Geometric realizations of cyclic actions on surfaces - II
aa r X i v : . [ m a t h . G T ] J u l GEOMETRIC REALIZATIONS OFCYCLIC ACTIONS ON SURFACES - II
ATREYEE BHATTACHARYA, SHIV PARSAD, AND KASHYAP RAJEEVSARATHY
Abstract.
Let Mod( S g ) denote the mapping class group of the closed ori-entable surface S g of genus g ≥
2. Given a finite subgroup H ≤ Mod( S g ),let Fix( H ) denote the set of fixed points induced by the action of H on theTeichm¨uller space Teich( S g ). The Nielsen realization problem, which was an-swered in the affirmative by S. Kerckhoff, asks whether Fix( H ) = ∅ , for anygiven H . In this paper, we give an explicit description of Fix( H ), when H is cyclic. As consequences of our main result, we provide alternative proofsfor two well known results, namely a result of Harvey on dim(Fix( H )), anda result of Gilman that characterizes irreducible finite order actions. Finally,we derive a correlation between the orders of irreducible cyclic actions and thefilling systems on surfaces. Introduction
Let S g be a closed orientable surface of genus g ≥ S g ) denotethe mapping class group of S g . Given a finite subgroup H ≤ Mod( S g ), let Fix( H )denote the set of fixed points induced by the natural action of H on the Teichm¨ullerspace Teich( S g ). The Nielsen realization problem asks whether Fix( H ) = ∅ , for anarbitrary finite subgroup H ≤ Mod( S g ). While this was proven for the cyclic caseby J. Nielsen [21] (the first complete proof was due to W. Fenchel [7, 8]), a gen-eral solution to the problem was asserted by S. Kerckhoff [15]. A natural questionthat remained was whether one can obtain an exact description of Fix( H ). Re-cently, in [22], a method to construct an explicit structure in Fix( H ) was deveoped.Extending the results in [22], in this paper, we obtain a comprehensive descrip-tion of all structures in Fix( H ), thereby giving a complete solution to the modularNielsen Realization problem for the case when H is an arbitrary cyclic subgroup ofMod( S g ).For g ≥
1, let H = h h i be a cyclic subgroup of Mod( S g ) of order n that actson S g yielding a quotient orbifold [25, Chapter 16] O h := S g /H of genus g ( h ).Following the nomenclature in [22], if O h has three cone points with at least onecone point of order n , then h is called a Type 1 action. In [22], it was shown that for g ≥
2, a Type 1 action h ∈ Mod( S g ) with g ( h ) = 0, which we call a spherical Type1 action (this is a special type of quasiplatonic [1] cyclic action), is realized as therotation by an angle θ h of a distinguished hyperbolic polygon P h (see Lemma 2.4for a description) with an appropriate side-pairing. While it is known that suchactions are irreducible (i.e. their generators are irreducible as mapping classes),we independently establish this fact by showing that this hyperbolic structure isunique (see Proposition 4.1). Furthermore, we extend the main result of [22] bygiving a precise description of how an arbitrary cyclic action h decomposes intospherical Type 1 actions, sphere-rotations, and permutations. On the other hand,we show that the action h can also be built inductively through finitely many r -compatibilities between pairs of such irreducible components. By an r -compatibility, Mathematics Subject Classification.
Primary 57M60; Secondary 57M50, 57M99.
Key words and phrases. surface, mapping class, finite order maps, Teichm¨uller space. we mean the identification of boundary components resulting from the deletion ofcyclically permuted disks around pairs of orbits of size r with the same local rotationangles induced by the action. This notion also includes the compatibility across apair of orbits induced by a cyclic action within the same surface, which we call a self r-compatibility . The last kind of compatibility is an n -compatibility which isrealized by pasting a cyclical permutation of n copies of the torus S to the action h . We will call this a toral addition , and the reverse process of removing sucha permutation component will be called a toral subtraction . (For more technicaldetails, see Section 2.) It is convenient to visualize an action h realized throughfinitely many of these compatibilities as a necklace with beads (see Section 3),where the beads represent the irreducible components, and two distinct beads areconnected with r strings, if there is an r -compatibility between the correspondingactions. This enables us to determine the size of a maximal reduction systemassociated with certain reducible actions (see Corollary 3.9). Using these ideas, weestablish our main result in Section 5, which describes the space of solutions to themodular Nielsen Realization Problem. Theorem 1 (Main Result) . Let H = h h i be an arbitrary cyclic action of order n on S g . Suppose that h is realized through putting together k spherical Type 1actions with a ′ pairwise r -compatibilities with r < n , k − a ′ − n -compatibilities, b self r -compatibilities with r < n , c toral additions, and d toral subtractions. Then F ix ( H ) ≈ M /M , where M ≈ k Y i =1 { [ P h i ] } × k + c − a ′ + b − Y j =1 ((0 , ℓ j ( h )] × R ) × R c − × R c − and M ≈ d Y j =1 ((0 , ℓ j ( h )] × R × R d − × R d − , where the ℓ j ( h ) and ℓ j ( h ) are positive constants determined by h (with the under-standing that when c (resp. d ) is zero, then the last two factors in M (resp. M )will disappear). As applications of our main theorem, we provide alternative proofs for the fol-lowing well known results due to Harvey [12, 17] and Gilman [10].
Corollary 1.
Let H = h h i be a cyclic action of order n on S g such that O h has c cone points. Then:(i) (Harvey) dim( Fix ( H )) = 6 g ( h ) + 2 c − , and(ii) (Gilman) h is irreducible if, and only if g ( h ) = 0 and c = 3 (or h is quasi-platonic). Moreover, for any maximal reduction system C for a reducible action h , we showthat the difference between the number of distinct orbits induced by h on C and S g \ C depends solely on g ( h ). Corollary 2.
Let C be a maximal reduction system for a reducible action h of order n on S g that induces r cone points on the quotient orbifold. Suppose that ℓ and k are the number of distinct orbits induced by h on C and S g \ C , respectively. Then ℓ = 3 g ( h ) − r and k = 2 g ( h ) − r. Finally, in Section 6, we draw an interesting parallel between irreducible cyclicactions on surfaces and filling systems on surfaces by appealing to the theory offat graphs [13, 16, 19]. It is well known [24] that a filling system C on S g with | S g \ C| = b corresponds to a 4-regular fat graph Γ C of genus g with b boundarycomponents. Moreover, it was shown in [22] that an automorphism ϕ of a fat graph EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 3 of genus g yields a cyclic action h ϕ on S g . This brings us to the final result in thepaper. Theorem 2.
