Split metacyclic actions on surfaces
aa r X i v : . [ m a t h . G T ] S e p SPLIT METACYCLIC ACTIONS ON SURFACES
NEERAJ K. DHANWANI, KASHYAP RAJEEVSARATHY, AND APEKSHA SANGHI
Abstract.
Let Mod( S g ) be the mapping class group of the closed ori-entable surface S g of genus g ≥
2. In this paper, we derive necessary andsufficient conditions under which two torsion elements in Mod( S g ) willhave conjugates that generate a finite split nonabelian metacyclic sub-group of Mod( S g ). As applications of the main result, we give a completecharacterization of the finite dihedral and the generalized quaternionicsubgroups of Mod( S g ) up to a certain equivalence that we will call weakconjugacy. Furthermore, we show that any finite-order mapping classwhose corresponding orbifold is a sphere, has a conjugate that lifts undercertain finite-sheeted regular cyclic covers of S g . Moreover, for g ≥
5, weshow the existence of an infinite dihedral subgroup of Mod( S g ) that isgenerated by the hyperelliptic involution and a root of a bounding pairmap of degree 3. Finally, we provide a complete classification of the weakconjugacy classes of the non-abelian finite split metacyclic subgroups ofMod( S ) and Mod( S ). Introduction
Let S g be the closed orientable surface of genus g ≥
0, Homeo + ( S g )be the group of orientation-preserving homeomorphisms on S g , and letMod( S g ) be the mapping class group of S g . Given F, G ∈ Mod( S g ) offinite order, a pair of conjugates F ′ , G ′ (of F, G resp.) may (or may not)generate a subgroup isomorphic to h F, G i . For example, consider the pe-riodic mapping classes F, G ∈ Mod( S ) represented by homeomorphisms F , G ∈
Homeo + ( S g ) (see [18] for details), as shown in the first subfigure ofFigure 1 below. F G π π F π π π π π G π G π G π F π π π π Figure 1.
Split metacyclic subgroups of Mod( S ) with con-jugate generators. Mathematics Subject Classification.
Primary 57M60; Secondary 57M50.
Key words and phrases. surface; mapping class; finite order maps; metacyclicsubgroups.
From Figure 1, it is apparent that h F, G i ∼ = D (i.e. the dihedral group oforder 8). For 1 ≤ i ≤
3, we consider the conjugates G i of G , represented bythe G i ∈ Homeo + ( S ) and for 1 ≤ j ≤
2, we consider the conjugates F j of F indicated in the (second and third) subfigures. In the second subfigure, wehave marked the fixed points of a conjugate F of F (with the same localrotation angles as F ). Also, note that the third subfigure is different from thefirst (as an imbedding S → R ), since has four pairs of tubes connecting thespheres, where in each pair, the tubes are aligned one behind the other. As itturns out, h F , G i ∼ = h F , G i ∼ = D , but since F and G commute, we have h F , G i ∼ = Z × Z . This example motivates the following natural question:Given F ′ , G ′ ∈ Mod( S g ) of orders n, m (resp.), can one derive equivalentconditions under which there exist conjugates F, G (of F ′ , G ′ resp.) suchthat h F, G i is a finite nonabelian split metacyclic subgroup of order m · n and twist factor k admitting the presentation h F, G | F n = G m = 1 , G − F G = F k i ∼ = Z n ⋊ k Z m . Considering that the finite abelian subgroups of Mod( S g ) have been exten-sively studied [6, 8, 11, 16], in the main result of this paper, we answer thisquestion in the affirmative for k = 1.Given a finite split (nonabelian) metacyclic subgroup H = h F, G i ofMod( S g ) as above, the Nielsen realization theorem [13, 17] asserts that wemay also view H as a subgroup of Homeo + ( S g ) with an associated H -actionon S g inducing the branched cover S g → S g /H . In order to establish themain result, we develop a notion of equivalence involving the finite split (non-abelian) metacyclic subgroups of Mod( S g ) called weak conjugacy . Two iso-morphic finite split metacyclic subgroups H = h F , G i and H = h F , G i of Mod( S g ) are said to be weakly conjugate if ( F , G ) is pairwise conjugateto ( F , G ), and the associated H i -actions on S g satisfy a certain equiva-lence (see Definition 2.10 for a rigorous definition). Even though this notionof equivalence of subgroups is (in general) weaker than conjugacy, it doesprovide us with an effective way of differentiating between isomorphic splitmetacyclic subgroups of Mod( S g ) by virtue of their topological realizations(i.e. by understanding how the associated topological actions on S g differfrom each other.)The main ingredient in the proof of the main result is the derivationnecessary and sufficient conditions for the weak conjugacy of split metacyclicsubgroups by establishing a correspondence between these weak conjugacyclasses and certain abstract tuples of integers called split metacyclic datasets (see Section 3). Furthermore, we note that the isomorphism class of afinite split metacyclic subgroup H is the union of the weak conjugacy classesthat H represents. Consequently, we obtain equivalent conditions underwhich two torsion elements in Mod( S g ) will have conjugates that generate afinite split metacyclic subgroup of Mod( S g ). This result is a generalizationof an analogous result from [8] for two-generator finite abelian subgroups.Moreover, given such a data set, these conditions also provide an immediateway of recovering the conjugacy classes of the pair of generators associatedwith the weak conjugacy class. The proof integrates ideas from the theoryof group actions on surfaces [13, 15] with elements of Thurston’s orbifoldtheory [20, Chapter 13]. In view of the Nielsen realization theorem, consider PLIT METACYCLIC ACTIONS ON SURFACES 3 representatives F , G ∈
Homeo + ( S g ) of F, G (resp.) with the same orders.Another key idea in the proof (of the main result) is the analysis of thegeometric properties of the automorphism induced by G in S g / hF i .In Section 4, we provide several applications of our main theorem. Thefirst application concerns the finite dihedral subgroups of Mod( S g ). Let D n = Z n ⋊ − Z be the dihedral group of order 2 n . We derive the followingcharacterization of dihedral subgroups of Mod( S g ) in Subsection 4.1. Proposition 1.
