distribution of to be the set
Let and be two permuting -actions on with and for . We say is equivalent to if and there is an orientation-preserving homeomorphism such that
for each , if , then and for all , and
is isotopic to relative to .
The equivalence class of a permuting -action is denoted by .
We now introduce the notion of an -data set, which encodes the signature of the quotient orbifold of a permuting -action and the turning angles around its distinguished fixed points. Furthermore, the -data set will be combined with the orbit distribution of the action to form a pair, which we will call a permuting -data set.
Given and , an -data set is a tuple
where and are integers, each is a residue class modulo , and each is a residue class modulo such that:
The number determined by the equation
is called the genus of the data set.
Fix an -data set of genus as above.
For each , we write
For each , choose a non-negative integer . Then the set is called an orbit distribution of .
Given an orbit distribution associated with an -data set , the pair is called a permuting -data set of genus , where .
be two -data sets as in Definition 3.5.
and are said to be equivalent if
Two permuting -data sets and are said to be equivalent if and are equivalent as above, and .
Note that equivalent data sets have the same genus.
Given and , equivalence classes of permuting -data sets of genus correspond to equivalence classes of permuting -actions on .
Let be a permuting -action on with quotient orbifold whose underlying surface has genus . If is a free action, then , and we simply write and . If is not free, let be the image in of the , for , and let be the other possible cone points of as in Figure 4.
Let be the generator of the orbifold fundamental group that goes around the point and let be the generators going around . Let and be the standard generators of the “surface part” of , chosen to give the following presentation of :
From orbifold covering space theory , we have the following exact sequence
where . The homomorphism is obtained by lifting path representatives of elements of . Since these do not pass through the cone points, the lifts are uniquely determined.
For , the preimage of consists of points cyclically permuted by . As in Notation 3.2, the rotation angle at each point is of the form where is a residue class modulo and . Lifting the , we have that . Similarly, lifting the gives where . Finally, we have
since is abelian, so
The fact that the data set has genus follows easily from the multiplicativity of the orbifold Euler characteristic for the orbifold covering :
Thus, gives a -data set
of genus , and hence forms a permuting -data set.
Consider another permuting -action in the equivalence class of with a distinguished fixed point set . Then by definition there exists an orientation-preserving homeomorphism such that for all and is isotopic to relative to . Therefore, , for , and since , the two actions will produce the same permuting -data sets.
Conversely, given a permuting -data set , we construct the orbifold and a representation . Any finite subgroup of is conjugate to one of the cyclic subgroups generated by or a , so condition (iv) in the definition of the data set ensures that the kernel of is torsion-free. Therefore, the orbifold covering corresponding to the kernel is a manifold, and calculation of the Euler characteristic shows that . Thus we obtain a -action on with distinguished fixed points . We now construct from in the following manner. For each pair , write . If , then choose orbits of size (if is a free action, this choice is trivial, but otherwise, such an orbit would always exist in a small neighbourhood around any fixed point of ). If , then there exists a cone point in of degree , so there exists an orbit of of size in . Once again, by considering a small neighbourhood of this orbit, we may choose distinct orbits and set
which in turn gives .
It remains to show that the resulting action on is determined upto our equivalence in . Suppose that two permuting -actions and have the same permuting -data set . encodes the fixed point data of the periodic transformation , so by a result of J. Nielsen  (or by a subsequent result of A. Edmonds [2, Theorem 1.3]), and have to be conjugate by an orientation-preserving homeomorphism . Let be the quotient orbifold of the action , and be the induced representations. Then induces a map such that as in [7, Theorem 2.1]. If is a loop around a cone point in , then is a loop around a cone point in , and these cone points are associated to the same pair in since . Hence, maps to and to as in Definition 3.4. Furthermore, by construction, and hence the permuting data set determines upto equivalence. ∎
4. Nonseparating multicurves
Recall that a multicurve is said to be nonseparating if it does not contain any pseudo-nonseparating submulticurves. In this section, we establish that a root of corresponds to a special kind of permuting action on the connected surface .
Let be a permuting -action on for . Two orbits are said to be equivalent (in symbols, ) if
if , then we further require that
In this section, we will only need the case when , but we will need the general case in Section 5.
Let be a nonseparating multicurve in . A permuting -action on is said to be nonseparating with respect to if there exists a -partition of such that
there exists mutually disjoint pairs of distinguished fixed points in such that modulo , for , and
there exists mutually disjoint pairs of distinguished nontrivial orbits in such that , for , and
Let be a nonseparating multicurve in . Then for , equivalence classes of permuting -actions on that are nonseparating with respect to correspond to the conjugacy classes in of -permuting roots of of degree .
First, we shall prove that a conjugacy class of an -permuting root of of degree yields an equivalence class of a permuting -action that is nonseparating with respect to . We assume without loss of generality that , with the implicit understanding that, when either of them is zero, the corresponding arguments may be disregarded.
Let be the -partition of associated with . Choose a closed tubular neighborhood of , and consider as in Definition 2.1. By isotopy, we may assume that , , and . Suppose that is a root of of degree , then by Remark 2.3, we may assume that preserves and takes to .
By the Nielsen-Kerckhoff theorem , is isotopic to a homeomorphism whose power is . So we may change by isotopy so that . We fill in the boundary circles of with disks and extend to a homeomorphism on by coning. Thus defines a effective -action on , where .
The -action fixes the centers and of the disks and , , of whose boundaries are the components of which are fixed by . The orientation of determines one for , so we may speak of directed angles of rotation about the centres of these disks. Since , it follows from [7, Theorem 2.1] that
as illustrated in Figure 5 below.