Let C be a filling of S g , for g ≥ , and let ϕ ∈ Aut (Γ C ) be of order n .Then h ϕ ∈ Mod ( S g ) is irreducible if, and only if, ( g, n ) = (1 , . Preliminaries A C n -action D on S g induces a branched covering S g → S g /C n , where thequotient orbifold O D := S g /C n has signature ( g ; n , . . . , n ℓ ) (see [6, 25]). Fromorbifold covering space theory, we obtain an exact sequence1 → π ( S g ) → π orb1 ( O D ) ρ → C n → , where π orb1 ( O D ) is a Fuchsian group [14] given by the presentation h α , . . . , α ℓ , x , y , . . . , x g , y g | α n = · · · = α n l ℓ = 1 , ℓ Y i =1 α i = g Y j =1 [ x j , y j ] i . The epimorphism π orb1 ( O D ) ρ −→ C n is called the surface kernel map [11], and has theform ρ ( α i ) = t ( n/n i ) c i , for 1 ≤ i ≤ ℓ , where C n = h t i and gcd( c i , n i ) = 1. The map ρ is often described by a ( g ; n , . . . , n ℓ )-generating vector [3, 9]. From a geometricviewpoint, a cone point of order n i lifts to an orbit of size n/n i on S g , and thelocal rotation induced by D around the points in the orbit is given by 2 πc − i /n i ,where c i c − i ≡ n i ). (For more details on the theory of finite group actionson surfaces, we refer the reader to [2, 4, 26].)Putting together the notions of orbifold signature and the generating vector, wecan obtain a combinatorial encoding of the conjugacy class of a cyclic action. Definition 2.1. A data set of degree n is a tuple D = ( n, g , r ; ( c , n ) , ( c , n ) , . . . , ( c ℓ , n ℓ )) , where n ≥ g ≥
0, and 0 ≤ r ≤ n − c i is a residue classmodulo n i such that:(i) r > ℓ = 0, and when r >
0, we have gcd( r, n ) = 1,(ii) each n i | n ,(iii) for each i , gcd( c i , n i ) = 1,(iv) for each i , lcm( n , . . . b n i , . . . , n ℓ ) = lcm( n , . . . , n ℓ ), and lcm( n , . . . , n ℓ ) = n ,if g = 0, and(v) ℓ X j =1 nn j c j ≡ n ).The number g determined by the Riemann-Hurwitz equation2 − gn = 2 − g + ℓ X j =1 (cid:18) n j − (cid:19) is called the genus of the data set, which we shall denote by g ( D ). Given a dataset D as above, we define n ( D ) := n, g ( D ) := g, r ( D ) = r, and g ( D ) := g . The quantity r ( D ) associated with a data set D will be non-zero if, and only if, D represents a free rotation of S g ( D ) by 2 πr ( D ) /n .The following lemma is a consequence of the classical results in [11, 20]. (For moredetails, see [5, 18, 23].) ATREYEE BHATTACHARYA, SHIV PARSAD, AND KASHYAP RAJEEVSARATHY
Lemma 2.2.
Data sets of degree n and genus g correspond to conjugacy classes of C n -actions on S g . From here on, in addition to following the nomencalture of data sets, we will appealto the theory developed in [22]. To begin with, we classify C n -actions on S g intothree broad categories. Definition 2.3.
Let D be a C n -action on S g . Then D is said to be a:(i) rotational action , if either r ( D ) = 0, or D is of the form( n, g ; ( s, n ) , ( n − s, n ) , . . . , ( s, n ) , ( n − s, n ) | {z } k pairs ) , for integers k ≥ < s ≤ n − s, n ) = 1, and k = 1, if andonly if n > Type 1 action , if ℓ = 3, and n i = n for some i .(iii) Type 2 action , if D is neither a rotational nor a Type 1 action.If g ( D ) = 0, then we call D a spherical action. The following lemma gives ageometric realization of spherical Type 1 actions. Lemma 2.4.
For g ≥ , a spherical Type 1 action D on S g can be realized explicitlyas the rotation θ D of a hyperbolic polygon P D with a suitable side-pairing W ( P D ) ,where P D is a hyperbolic k ( D ) -gon with k ( D ) := (cid:26) n, if n , n = 2 , and n, otherwise, and for ≤ m ≤ n − , W ( P D ) = n Y i =1 a i − a i with a − m +1 ∼ a z , if k ( D ) = 2 n, and n Y i =1 a i with a − m +1 ∼ a z , otherwise, where z ≡ m + qj (mod n ) , q = ( n/n ) c − , and j = n − c . Definition 2.5.
Let D = ( n, g ; ( c , n ) , ( c , n ) , . . . , ( c ℓ , n ℓ )) be a C n -action on S g . For a given g ′ ≥
1, one can obtain a new action from D by removing cyclicallypermuted (mutually disjoint) disks around points in an orbit of size n , and thenattaching n copies of the surface S g ′ , along the resultant boundary components.The resultant action, which is uniquely determined up to conjugacy, is denoted by J D, g ′ K , where J D, g ′ K := ( n, g + g ′ ; ( c , n ) , ( c , n ) , . . . , ( c ℓ , n ℓ )) . Given an action of type J D, g ′ K for some g ′ ≥
1, one can reverse the constructionprocess described above to recover the action D . We denote this reversal processby J D, g ′ K (i.e. J D, g ′ K = D ).It is easy to see that a construction of type J D, g ′ K and J D, g ′ K for some g ′ > , can be realized by g ′ inductively performed constructions of type J D, K (or toraladditions ) and J D, K (or toral subtractions ), respectively. We will now describe aconstruction of a new C n -action from a pair of existing C n -actions across a pair ofcompatible orbits of size m , where m is a proper divisor of n . Definition 2.6.
For i = 1 ,
2, two actions D i = ( n, g i, ; ( c i, , n i, ) , ( c i, , n i, ) , . . . , ( c i,ℓ i , n i,ℓ i ))are said to form an ( r, s )- compatible pair D = L D , D , ( r, s ) M if there exists 1 ≤ r ≤ ℓ and 1 ≤ s ≤ ℓ such that EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 5 (i) n ,r = n ,s = m , and(ii) c ,r + c ,s ≡ m ).The number 1 + g ( D ) − g ( D ) − g ( D ) will be denoted by A ( D ) . The following lemma provides a combinatorial recipe for constructing a new actionfrom an ( r, s )-compatible pair of existing actions.
Lemma 2.7.