Let F ∈ Mod( S g ) be of order n . Then there exists aninvolution G ∈ Mod( S g ) such that h F, G i ∼ = D n if and only if F and F − are conjugate in Mod( S g ) . It is worth mentioning here that dihedral actions on Riemann surfaces havebeen classified in [7]. Furthermore, for g ≥
2, in Subsection 4.2, we describea special family of split metacyclic actions on S g − that yield, and alsocorrespond to generalized quaternionic actions on S g (see Proposition 4.7).We plan to undertake the generalization of this result to arbitrary finitemetacyclic groups in future works.Let S g,r be the surface of genus g with r punctures. Let p : S ˜ g → S g,r bea branched cover with deck transformation group hF i , where F is periodic.Let LMod p ( S g,r ) (resp. SMod p ( S ˜ g )) be the liftable (resp. symmetric) map-ping class groups of p . In Subsection 4.3, we analyze the cyclic subgroupsof Mod( S g,r ) that lift to finite split metacyclic groups under certain coversof type p . For a periodic mapping class F ∈ Mod( S g ), let O hFi := S g / hF i denote the corresponding orbifold of F . It is known [10] that F is irreducibleif and only if O hFi ≈ S , . An irreducible mapping class F ∈ Mod( S g ) oforder n is said to be of Type 1 if O hFi has a cone point of order n . In thefollowing application, we describe how finite cyclic subgroups of Mod( S , )can lift (under p ) to finite split metacyclic subgroups of Mod( S ˜ g ), when F is irreducible. Corollary 1. p : S ˜ g → S , be a branched cover with deck transformationgroup hF i , where F is an irreducible mapping class of order n . Suppose thata non-trivial element G ′ ∈ Mod( S , ) has a conjugate G ∈ LMod p ( S , ) .Then:(i) F is of Type 1,(ii) | G | = 2 or , and(iii) G has lift ˜ G ∈ SMod( S ˜ g ) such that either h F, ˜ G i ∼ = Z n ⋊ k Z or h F, ˜ G i ∼ = Z n ⋊ k Z , for some k ∈ Z × n . When p is a regular cyclic cover (i.e the hF i -action on S ˜ g is free), we have r = 0 and ˜ g = n ( g −
1) + 1. In this context, we show the following.
Proposition 2.
For g, n ≥ , let p : S n ( g − → S g be a regular cover withdeck transformation group Z n = hF i . Then any involution G ′ ∈ Mod( S g ) has a conjugate G ∈ LMod( S g ) with a lift ˜ G ∈ SMod( S n ( g − ) such that h F, ˜ G i ∼ = D n . Moreover, we provide sufficient conditions for the liftability (under p ) of aperiodic mapping class whose corresponding orbifold is a sphere (see Propo-sitions 4.14 - 4.15). As a consequence, we obtain the following corollary. N. K. DHANWANI, K. RAJEEVSARATHY, AND A. SANGHI
Corollary 2.
For g ≥ and prime n , let p : S n ( g − → S g be a regular n -sheeted with deck transformation group hF i ∼ = Z n . Let G ′ ∈ Mod( S g ) beof order m such that the genus of O hG ′ i is zero. Then G ′ has a conjugate G ∈ LMod p ( S g ) with a lift ˜ G ∈ SMod p ( S g ) such that h F, ˜ G i ∼ = Z m ⋊ k Z n ifthere exists k ∈ Z × n such that | k | = m . An infinite dihedral group is a group that admits a presentation of theform h x, y | y = 1 , y − xy = x − i . By a root of a mapping class F ∈ Mod( S g ) of degree n , we mean a G ∈ Mod( S g ) such that G n = F . As a final application, in Subsection 4.4, weuse the theory developed in [19], to construct roots of bounding pair mapsin Mod( S g ) of degree 3, for g ≥
5, which together with the hyperellipticinvolution, generate infinite dihedral subgroups of Mod( S g ). Proposition 3.