Given a pair of cyclic actions as in Definition 2.6, we have L D , D , ( r, s ) M = ( n, g , + g , ; ( c , , n , ) , . . . , \ ( c ,r , n ,r ) , . . . , ( c ,ℓ , n ,ℓ ) , − ( c , , n , ) , . . . , \ ( c ,s , n ,s ) , . . . , ( c ,ℓ , n ,ℓ )) , where A L D , D , ( r, s ) M = nn ,r . It is always possible to construct a new C n action from a pair of C n actions D i asin Definition 2.6 across a pair of orbits of size n . Definition 2.8.
Given actions D i as in Definition 2.6, we define L D , D M := ( n, g , + g , ; ( c , , n , ) , . . . , ( c ,ℓ , n ,ℓ ) , − ( c , , n , ) , . . . , ( c ,ℓ , n ,ℓ )) , where g ( L D , D M ) = g ( D ) + g ( D ) + n − A L D , D M := n − Definition 2.9.
For ℓ ≥
4, let D = ( n, g ; ( c , n ) , ( c , n ) , . . . , ( c ℓ , n ℓ )) , be a C n -action. Then D is said yield an ( r, s )- self compatible action D ′ = J D, ( r, s ) K , ifthere exist 1 ≤ r < s ≤ ℓ such that(i) n r = n s = m , and(ii) c r + c s ≡ m ).The number g ( D ′ ) − g ( D ) will be denoted by A ( D ′ ).The following result gives an explicit realization of the ( r, s )- self compatible actionyielded by an action D as above. Lemma 2.10.
Let D be an ( r, s ) -self compatible C n -action as in Definition 2.9.Then we have J D, ( r, s ) K = ( n, g + 1; ( c , n ) , . . . , \ ( c r , n r ) , . . . , \ ( c r , n s ) , . . . , ( c ℓ , n ℓ )) , where g ( J D, ( r, s ) K ) = g ( D ) + n/n r . Finally, we state the main topological result of [22], which will be used extensivelyin this paper.
Lemma 2.11.
For g ≥ , a Type 2 action on S g can be constructed from finitelymany compatibilities of the following types between spherical Type 1 actions:(i) J D, ( r, s ) K ,(ii) J D, g ′ K , J D, g ′ K ,(iii) L D , D , ( r, s ) M , and(iv) L D , D M . ATREYEE BHATTACHARYA, SHIV PARSAD, AND KASHYAP RAJEEVSARATHY Decomposing cyclic actions into irreducibles
In this section, we generalize Lemma 2.11 to obtain a topological description ofthe decomposition of an arbitrary cyclic action into irreducible components. Weshow that this decomposition can be visualized as a “necklace with beads”, wherethe beads are the irreducible components, and strings that connect a pair of beadssymbolize the compatibility between them. We will now present an example thatcaptures this idea.
Example 3.1.
Consider the spherical Type 1 actions D = (42 ,
0; (2 , , (19 , , (19 , D = (42 ,
0; (5 , , (13 , , (23 , D = (42 ,
0; (1 , , (8 , , (23 , D =(42 ,
0; (1 , , (11 , , (13 , D = (42 ,
0; (13 , , (10 , , (25 , D =(42 ,
0; (19 , , (17 , , (29 , L D , D , (3 , M , L D , D , (2 , M , L D , D M , L D , D , (2 , M , and L D , D , (3 , M , together realize the action D ′ = (42 ,
0; (2 , , (19 , , (5 , , (23 , , (1 , , (1 , , (13 , , (13 , , (19 , , (29 , on S . A visual interpretation of this realization is shown in Figure 1 below,where the number of lines connecting D i to D j are the sizes of the compatibleorbits. (Note that the number 42 refers to the number of lines connecting D D .) D D D D D D Figure 1.
A visualization of the action D ′ . Remark 3.2.
While we realize new actions from successive compatibilities acrossactions (represented by data sets), for simplicity, we assume from here on thatthe original indexing of the pairs (that correspond to cone points) in the data setsremains unaltered.Fixing the notation, J D, K := D, J D, K := D, and L D , D , (0 , M := L D , D M ,we formalize this idea in the following definition. Definition 3.3.
For 1 ≤ i ≤ k , let D i be a collection of irreducible Type 1 actionsof order n on S g i .(i) The D i are said to form a linear k -chain T = ( D , . . . , D k ) if for 1 ≤ i ≤ k − r i and s i such that actions given by D ′ = L D , D , ( r , s ) M , and D ′ j = L D ′ j , D j +1 , ( r j , s j ) M , for 2 ≤ j ≤ k − , are well defined.(ii) If in addition to (i), there exist positive integers r k and s k such that D ′ k = L D ′ k − , D ′ , ( r k , s k ) M is also well-defined, then T is said to be a closed linear k -chain .Given a k -chain T as above, we define C ( T ) = { ( r , s ) , . . . , ( r k − , s k − ) } , f ( T ) := |{ j : ( r j , s j ) = (0 , }| , A T := A ( D ′ k ), if T is closed, and D T := ( D ′ k , if T is closed D ′ k − , otherwise.It is implicit in Definition 3.3 that for 1 ≤ i < j ≤ k , the tuple ( D i , D i +1 , . . . , D j )forms a linear ( j − i + 1)-chain. In Example 3.1 above, ( D i , D i +1 , . . . , D j ), for 1 ≤ i < j ≤ T = ( D , . . . , D ),we have C ( T ) = ((3 , , (2 , , (0 , , (2 , , (3 , EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 7
Example 3.4.
In Example 3.1, we can simultaneously add the self compatibilities J D ′ , (1 , K , J D ′ , (2 , K , J D ′ , (5 , K , and J D ′ , (7 , K to realize the C -action on S given by D ′′ = (42 ,
4; (5 , , (1 , . An illustration of this realization is givenin Figure 2 below. D D D D D D Figure 2.
A visualization of the action D ′′ .Furthermore, we perform 3 successive toral subtractions to obtain a realization ofthe action D = (42 ,
1; (5 , , (1 , S .This leads us to the following definition. Definition 3.5.
For 1 ≤ i ≤ k , let D i be a collection of spherical Type 1 actionsof order n on S g i . Then the D i are said to form a necklace with k beads N := (( D , . . . , D k ); (( x , y ) , . . . , ( x m , y m )); ( g ′ , g ′′ )) , where g ′ ≥
0, and 0 ≤ g ′′ ≤ g ′ + m are integers such that:(i) When k = 1, D is either a Type 1 action or the action D = ( n,
0; ( k, n ) , ( n − k, n )) (i.e. a rotation of S by 2 πk/n .)(ii) When k ≥ D i is a irreducible Type 1 action on S g i ,(b) the tuple D T N := ( D , . . . , D k ) defines a linear k -chain,(c) if m >
1, then 0 < r ′ j , s ′ j ≤ [( k + 2 + f ( D T N ) / ≤ j ≤ m , thepairs ( r ′ j , s ′ j ) are coordinate wise distinct with D T = L D T N , ( r ′ , s ′ ) M and D jT = L D j − T , ( r ′ j , s ′ j ) M , for 2 ≤ j ≤ m, being well-defined.(d) for 1 ≤ i < j ≤ m , denoting T i,j = ( D i , D i +1 , . . . , D j ), we require T x j ,y j to be a closed linear chain for 1 ≤ j ≤ m such that A T xj,yj = A ( D jT ).(iii) Both D ′N := J D m − T , g ′ K and D N := J D ′N , g ′′ K are well defined actions.It follows by definition that if we replace the ( g ′ , g ′′ ) with a pair ( g ′ + p, g ′′ + p ),where p is a natural number, then the necklace remains unchanged. So for the casewhen g ′ = g ′′ , we simply omit the pair ( g ′ , g ′′ ). Moreover, we allow m = 0 in anecklace N , in which case, we simply write N := ( T N ; ( g ′ , g ′′ )) . Example 3.6.
Going back to Example 3.4, we see that the action D is realized asa necklace with 6 beads N = (( D , . . . , D ); ((1 , , (2 , , (5 , , (7 , , . It is interesting to note that the subnecklaces (( D , D , D ); ((1 , , D , D , D ); ((4 , , ATREYEE BHATTACHARYA, SHIV PARSAD, AND KASHYAP RAJEEVSARATHY
Proposition 3.7.
Given an arbitrary cyclic action D of order n on S g , there existsa necklace N with k beads, for some k ≥ , such that D N = D .Proof. If D is a Type 1 action, then we can see that D N = D , for N = (( J D, g ( D ) K ); ; ( g ( D ) , . Moreover, it follows from an inductive applicationof Lemma 2.11 that the result holds true for an arbitrary Type 2 action D .It remains to show that there is a necklace that realizes every rotational action.But this follows from the fact that a free D = ( n, g + 1 , r ; ) is realized by N =(( J D ′ , (1 , K ); ; ( g , D ′ = ( n,
0; ( r, n ) , ( n − r, n )) is a rotation of the sphereby 2 πr/n . Finally, a non-free rotation D = ( n, g ; ( k , n − k ) , . . . , ( k r , n − k r )) isrealized by N = (( D ′ ); ; ( g , (cid:3) Remark 3.8.
It is important to note that given an action D , there could existtwo distinct necklaces N and N such that D N = D = D N . For example,consider the action D = (5 ,
1; (1 , , (2 , , (2 , S . This can be realized by thenecklace N = (( D ′ ); ; (1 , D ′ = (5 ,
0; (1 , , (2 , , (2 , D N = D , for N = (( D , D , D ′ ); ((1 , , where D = (5 ,
0; (1 , , (1 , , (3 , , and D = (5 ,
0; (2 , , (4 , , (4 , Corollary 3.9.
Let D a cyclic action for order n on S g such that D = D N forsome necklace N , as in Definition 3.5 with g ′′ = 0 .(i) If g ′ = 0 , then there is a maximal reduction system C for D such that |C| = g − k X i =1 g ( D i ) + k − . (ii) If g ′ = 0 , then there is a maximal reduction system C for D such that |C| = g − k X i =1 g ( D i ) + k − n (2 g ′ − . Structures realizing compatibilities
In this section, we classify the structures that realize the individual componentsand compatibilities that constitute a necklace, as described in Definition 3.5. Webegin by describing the structures that realize spherical Type 1 actions, which formthe beads of the necklace.4.1.
Spherical Type 1 actions.
In this subsection, we show that the structure P D (described in Lemma 2.4) that realizes a Type 1 action D is unique. Proposition 4.1. If D is a spherical Type 1 action, then Fix ( h D i ) = {P D } is asingleton.Proof. First consider the case when n i = 2 for some i . Then D can be realized asa rotation of the regular hyperbolic n -gon P D (as in Lemma 2.4), with all interiorangles equals to 2 π/n . It follows from basic hyperbolic trigonometry that such ahyperbolic polygon is unique, which proves the result for this case.When n , n = 2, P D is a semi-regular hyperbolic 2 n -gon with side length ℓ , and alternate interior angles of measure 2 π/n and 2 π/n , respectively. Let { P , . . . , P n − } be the vertices of P D and O denotes the fixed point at the center,as shown in Figure 3 below. EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 9 P i P i +1 P i +2 O Figure 3.
The polygon P D for a C -action on S .As the rotation of P D by θ D is an isometry, it follows that | OP i | = | OP i +2 | , forall i . Hence, the hyperbolic SSS congruence implies that the triangles P i OP i +1 aremutually congruent to each other, with ∠ P i OP i +1 = π/n , ∠ OP i P i +1 = 2 π/n , and ∠ OP i +1 P i = 2 π/n . Thus P D is uniquely determined, and the assertion follows. (cid:3) Remark 4.2.
Let D be a reducible action on S g , and let C be a maximal reductionsystem for D . By extending C to a pants decomposition P of S g , we see thatdim(Fix( h D i )) ≥ |C| >
0. Conversely, suppose that dim(Fix( h D i )) >
0, we canreverse the above argument to show that D is reducible.The following corollary is immediate from Proposition 4.1 and Remark 4.2. Corollary 4.3.
A spherical Type 1 action D is irreducible. We could provide an alternative approach to the proof of Proposition 4.1 by under-standing the action induced by D in Teich( S g ), which we will denote by D . Weillustrate this idea using the following example. Example 4.4.
Consider the spherical Type 1 C -action D on S realized as therotation of the regular hyperbolic 14-gon by 2 π/
14 radians, as shown in Figure 5below. The two separating curves c = abca − b − c − and c = def d − e − f − g ba cfe dd ea gb fc R π R R Figure 4.
An order 14 action on S .(marked in red and blue resp.) cut the surface into three disjoint componentsmarked by the regions R , R and R . It is apparent that the nonseparating curves c = ab , c = de , c = g − abc , and c = ga − b − c − (marked in pink, brown, greenand orange resp.) together with s , s form a pants decomposition P of S . Since D ( c ) = c , D ( c ) = c , and D ( c ) = c , we can associate Fenchel-Nielsen coordinates ( ℓ i , θ i ) to each c i ∈ P , and concludethat D has a description as follows: D (( ℓ i , θ i )) = ( ℓ i +1 , θ i +1 ) , for i = 1 , , and D (( ℓ j , θ j )) = ( ℓ j +1 , θ j +1 ) , for j = 5 . (4.1)It is now apparent that both D and D (which is induced by an irreducible Type1 action of order 7) are not permutations of the coordinates of Teich( S g ). WhileEquation 4.1 does not readily imply that D has a unique fixed point, it is possibleto conclude the same by considering the action of h D i on other curves in the regions R i (for example, g − abga − b − ⊂ R ). Remark 4.5.