For g ≥ , there exists an infinite dihedral subgroup of Mod( S g ) that is generated by the hyperelliptic involution and a root of abounding pair map of degree . In Section 5, we give complete classifications of the weak conjugacy classesof finite split metacyclic subgroups of Mod( S ) and Mod( S ). It may benoted that similar classifications for 2 ≤ g ≤ S ) and Mod( S ).2. Preliminaries
Fuchsian groups.
Let Homeo + ( S g ) denote the group of orientation-preserving homeomorphisms on S g , and let H <
Homeo + ( S g ) be a finitegroup. A faithful and properly discontinuous H -action on S g induces abranched covering S g → O H := S g /H with ℓ cone points x , . . . , x ℓ on the quotient orbifold O H ≈ S g (which wewill call the corresponding orbifold ) of orders n , . . . , n ℓ , respectively. Thenthe orbifold fundamental group π orb1 ( O H ) of O H has a presentation given by(1) * α , β , . . . , α g , β g , ξ , . . . , ξ ℓ | ξ n , . . . , ξ n ℓ ℓ , ℓ Y i =1 ξ i g Y i =1 [ α i , β i ] + . In classical parlance, π orb1 ( O H ) is also known as a Fuchsian group [12, 15]with signature Γ( O H ) := ( g ; n , . . . , n ℓ ) , and the relation Q ℓi =1 ξ i Q g i =1 [ α i , β i ] appearing in its presentation is calledthe long relation . From Thurston’s orbifold theory [20, Chapter 13], weobtain exact sequence(2) 1 → π ( S g ) → π orb1 ( O H ) φ H −−→ H → . In this context, we will require the following result due to Harvey [11].
PLIT METACYCLIC ACTIONS ON SURFACES 5
Lemma 2.1.
A finite group H acts faithfully on S g with Γ( O H ) = ( g ; n , . . . , n ℓ ) if and only if it satisfies the following two conditions:(i) g − | H | = 2 g − ℓ X i =1 (cid:18) − n i (cid:19) , and(ii) there exists a surjective homomorphism φ H : π orb1 ( O H ) → H that pre-serves the orders of all torsion elements of Γ . Cyclic actions on surfaces.
For g ≥
1, let F ∈ Mod( S g ) be of order n . The Nielsen-Kerckhoff theorem [13, 17] asserts that F is represented by a standard representative F ∈
Homeo + ( S g ) of the the same order. We refer toboth F and the group it generates, interchangeably, as a Z n -action on S g .Each cone point x i ∈ O hFi lifts to an orbit of size n/n i on S g , and the localrotation induced by F around the points in each orbit is given by 2 πc − i /n i ,where gcd( c i , n i ) = 1 and c i c − i ≡ n i ). Further, it is known (see [11]and the references therein) that the exact sequence in 2.2 takes the followingform 1 → π ( S g ) → π orb1 ( O hFi ) φ hFi −−−→ hF i → , where φ hFi ( ξ i ) = F ( n/n i ) c i , for 1 ≤ i ≤ ℓ . We will now introduce a tuple ofintegers that encodes the conjugacy class of a Z n -action on S g . Definition 2.2. A data set of degree n is a tuple D = ( n, g , r ; ( c , n ) , . . . , ( c ℓ , n ℓ )) , where n ≥ g ≥
0, and 0 ≤ r ≤ n − c i ∈ Z × n i suchthat:(i) r > ℓ = 0 and gcd( r, n ) = 1, whenever r > n i | n ,(iii) lcm( n , . . . b n i , . . . , n ℓ ) = N , for 1 ≤ i ≤ ℓ , where N = n , if g = 0, and(iv) ℓ X j =1 nn j c j ≡ n ).The number g determined by the Riemann-Hurwitz equation(3) 2 − gn = 2 − g + ℓ X j =1 (cid:18) n j − (cid:19) is called the genus of the data set, denoted by g ( D ).Note that quantity r (in Definition 2.2) will be non-zero if and only if D represents a free rotation of S g by 2 πr/n , in which case, D will take the form( n, g , r ; ). We will not include r in the notation of a data set, whenever r = 0.By the Nielsen-Kerckhoff theorem, the canonical projection Homeo + ( S g ) → Mod( S g ) induces a bijective correspondence between the conjugacy classesof finite-order maps in Homeo + ( S g ) and the conjugacy classes of finite-ordermapping classes in Mod( S g ). This leads us to the following lemma (thatfollows from [19, Theorem 3.8] and [11]), which allows us to use data sets todescribe the conjugacy classes of cyclic actions on S g . N. K. DHANWANI, K. RAJEEVSARATHY, AND A. SANGHI
Lemma 2.3.