The argument in Example 4.4 can be generalized to a polygon oftype P D , which realizes an spherical Type 1 action D . In particular, for any suchaction D of order n one can find a pants decomposition P consisting of 3 g − γ , . . . , γ g − such that for each γ i (1 ≤ i ≤ g −
4) there exist γ j (1 ≤ j ≤ g − < k ij < n with D k ij ( γ i ) = γ j . However, for the sake of brevity, we abstainfrom giving the details here.Moreover, for a spherical Type 1 action to induce a permutation on Teich( S g ),there must exist a nonseparating curve c ∈ S g whose orbit under D determines amulticurve of size n (i.e D has a permutation component). However, it is apparentfrom the irreducibility of D that such an orbit cannot exist, which we formally stateas the concluding result of this subsection. Corollary 4.6.
Let D be a spherical Type 1 action on S g . Then D is not apermutation of the coordinates of Teich ( S g ) . Compatibilities of type L D , D , ( r, s ) M and J D, ( r, s ) K . Consider an ir-reducible Type 1 action D on S g , and a D -orbit of size k. Removing k mutuallydisjoint cyclically permuted (by the action of D ) discs around the points in thisorbit, we obtain a homeomorphic copy of S g,k with a homeomorphism ˆ D induced by D , which cyclically permutes the components of ∂S g,k . Note that Teich( S g ) can beviewed as a subspace of Teich( S g,k ) in the following manner. The Fenchel-Nielsencoordinates of an arbitrary structure ξ ∈ Teich( S g ) are given by ξ = Q g − i ( ℓ i , θ i ) , where the pair ( ℓ i , θ i ) denote the length and twist parameters contributed by the i -th curve of a pants decomposition P of S g where i = 1 , . . . g − P can always beextended to a pants decomposition ˆ P of S g,k where the first 3 g − P belong to P . As there are 3 g − k non-boundary curves in ˆ P , anarbitrary ˆ ξ ∈ Teich( S g,k ) can be decomposed as ˆ ξ = g − k Y i ( ℓ i , θ i ) × k Y j =1 ℓ b j , where ℓ b j denotes the length parameter of the j -th boundary component (for j = 1 , . . . , k ) of S g,k . In light of the above decomposition of ˆ ξ , two natural questions that arise are:“Does there exist an endomorphism ˆ D : Teich( S g,k ) → Teich( S g,k ) such thatˆ D | Teich( S g ) = D ? Moreover, is ˆ D | Teich( S g,k ) \ Teich( S g ) a permutation?” We willshow shortly that these questions do not always have positive answers. Consider EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 11 the decomposition Teich( S g,k ) ≈ T NB × R k + , where T NB = { g − k Y i ( ℓ i , θ i ) } and R k + ≈ { k Y j =1 ℓ b j } . The action of D implies that ˆ D , if it exists, should preserve the above decom-position of Teich( S g,k ), and furthermore, ˆ D (cid:16)Q kj =1 ℓ b j (cid:17) = Q kj =1 ℓ b σk ( j ) where σ k = (12 . . . k ) . The following result shows that ˆ D is completely determined by D if, and only if, k is a proper divisor of n . Theorem 4.7.
Let D be a spherical Type 1 action on S g of order n with a D -orbit of size k . Then D never extends to an endomorphism of Teich ( S g,k ) , whichinduces an order n permutation of the coordinates of Teich ( S g,k ) \ Teich ( S g ) . Inparticular, the extended action ˆ D is completely determined by D if, and only if, k is a proper divisor of n .Proof. As D is an spherical Type 1 action, we may assume (see Example 4.4) thatthere exists a pants decomposition P of S g with s separating curves α , . . . , α s and r non-separating curves β , . . . , β r such that for each 1 ≤ i ≤ s −
1, thereexist 1 ≤ j ≤ s ( j = i ) and 1 < M ij < n with D M ij ( α i ) = α j . Similarly, foreach 1 ≤ i ≤ r −
1, there exist 1 ≤ j ≤ r ( j = i ) and 1 < N ij < n such that D N ij ( β i ) = β j . Without loss of generality, we may assume that D N ,r ( β ) = β r .In order that D extends to an endormorphism of Teich( S g,k ), P should extendto a pants decomposition ˆ P of S g,k as in the discussion above, with k new non-boundary curves γ , . . . , γ k and k boundary curves γ ′ , . . . , γ ′ k such that ˆ D ( γ ′ i ) = γ ′ i +1 , for each i. We may assume that γ is a nonseparating curve isotopic to β r in S g , and thus ˆ D M ( γ ) = β (since D M ( γ ) = β ), and the isotopy class of β remain unaltered in S g,k , as illustrated in Figure 5 below. In the case when k = n ,it is apparent that the curve P i γ ′ i ∈ H ( S g,k ) (indicated by the dotted curve inthe polygon, and the curve γ in the bounded surface in Figure 5 below) is leftinvariant by the action of D . β γ β r γ γ γ γ ′ γ ′ γ ′ γ ′ P γ ′ i Figure 5.
Extension of a pants decomposition of S g .Hence, D has to induce an order n rotation of the component S ′ of S g \ γ home-omorphic to S ,k +1 , which cyclically permutes its k boundary components γ ′ i andfixes the k + 1-th boundary component, namely, γ . This obviates the possibilityof such an extension in this case, as D | S ′ can never induce an order k permuationof the γ i .Furthermore, it is clear from the structure P D that when k is a proper divisorof n , then γ cannot be left invariant by the action of D . Consequently, the action of ˆ D on the γ i is completely determined by the action of D on P , and hence theresult follows. (cid:3) Remark 4.8.
Let (
X, ξ ) be a closed hyperbolic surface with an isometry D of finiteorder. Let B p ( r ) denote the closed disc of radius r centered at any point p ∈ X. Here, r is bounded above by the injectivity radius r ξ ( p ) at p . If D ( p ) = p and D ( B p ( r )) = B p ( r ) such that D | B p ( r ) becomes a rotation about p , then r ≤ r M ,where r M = sup p ∈ X r ξ ( p ) . Note that this is a consequence of the derivative of D at p being a rotation about the origin in T p X , and the fact that the exponential mapis a radial isometry.The following result describes the structures that realize compatibilities of type L D , D , ( r, s ) M . Corollary 4.9.