For g ≥ and n ≥ , data sets of degree n and genus g correspond to conjugacy classes of Z n -actions on S g . We will denote the data set corresponding to the conjugacy class of a periodicmapping class F by D F . For compactness of notation, we also write a dataset D (as in Definition 2.2) as D = ( n, g , r ; (( d , m ) , α ) , . . . , (( d ℓ ′ , m ℓ ′ ) , α ℓ ′ )) , where ( d i , m i ) are the distinct pairs in the multiset S = { ( c , n ) , . . . , ( c ℓ , n ℓ ) } ,and the α i denote the multiplicity of the pair ( d i , m i ) in the multiset S = { ( c , n ) , . . . , ( c ℓ , n ℓ ) } . Further, we note that every cone point [ x ] ∈ O hFi corresponds to a unique pair in the multiset S appearing in D F , which wedenote by P x := ( c x , n x ).Given u ∈ Z × m and G ∈ H ≤ Homeo + ( S g ) be of order m , let F G ( u, m )denote the set of fixed points of G with induced rotation angle 2 πu − /m .Let C H ( G ) be the centralizer of an G ∈ H and ∼ denote the conjugationrelation between any two elements in H . We conclude this subsection bystating the following result from the theory of Riemann surfaces [4], whichwe will use in the proof of our main theorem. Lemma 2.4.
Let
H <
Homeo + ( S g ) with Γ( O H ) = ( g ; n , . . . , n ℓ ) , and let G ∈ H be of order m . Then for u ∈ Z × m , we have | F G ( u, m ) | = | C H ( G ) | · X ≤ i ≤ ℓm | n i G∼ φ H ( ξ i ) niu/m n i . Hyperbolic structures realizing cyclic actions.
Given a finite sub-group
H <
Mod( S g ), let Fix( H ) denote the subspace of fixed points in theTeich ¨muller space Teich( S g ) under the action of H . When H is cyclic, amethod for constructing the hyperbolic metrics representing the points inFix( H ) was described in [1] and [18], thereby yielding explicit solutions tothe Nielsen realization problem [13, 17]. This method involved the decompo-sition of an arbitrary periodic element in Mod( S g ) (that is not realizable asa rotation of S g ) into irreducible Type 1 components, which are uniquely re-alized as rotations of certain special hyperbolic polygons with side-pairings.A mapping class that is not reducible is called irreducible . Let F ∈ Mod( S g ) be of order n . Gilman [10] showed that F is irreducible if andonly if Γ( O hFi ) has the form (0; n , n , n ) (i.e. the quotient orbifold O hFi is a sphere with three cone points.) Following the nomenclature in [1, 18], F is rotational if F is a rotation of S g about an axis under an appropriateisometric embedding of S g ֒ → R . A non-rotational F is said to be of Type1 if Γ( O hFi ) = ( g ; n , n , n ), otherwise, it is called a Type 2 action. Thefollowing result describes the unique hyperbolic structure that realizes anirreducible Type 1 action.
Theorem 2.5.
For g ≥ , consider a irreducible Type 1 action F ∈ Mod( S g ) with D F = ( n,
0; ( c , n ) , ( c , n ) , ( c , n )) . PLIT METACYCLIC ACTIONS ON SURFACES 7
Then F can be realized explicitly as the rotation θ F = 2 πc − n of a hyperbolicpolygon P F with a suitable side-pairing W ( P F ) , where P F is a hyperbolic k ( F ) -gon with k ( F ) := ( n, if n , n = 2 , and n, otherwise,and for ≤ m ≤ n − , W ( P F ) = n Y i =1 a i − a i with a − m +1 ∼ a z , if k ( F ) = 2 n, and n Y i =1 a i with a − m +1 ∼ a z , otherwise,where z ≡ m + qj (mod n ) with q = ( n/n ) c − and j = n − c . Further, it was shown that the process of realizing an arbitrary non-rotational action F of order n using these unique hyperbolic structures real-izing irreducible Type 1 components involved two broad types of processes.(a) k -compatibility. In this process, for i = 1 ,
2, we take a pair of irreducibleType 1 mapping classes F i ∈ Mod( S g i ) such that the hF i i -action on S g i induces a pairs of compatible orbits of size k (where the inducedlocal rotation angles add upto 0 modulo 2 π ). We remove (cyclicallypermuted) hF i i -invariant disks around points in the compatible orbitsand then identify the resulting boundary components realizing a periodicmapping class F ∈ Mod( S g + g + k − ). An analogous construction canalso be performed using a pair of orbits induced by a single hF ′ i -actionon S g to realize a periodic mapping class F ∈ Mod( S g + k ).(b) Permutation additions and deletions.
The addition of a permutationcomponent involves the removal of (cyclically permuted) invariant disksaround points in an orbit of size n induced by an hF i -action on S g andthen pasting n copies of S g ′ (i.e. S g ′ with one boundary component)to the resultant boundary components. This realizes a action on S g + ng ′ with the same fixed point and orbit data as F . The reversal of thisprocess is called a permutation deletion. Thus, in summary, we have the following:
Theorem 2.6.
For g ≥ , a non-rotational periodic mapping class in Mod( S g ) can be realized through finitely many k -compatibilities, permuta-tion additions, and permutation deletions on the unique structures of type P F realizing irreducible Type 1 mapping classes. A final, but yet vital ingredient in the realization of split metacyclic actionsis the following elementary lemma, which is a direct generalization of [8,Lemma 6.1].
Lemma 2.7.
Let H = h F, G i be a finite metacyclic subgroup of Mod( S g ) .Then Fix( H ) = Fix( h F i ) ∩ Fix( h G i ) . N. K. DHANWANI, K. RAJEEVSARATHY, AND A. SANGHI
Split metacyclic actions on surfaces.
Given integers m, n ≥
2, and k ∈ Z × n such that k m ≡ n ), a finite split metacyclic action of order mn (written as m · n ) on S g is a tuple ( H, ( G , F )), where H <
Homeo + ( S g ),and H = hF , G | F n = G m = 1 , G − F G = F k i . We will call the multiplicative class k the twist factor of the split metacyclicaction ( H, ( G , F )). As we are only interested in nonabelian split metacyclicsubgroups, we will assume from here on that k = 1. Note that in classicalnotation H ∼ = Z n ⋊ k Z m . As hF i ⊳ H , it is known [4, 21] that G wouldinduce a ¯ G ∈
Homeo + ( O hFi ) that preserves the set of cone points in O hFi along with their orders. We will call ¯ G , the induced automorphism on O hFi by G , and we formalize this notion in the following definition. Definition 2.8.
Let
H <
Homeo + ( S g ) be a finite cyclic group with | H | = n .We say an ¯ F ∈
Homeo + ( O H ) is an automorphism of O H if for [ x ] , [ y ] ∈ O H , k ∈ Z × n and ¯ F ([ x ]) = [ y ], we have:(i) n x = n y , and(ii) kc x = c y .We denote the group of automorphisms of O H by Aut k ( O H ).We note that the concept of an induced orbifold automorphism in Defini-tion 2.8 is more general than the one that was used in [8], which requireda more rigid condition that c x = c y . The following lemma, which providessome basic properties of the induced map ¯ G , is a split metacyclic analogof [8, Lemma 3.1]. Lemma 2.9.
Let G , F ∈
Homeo + ( S g ) be maps of orders m, n , respectively,such that G − F G = F k , and let H = hF i . Then:(i) G induces a ¯ G ∈
Aut k ( O H ) such that O H / h ¯ Gi = S g / hF , Gi , (ii) | ¯ G| divides |G| , and(iii) | ¯ G| < m if and only if F l = G k , for some < l < n and < k < m . We will now formalize the notion of weak conjugacy from Section 1.
Definition 2.10.
Two finite split metacyclic actions ( H , ( G , F )) and( H , ( G , F )) of order m · n and twist factor k are said to be weakly con-jugate if there exists an isomorphism, ψ : π orb1 ( O H ) ∼ = π orb1 ( O H ) and anisomorphism χ : H → H such that the following conditions hold.(i) χ (( G , F )) = ( G , F ).(ii) For i = 1 ,
2, let φ i : π orb1 ( O H i ) → H i be the surface kernel (appearingin the exact sequence (2.2) in Section 2). Then ( χ ◦ φ H )( g ) = ( φ H ◦ ψ )( g ) , whenever g ∈ π orb1 ( O H ) is of finite order.(iii) The pair ( G , F ) is conjugate (component-wise) to the pair ( G , F ) inHomeo + ( S g ) . The notion of weak conjugacy defines an equivalence relation on split meta-cyclic actions on S g and the equivalence classes thus obtained will be called weak conjugacy classes . PLIT METACYCLIC ACTIONS ON SURFACES 9
Remark 2.11.
By virtue of the Nielsen-Kerckhoff theorem, the notion ofweak conjugacy in Definition 2.10 naturally extends to an analogous notionin Mod( S g ) via the natural association( hF , Gi , ( G , F )) ↔ ( h F, G i , ( G, F )) . For simplicity, we will now introduce the following notation.
Definition 2.12.
Let
F, G ∈ Mod( S g ) be of finite order. Then for some k ∈ Z × n \ { } , we say (in symbols) that J F, G K k = 1 if there exists conjugates F ′ , G ′ (of F, G resp.) such that ( F ′ ) − G ′ F ′ = ( G ′ ) k .We conclude this subsection with the following crucial remark. Remark 2.13.