Let D = L D , D , ( r, s ) M , where the D i are spherical Type 1 ac-tions.(i) If ( r, s ) = (0 , , thenFix ( h D i ) ≈ { [ P D ] } × { [ P D ] } × (0 , ℓ ( D )] × R , where ℓ ( D ) is a positive constant determined by D .(ii) If ( r, s ) = (0 , , thenFix ( h D i ) ≈ { [ P D ] } × { [ P D ] } × Y j =1 ((0 , ℓ j ( D )] × R ) , where for each j , ℓ j ( D ) is a positive constant determined by D .Proof. We will only prove (i), as (ii) will follow from a similar argument. ByTheorem 4.7, it is apparent that the action induced by the D i on S g i ,k is com-pletely determined by the action of D i on the S g i . So any structure that realizes L D , D , ( r, s ) M as an isometry, is uniquely determined by the structures P D i , andone additional length and twist parameter contributed by the isometric boundarycomponents (cyclically permuted by the D i ) of S g i ,k .Let ℓ denote the length of each boundary component of S g i ,k . It remains toshow that ℓ ≤ ℓ ( D ), where ℓ ( D ) is a positive constant determined by D . To seethis, consider the unique hyperbolic surface ( X i , ξ ih ) (for i = 1 ,
2) realizing D i as an isometry. For each i , let { p ij } ≤ j ≤ k ⊂ X i be the points in a distinguishedcompatible D i -orbit of size k . Let B ij ( r i ) := B p ij ( r i ) denote mutually disjointcyclically permuted disks under D i . Since D ki ( B ij ( r i )) = B ij ( r i ) , it follows fromRemark 4.8 that r i ≤ r M i . Thus the circumference c ij of each B ij ( r i ) satisfies c ij = 2 π sinh( r i ) ≤ π sinh( r M i ) = L i (say) . Let L = min( L , L ) , and r D = min( r M , r M ). Removing { B ij ( r ) } ≤ j ≤ k (where r ≤ r D and the circumference c ( r ) of B ij ( r ) satifies c ( r ) ≤ L ) from each X i , andgluing the surfaces X i \ ∪ j B p ij ( r ) along their boundary components, we obtain adiffeomorphic copy X of S g + g + k − with a C n action D , and a reduction system C consisting of k nonseparating curves. Moreover, X admits a canonical Riemannianmetric ξ realizing D as an isometry with each curve of C having length c ( r ). By theuniformization theorem, there is a unique hyperbolic metric ξ h = e f ξ on X , alsorealizing D as an isometry, where f = f ( ξ , ξ ) is a smooth real valued function on X . The result (i) now follows from the observation that under ξ h , each curve of C has length ℓ h = ℓ h ( c ( r ) , f ) ≤ ℓ ( D ) where ℓ ( D ) = ℓ ( L, f ) is a unique constant (as
L, f are uniquely determined by D ). (cid:3) EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 13
Considering the similarities between the compatibilities J D ′ , ( r, s ) K and L D , D , ( r, s ) M , it is quite evident that the structures that realize J D ′ , ( r, s ) K shouldalso arise analogously, and so we have the following. Corollary 4.10.
Let D = J D ′ , ( r, s ) K be an action of order n on S g . Then,Fix ( h D i ) ≈ Fix ( h D ′ i ) × (0 , ℓ ( D ′ )] × R , where ℓ ( D ′ ) is a positive constant determined by D ′ . Compatibilities of type J D, g K and J D, g K . Let D be an action of order n on S g . As we saw earlier, an action of type J D, g K is realized by pasting apermutation component (that cyclically permutes n isometric copies of S g , ) tothe action D . As we saw earlier, the action J D, g K can also be realized iterativelyfrom g compatibilities of type J D, K . Besides, the arguments in Theorem 4.7would imply that each copy of S , (that is attached in a J D, K type construction)contributes 2 additional length parameters, and 1 twist parameter. Furthermore,following the arguments in Corollary 4.9, we can show that one of the length param-eters (contributed by ∂ ( S , , ) is bounded by a positive constant that is determineduniquely by the action on which the permutation component is pasted. Hence,when the compatibility J D, g K is completed, a total of 3 g − h D i ), and so we havethe following result. Corollary 4.11.
Let D be a cyclic action of order n on S g . Suppose that theactions J D, g K and J D, g K are well defined, for some g , g ≥ . Then(i) Fix ( h J D, g K i ) ≈ Fix ( h D i ) × g Y i =1 ((0 , ℓ i ( D )] × R ) × g − Y i =1 ( R + × R ) , where each ℓ i ( D ) is a positive constant determined by the action J D, g K .(ii) Fix ( h J D, g K i ) ≈ Fix ( h D i ) / g Y i =1 ((0 , ℓ i ( D )] × R ) × g − Y i =1 ( R + × R ) ! , where each ℓ i ( D ) is a positive constant determined by the action J D, g K . Structures that realize arbitrary actions
In this section, we will piece together the structures detailed in the Section 4(that realize various kinds of compatibilities) to describe the structures that willrealize arbitrary cyclic actions. Recalling that for an arbitrary cyclic action D ,there exists a necklace(*) N = (( D , . . . , D k ); (( x , y ) , . . . , ( x m , y m )); ( g ′ , g ′′ ))as in Definition 3.5, such that D N = D (see Proposition 3.7), we will now state themain result in this paper. Theorem 5.1 (Main Theorem) . Let D be a cyclic action of order n on S g , andlet N be a necklace as in ( ∗ ) such that D N = D . Then Fix ( h D i ) ≈ M /M , where M = k Y i =1 {P D i } × g ′ + k +2 f ( T N )+ m − Y i =1 ((0 , ℓ ′ i ( D )] × R ) × g ′ − Y i =1 ( R + × R ) and M = g ′′ Y i =1 ((0 , ℓ ′′ i ( D )] × R ) × g ′′ − Y i =1 ( R + × R ) , where the ℓ ′ j ( D ) and ℓ ′′ j ( D ) are positive constants determined by D . Consequently, dim( Fix ( h D i ) = 6( g ′ − g ′′ ) + 2 k + 4 f ( T N ) + 2 m − . The proof of this theorem is a direct consequence of Theorem 4.7 and Corollaries4.9, 4.10 and 4.11. In classical parlance, Fix( h D i ) is also known as the branchedlocus of D . An immediate consequence of Theorem 5.1, is the following result dueto Harvey [12, 17]. Corollary 5.2.