Let
H <
Mod( S g ) be a finite split metacyclic subgroup, andlet I ( H ) denote the isomorphism class of H . By Remark 2.11, we have I ( H ) = { H ′ : H ′ ∼ = H and ( H ′ , ( G ′ , F ′ )) represents a weak conjucacy classfor some F ′ , G ′ ∈ H ′ such that H = h F ′ , G ′ i} . Consequently, periodic mapping classes
F, G ∈ Mod( S g ) satisfy J F, G K k = 1if and only if there exists conjugates F ′ , G ′ (of F, G resp.) such that thetriple ( h F ′ , G ′ i , G ′ , F ′ ) represents a weak conjugacy class of a finite splitmetacyclic subgroups (of twist factor k ) of Mod( S g ).3. Main theorem
In this section, we establish the main result of the paper by derivingequivalent conditions under which torsion elements
F, G ∈ Mod( S g ) wouldsatisfy J F, G K k = 1. We will introduce an abstract tuple of integers thatwill capture the weak conjugacy class of a finite split metacyclic subgroupof Mod( S g ). Definition 3.1. A split metacyclic data set of degree m · n , twist factor k ,and genus g ≥ m · n, k ) , g ; [( c , n ) , ( c , n ) , n ] , · · · , [( c ℓ , n ℓ ) , ( c ℓ , n ℓ ) , n ℓ ]) , where m, n ≥
2, the n ij are positive integers for 1 ≤ i ≤ ℓ, ≤ j ≤
2, the c ij ∈ Z × n ij , and k ∈ Z × n such that k m ≡ n ), satisfying the followingconditions.(i) 2 g − mn = 2 g − ℓ X i =1 (cid:18) − n i (cid:19) . (ii) (a) For each i, j , n i | m , n i | n , either gcd( c ij , n ij ) = 1 or c ij = 0,and c ij = 0 if and only if n ij = 1.(b) For each i , n i = n i · β i , where β i is least positive integer such that γ i X i ′ =0 k c i mni i ′ ≡ n ) , where γ i = c i nn i n i β i − ℓ X i =1 c i mn i ≡ m ). (iv) Defining A := ℓ X i =1 c i nn i ℓ Y s = i +1 k c s mns and d := ( n, k − A ≡ ( n ) , if g = 0 , and dθ (mod n ) , for θ ∈ Z n , if g ≥ . (v) If g = 0, there exists ( p , . . . , p ℓv ) , ( q , . . . , q ℓv ) ∈ Z ℓv and v ∈ N suchthat the following conditions hold.(a) ℓv X i ′ =1 p i ′ c i mn i ≡ m ) and ℓv X i ′ =1 c i nn i p i ′ X s =1 k c i mni ( p i ′ − s ) ! ℓv Y t ′ = i ′ +1 k c t mnt ! ≡ n ) . (b) ℓv X i ′ =1 q i ′ c i mn i ≡ m ) and ℓv X i ′ =1 c i nn i q i ′ X s =1 k c i mni ( q i ′ − s ) ! ℓv Y t ′ = i ′ +1 k c t mnt ! ≡ n ) , where i ≡ ( i ′ (mod ℓ ) , if i ′ = aℓ,ℓ otherwise, t ≡ ( t ′ (mod ℓ ) , if t ′ = aℓ, and ℓ, otherwise.(vi) If g = 1, there exists ( p , . . . , p ℓv ) , ( q , . . . , q ℓv ) ∈ Z ℓv and m ′ , n ′ ∈ Z , v ∈ N such that m ′ | m and n ′ | n , satisfying the following condi-tions.(a) ℓv X i ′ =1 p i ′ c i mn i ≡ m ′ (mod m ) and ℓv X i ′ =1 c i nn i p i ′ X s =1 k c i mni ( p i ′ − s ) ! ℓv Y t ′ = i ′ +1 k c t mnt ! ≡ n ) . (b) ℓv X i ′ =1 q i ′ c i mn i ≡ m ) , ℓv X i ′ =1 c i nn i q i ′ X s =1 k c i mni ( q i ′ − s ) ! ℓv Y t ′ = i ′ +1 k c t mnt ! ≡ n ′ (mod n ) , where i ≡ ( i ′ (mod ℓ ) , if i ′ = aℓ,ℓ, otherwise, t ≡ ( t ′ (mod ℓ ) if t ′ = aℓ, and ℓ otherwise.(c) Denoting d = gcd( m/m ′ , m ′ ) and d = gcd( n/n ′ , n ′ ), we have A ≡ − βk α + β (mod n ), where α | mm ′ d , β | nn ′ d , and we take α = 1, when m ′ = 0, and β = 1, when n ′ = 0. PLIT METACYCLIC ACTIONS ON SURFACES 11
We will now show that the split metacyclic data sets of genus g are inone-to-one with the weak conjugacy classes of split metacyclic subgroups ofMod( S g ). Proposition 3.2.
For integers n, m, g ≥ , the split metacyclic data sets ofdegree m · n with twist factor k and genus g correspond to the weak conjugacyclasses of Z n ⋊ k Z m -actions on S g .Proof. Let D be a split metacyclic data set of degree m · n with twist fac-tor k and genus g (as in Definition 3.1 above). We need to show that D corresponds to the weak conjugacy class of a Z n ⋊ k Z m -action on S g rep-resented by ( H, ( G , F )). To this effect, we first establish the existence of aorder-preserving epimorphism φ H : π orb ( O H ) → H . Let the presentationsof H and Γ( O H ) be given by Z n ⋊ k Z m = hF , G | F n = G m = 1 , G − F G = F k i and h α , β , · · · , α g , β g , ξ , · · · , ξ ℓ | ξ n = · · · = ξ n ℓ ℓ = ℓ Y i =1 ξ i g Y i =1 [ α i , β i ] = 1 i , respectively. We consider the map ξ i φ H c i mni F c i nni , for 1 ≤ i ≤ ℓ. As |G c i mni | = n i and |F c i nni | = n i , condition (ii) of Definition 3.1 wouldimply that φ H is an order-preserving map. For clarity, we break the argu-ment for the surjectivity of φ H into three cases.First, we consider the case when g = 0. Conditions (iii) and (iv) showthat φ H satisfies the long relation Q ℓi =1 ξ i = 1 and the surjectivity of φ H follows from condition (v).When g ≥ π orb ( O H ) has additional hyperbolic generators (viewingthem as isometries of the hyperbolic plane), namely the α i and the β i .Extending φ H by mapping α φ H G, β φ H F yields an epimorphism.Moreover, by carefully choosing the α i and the β i , for i ≥
2, conditions (iii)and (iv) would together ensure that the long relation Q ℓi =1 ξ i Q g i =1 [ α i , β i ] = 1is satisfied.When g = 1, π orb ( O H ) has two additional hyperbolic generators, namelythe α and the β . We extend φ H by defining α φ H G α and β φ H β ,and apply conditions (iv) and (vi) to obtain the desired epimorphism.It remains to show that D determines F , G ∈
Homeo + ( S g ) up to conjugacy(i.e. condition (iii) of Definition 2.10). Let D ¯ G = ( m, g ; ( c , n ) , . . . , ( c ℓ , n ℓ ))represent the conjugacy class of the action ¯ G induced on the orbifold O hFi by the action G ∈
Homeo + ( S g ). We note that by Lemma 2.9, Γ( O hFi ) hasthe form ( g ; n n , . . . n n | {z } mn times , . . . , n r n r , . . . , n ℓ n ℓ | {z } mnℓ times ) , where if n i /n i = 1, for some 1 ≤ i ≤ ℓ , then we exclude it from thesignature, and g = g ( D ¯ G ) is determined by Equation (3) of Definition 2.2. So, we get D F = ( n, g ; ( d , n n ) , . . . , ( d mn , n n ) , . . . , ( d ℓ , n ℓ n ℓ ) , . . . , ( d ℓ mn , n ℓ n ℓ ))where d i n i ≡ c i nn i n i X j ′ =1 k c i mni ( j ′ − (mod n ) ,d ij i ≡ d i k γ ( j i − (mod n i n i ) 1 ≤ i ≤ l, ≤ j i ≤ mn i , and γ is the least positive integer such that | k | | γ mn i . Moreover, by applyingLemma 2.4, we see that D G = ( m, g ; (( u ij , m i ) , m i | ℧ G mmi ( u ij , m i ) | m ) : u ij ∈ Z × m i , m i | m, and gcd( u ij , m i ) = 1) , where | ℧ G mmi ( u ij , m i ) | = | F G mmi ( u ij , m i ) |− X m i ′ ∈ N m i ′ = m i m i | m i ′ | m X ( u i ′ j ′ ,m i ′ )=1 u ij ≡ u i ′ j ′ (mod m i ) | ℧ G mmi ′ ( u i ′ j ′ , m i ′ ) | and g is determined by Equation (3) of Definition 2.2.Conversely, consider the weak conjugacy class of Z n ⋊ k Z m -actions on S g represented by ( H, ( G, F )), where H = hF , Gi . So, Lemma 2.1 would implythat there exists a surjective homomorphism φ H : π orb ( O H ) → H : ξ i φ H c i mni F c i nni , for 1 ≤ i ≤ ℓ, which is order-preserving on the torsion elements. This yields a split meta-cyclic data set of degree m · n with twist factor k and genus g as in Defini-tion 3.1. By Lemma 2.1, this tuple satisfies condition (i) of Definition 3.1,while condition (ii) follows from the fact that φ H is order-preserving on tor-sion elements. Conditions (iii)-(iv) follow from the long relation satisfiedby π orb ( O H ), and condition (v)-(vi) are implied by the surjectivity of φ H .Thus, we obtain the split metacyclic data set of degree m · n with twist factor k and genus g , and the result follows. (cid:3) We denote the data sets D F and D G (representing the cyclic factors of H ) derived from the split metacyclic data set D appearing in the proof ofProposition 3.2 by D and D , respectively. Thus, our main theorem willnow follow from Remark 2.13 and Proposition 3.2. Theorem 3.3 (Main theorem) . Let
F, G ∈ Mod( S g ) be of orders n, m ,respectively. Then [[ F, G ]] k = 1 if and only if there exists a split metacyclicdata set D of degree m · n , twist factor k , and genus g such that D = D F and D = D G . We conclude this section with an example of a split metacyclic action oforder 16 on S . PLIT METACYCLIC ACTIONS ON SURFACES 13
Example 3.4.
The split metacyclic data set D = ((4 · , − ,
1; [(0 , , (1 , , Z ⋊ − Z -action on S representedby ( hF , Gi , ( G , F )), where D F = (4 ,
1; (1 , , (1 , , (1 , , (1 , D G = (4 , ,
1; ) . The geometric realization of this action is illustrated in Figure 2 below. (1 , , , , G π (1 , ,
2) (1 ,
2) (1 , , ,
4) (1 , , ,
4) (3 , ,
4) (3 , Figure 2.
Realization of a Z ⋊ − Z -action in Mod( S ).Note that the pairs of integers appearing in Figure 2 represent the com-patible orbits involved in the realization of F . Here, the action F is re-alized via two 1-compatibilities between the action F ′ on two copies of S with D F ′ = (4 ,
0; ((1 , , , (1 , , (3 , F ′ is re-alized by a 1-compatibility between the action F ′′ on two copies of S with D F ′′ = (4 ,
0; (1 , , (1 , , (3 , Applications
Dihedral groups.