Let D be a cyclic action of order n on S g such that O D has c conepoints. Then dim( Fix ( h D i )) = 6 g ( D ) + 2 c − . Proof.
This follows directly from Theorem 5.1 by observing that g ( D N ) = g ′ − g ′′ + m and the number of cone points in O D N = k + 2 f ( T N ) − m + 2 . (cid:3) Example 5.3.
For the necklace structure realizing the action D in Example 3.4,we see that k = 6, m = 4, f ( T N ) = 1, and ( g ′ , g ”) = (0 , h D i ) ≈ M /M , where M = Y i =1 (0 , ℓ ′ i ( D )] ! × R and M = Y i =1 (0 , ℓ ′′ i ( D )] × R ! × R × R , and so we have dim(Fix( h D i )) = 20 −
16 = 4.Corollary 5.2 leads us to the following result due to Gilman [10] that characterizesirreducible cyclic actions.
Corollary 5.4.
A cyclic action D on S g is irreducible if, and only if g ( D ) = 0 and O D is an orbifold with three cone points.Proof. Consider an action D on S g of the form D = ( n,
0; ( c , n ) , ( c , n ) , ( c , n )) , and let N be any necklace with k beads such that D N = D . It follows fromCorollary 5.2 that dim(Fix( h D i )) = 0. Therefore, by Remark 4.2, we conclude that D is irreducible.Conversely, suppose that D is irreducible. Then g ( D ) = 0, as otherwise, D would have a nontrivial permutation component. By Remark 4.2, it followsdim(Fix( h D i )) = 0, and so Corollary 5.2 would imply that O D has exactly 3 conepoints, and the assertion follows. (cid:3) Let C be a maximal reduction system for a reducible action D on S g . The concludingresult of this section, which is a direct consequence of Theorem 5.1, shows that thedifference between the number of orbits induced by D on C and S g \ C depends onlyon g ( D ). Corollary 5.5.
Let C be a maximal reduction system for a reducible action D oforder n on S g . Suppose that ℓ and k are the number of distinct orbits induced by D on C and S g \ C , respectively. Then ℓ = 3 g ( D ) − r and k = 2 g ( D ) − r. It is quite apparent that the numbers ℓ and k in Corollary 5.5 are precisely thenumber of curves and the number of pants, respectively, in a pants decompositionof the surface of genus g ( D ) with r punctures.6. Relation between cyclic actions and filling systems
A collection Ω = { γ , . . . , γ s } of simple closed curves on S g is called a filling ofsize s if S g \ Ω is a disjoint union of k topological disks, for some k ≥
1. A fillingΩ of S g is called minimal , if k = 1. It was shown in [24] that a filling Ω of S g of size s corresponds to a 4-regular fat graph of genus g with s standard cyclesand b boundary components for some b ≥
1. We call such a graph a filling graphof genus g with b boundary components, and when b = 1 it is called a minimal EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 15 filling graph . As an automorphism h of a fat graph of genus g yields a cyclic action D h on S g [22], a natural question is whether one can classify the cyclic actions on S g , which corresponds to automorphisms of some filling graph of genus g . To thiseffect, we establish the following result. Theorem 6.1.
Let Γ be a filling graph of genus g ≥ , and let h ∈ Aut (Γ) be oforder n . Then D h ∈ Mod ( S g ) is irreducible if, and only if, ( g, n ) = (1 , . In order to prove Theorem 6.1, we need the following technical lemmas. The firstresult is a direct application of Corollary 5.4 and the Riemann-Hurwitz equation.
Lemma 6.2.
Let H = h h i be an irreducible C n -action on S g . Then we have, g + 1 ≤ n ≤ g + 2 . Lemma 6.3.
Let Γ be a minimal filling graph of genus , and let h ∈ Aut (Γ) be oforder n . Then n divides .Proof. It follows from the work of Sanki [24] that there exist unique minimal fillinggraphs Γ and Γ corresponding to minimal fillings of sizes 3 and 4, respectively.Moreover, it was shown that ∂ Γ = e e − e e − e − e e − e e − e e e − and ∂ Γ = f f f f − f − f − f − f f f f − f − . It is easy to see that Aut(Γ ) ∼ = Z , wherethe generator corresponds to the hyperelliptic involution on S , and Aut(Γ ) ∼ = Z ,whose generator corresponds to the action D = (4 ,
0; (1 , , (1 , , (1 , , S . (cid:3) Lemma 6.4.
Let g ≥ and let D be an irreducible C n -action on S g .(i) If n = 4 g − , then ( g, n ) = (2 , .(ii) If n = 4(2 g − / , then ( g, n ) = (5 , or (8 , .Proof. Let h be of order n = (4 g − D is of form ( n,
0; ( c , n ) , ( c , n ) , ( c , n )) . By the Riemann-Hurwitz equation, we have n + n = , which implies that,( n , n ) = (4 ,
4) or (3 , i ) that ( g, n ) =(2 , (cid:3) Lemma 6.5.
Let g ≥ and let D be an irreducible C n -action on S g . Suppose that D can be realized by an automorphism of some minimal filling graph of genus g .Then ( g, n ) / ∈ { (5 , , (8 , } .Proof. When ( g, n ) = (5 , D must have the form (12 ,
0; ( c , , ( c , , ( c , D be realized by an automorphism of some minimal fillinggraph Γ. Let V (Γ) denote the vertex set of Γ. Then | V (Γ) | = 9, and the actionpartitions V (Γ) into k disjoint orbits of sizes t , . . . , t k . By the given condition, k = 2 and 9 = t + t = 12 / /
12 = 3, which is a contradiction. A similarargument works for the case when ( g, n ) = (8 , (cid:3) We will first establish Theorem 6.1 for the case when Γ is a minimal filling graph.
Proposition 6.6.
Let Γ be a minimal filling graph of genus g ≥ , and let h ∈ Aut (Γ) be of order n . Then D h ∈ Mod ( S g ) is irreducible if, and only if, ( g, n ) =(1 , .Proof. By hypothesis, D can be realized as an automorphism of a minimal fillinggraph of genus g , which implies that h can be described as a rotation of a (8 g − n | (8 g − g = 1readily follows by a direct application of the Riemann-Hurwitz equation.Suppose that g ≥ | (2 g − n ∈{ (4 g − , g − / } . Similarly, if g ≥ ∤ (2 g − n = (4 g − (cid:3) We are now ready to prove Theorem 6.1.
Proof (of Theorem 6.1).
Suppose that Γ has b boundary components. The casewhen b = 1 follows from Proposition 6.6. When b ≥
2, a simple Euler characteristicargument shows that Γ has v = 2 g − b vertices, and so n | g − b ). Sincethe b boundary components correspond to an orbit of size n/b under D h , it impliesthat b ≤ n/ ≤ (4 g + 2) / g + 1.If n = 4(2 g − b ) /k , then applying n ≥ g + 1 ≥ b , we get 2 ≤ k ≤ n = 4 v/k , for 2 ≤ k ≤
8, where v = 2 g − b . As D h is irreducible, it has the form ( n,
0; ( c , n ) , ( c , n ) , ( c , n )), which implies b = n/n i , for some i . Further, we observe that n/n i + n/n j ≤ n/
6, for i = j .For k = 4 (resp. 8), we have v = n (resp. 2 n ), which is impossible. For k = 2, wehave v = n/
2, which implies n/n i + n/n j = n/ i = j , which yields twosolutions ( n i , n j ) = (4 ,
4) or (3 , n i , n j ) = (3 ,
6) implies that b = 1,which is a contradiction. So, the only possibility that survives is ( n i , n j ) = (4 , g, n ) = (1 , k = 3 , , and 7.Finally, it remains to examine the case when k = 5, that is, v = 5 n/
4. In thiscase, we have n/n i + n/n j = n/
4, for some i = j , whose solutions are ( n i , n j ) =(5 , , (6 ,
12) or (8 , b ≥
2, the only feasible solution is ( n i , n j ) = (8 , b = 4 and g = 2. But this is impossible as 5 ∤ g − b , which completesthe proof. (cid:3) References [1] Robert Benim and Aaron Wootton. Enumerating quasiplatonic cyclic group actions.
RockyMountain J. Math. , 43(5):1459–1480, 2013.[2] Thomas Breuer.
Characters and automorphism groups of compact Riemann surfaces , vol-ume 280 of
London Mathematical Society Lecture Note Series . Cambridge University Press,Cambridge, 2000.[3] S. Allen Broughton. Classifying finite group actions on surfaces of low genus.
J. Pure Appl.Algebra , 69(3):233–270, 1991.[4] S. Allen Broughton and Aaron Wootton. Finite abelian subgroups of the mapping class group.
Algebr. Geom. Topol. , 7:1651–1697, 2007.[5] Allan L. Edmonds. Surface symmetry. I.
Michigan Math. J. , 29(2):171–183, 1982.[6] Benson Farb and Dan Margalit.
A primer on mapping class groups , volume 49 of
PrincetonMathematical Series . Princeton University Press, Princeton, NJ, 2012.[7] W. Fenchel. Estensioni di gruppi discontinui e trasformazioni periodiche delle superficie.
AttiAccad. Naz Lincei. Rend. Cl. Sci. Fis. Mat. Nat.(8) , 5:326–329, 1948.[8] W. Fenchel. Remarks on finite groups of mapping classes.
Mat. Tidsskr. B. , 1950:90–95, 1950.[9] Jane Gilman. On conjugacy classes in the Teichm¨uller modular group.
Michigan Math. J. ,23(1):53–63, 1976.[10] Jane Gilman. Structures of elliptic irreducible subgroups of the modular group.
Proc. LondonMath. Soc. (3) , 47(1):27–42, 1983.[11] W. J. Harvey. Cyclic groups of automorphisms of a compact Riemann surface.
Quart. J.Math. Oxford Ser. (2) , 17:86–97, 1966.[12] W. J. Harvey. On branch loci in Teichm¨uller space.
Trans. Amer. Math. Soc. , 153:387–399,1971.[13] Gareth A. Jones and David Singerman. Theory of maps on orientable surfaces.
Proc. LondonMath. Soc. (3) , 37(2):273–307, 1978.[14] Svetlana Katok.
Fuchsian groups . Chicago Lectures in Mathematics. University of ChicagoPress, Chicago, IL, 1992.[15] Steven P. Kerckhoff. The Nielsen realization problem.
Ann. of Math. (2) , 117(2):235–265,1983.[16] Sergei K. Lando and Alexander K. Zvonkin.
Graphs on surfaces and their applications , vol-ume 141 of
Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2004.[17] C. Maclachlan and W. J. Harvey. On mapping-class groups and Teichm¨uller spaces.
Proc.London Math. Soc. (3) , 30(part 4):496–512, 1975.
EOMETRIC REALIZATIONS OF CYCLIC ACTIONS ON SURFACES - II 17 [18] Darryl McCullough and Kashyap Rajeevsarathy. Roots of Dehn twists.
Geom. Dedicata ,151:397–409, 2011.[19] M. Mulase and M. Penkava. Ribbon graphs, quadratic differentials on Riemann surfaces, andalgebraic curves defined over Q . Asian J. Math. , 2(4):875–919, 1998.[20] Jakob Nielsen.
Die Struktur periodischer Transformationen von Fl¨achen , volume 11. Levin& Munksgaard, 1937.[21] Jakob Nielsen. Abbildungsklassen endlicher Ordnung.
Acta Math. , 75:23–115, 1943.[22] Shiv Parsad, Kashyap Rajeevsarathy, and Bidyut Sanki. Geometric realizations of cyclicactions on surfaces.
Journal of Topology and Analysis , doi:10.1142/S1793525319500365, 2018.[23] Kashyap Rajeevsarathy and Prahlad Vaidyanathan. Roots of dehn twists about multicurves.
Glasgow Mathematical Journal , doi:10.1017/S0017089517000283:1–29, 2017.[24] Bidyut Sanki. Filling curves on closed surfaces.
Journal of Topology and Analysis ,doi:10.1142/S1793525318500309, 2018.[25] William P. Thurston.
Three-dimensional geometry and topology. Vol. 1 , volume 35 of
Prince-ton Mathematical Series . Princeton University Press, Princeton, NJ, 1997. Edited by SilvioLevy.[26] Heiner Zieschang.
Finite groups of mapping classes of surfaces , volume 875 of
Lecture Notesin Mathematics . Springer-Verlag, Berlin, 1981.
Department of Mathematics, Indian Institute of Science Education and ResearchBhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India
E-mail address : [email protected] URL : https://sites.google.com/iiserb.ac.in/homepage-atreyee-bhattacharya/home?authuser=1 Department of Mathematics, Indian Institute of Science Education and ResearchBhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India
E-mail address : [email protected] Department of Mathematics, Indian Institute of Science Education and ResearchBhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India
E-mail address : [email protected] URL : https://home.iiserb.ac.in/ ee