Let D n = Z n ⋊ − Z be the dihedral group of order2 n . We will call a split metacyclic data set of degree 2 · n and twist factor − dihedral data set . A simple computation reveals that a dihedral dataset ((2 · n, − , g ; [( c , n ) , ( c , n ) , n ] , · · · , [( c ℓ , n ℓ ) , ( c ℓ , n ℓ ) , n ℓ ]) , would have the property that ( c j , n j ) ∈ { (0 , , (1 , } , for 1 ≤ j ≤ ℓ . Thefollowing is an immediate consequence of Proposition 3.2. Corollary 4.1.
For g ≥ and n ≥ , dihedral data sets of degree · n andgenus g correspond to the weak conjugacy classes of D n -actions on S g . The following proposition provides an alternative characterization of a D n -action in terms of the generator of its factor subgroup of order n . Proposition 4.2.
Let F ∈ Mod( S g ) be of order n . Then there exists aninvolution G ∈ Mod( S g ) such that h F, G i ∼ = D n if and only if D F has theform (*) ( n, g , r ; (( c , n ) , ( − c , n ) , . . . , ( c s , n s ) , ( − c s , n s )) . Proof.
Suppose that D F has the form ( ∗ ). Then O hFi is an orbifold ofgenus g with 2 s cone points [ x ] , [ y ] , . . . , [ x s ] , [ y s ], where P x i = ( c i , n i )and P y i = ( − c i , n i ), for 1 ≤ i ≤ s . Up to conjugacy, let ¯ G ∈
Aut k ( O hFi )be the hyperelliptic involution so that ¯ G ([ x i ]) = [ y i ], for 1 ≤ i ≤ s . Toprove our assertion, it would suffice to show the existence of an involution G ∈
Homeo + ( S g ) that induces ¯ G . This amounts to showing that there existsa split metacyclic data set D of degree 2 · n with twist factor − H, ( G , F )) so that D G has degree 2. Consider thetuple D = ((2 · n, − ,
0; [(1 , , (0 , , , . . . , [(1 , , (0 , , | {z } t − , [(1 , , ( c ( t − , n ( t − ) , , , ( c t , n t ) , , [(0 , , ( c , n ) , n ] , . . . , [(0 , , ( c s , n s ) , n s ]) , where t = 2 g + 2 , (( c ( t − , n ( t − ) , ( c t , n t )) = (cid:26) ((0 , , (1 , − P si =1 c i nn i (mod n ))) , if g = 0 , and((1 , n ) , (1 , − P si =1 c i nn i (mod n ))) , if g > . It follows immediately that D satisfies conditions (i)-(iv) of Definition 3.1.As t ≥
2, by taking v = 1, we may choose ( p , . . . , p t + s ) = (1 , , . . . ,
0) toconclude that D also satisfies condition (v)(a). Since t = 2 ⇐⇒ g = 0,and when g = 0, we have that lcm( n , . . . , n s ) = n , from which condition(v)(b) follows. Finally, for the case when g = 0, (v)(b) follows by choos-ing ( q , . . . , q t − , q t − , . . . , q t + s ) = (0 , . . . , − , , . . . , D is a split metacyclic data set. Further, a direct application of Theo-rem 3.3 would show that D indeed encodes the weak conjugacy representedby ( H, ( G , F )), as desired.The converse follows immediately from Remark 2.11 and Proposition 3.2. (cid:3) We now provide a couple of examples of dihedral actions along with theirrealizations.
Example 4.3.
Consider the Z ⋊ − Z -action hF , Gi on S illustrated inFigure 3 below, where D F = (3 ,
1; (1 , , (2 , D G = (2 ,
1; (1 , , (1 , , (1 , , (1 , . FG π π Figure 3.
Realization of a D -action in Mod( S ). PLIT METACYCLIC ACTIONS ON SURFACES 15
The weak conjugacy class of the action ( hF , Gi , ( G , F )) is encoded by D = ((2 · , − ,
0; [(1 , , (0 , , , [(1 , , (0 , , , [(1 , , (0 , , , [(1 , , (1 , , , [(0 , , (2 , , . Example 4.4.
Consider the Z ⋊ − Z -actions hF , Gi and hF , G ′ i on S illustrated in Figure 4 below, where D F = (4 ,
0; (1 , , (3 , , (1 , , (3 , D G = (2 ,
1; (1 , , (1 , , (1 , , (1 , D G ′ = (2 , ,
1; ) . F G π π G ′ π Figure 4.
Realization of a D -action in Mod( S ).The weak conjugacy classes ( hF , Gi , ( G , F )) (resp. ( hF , G ′ i , ( G ′ , F ))) areencoded by((2 · , − ,
0; [(1 , , (0 , , , [(1 , , (0 , , , [(0 , , (1 , , , [(0 , , (3 , , · , − ,
0; [(1 , , (1 , , , [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (3 , , , respectively.4.2. Generalized quaternions.
For n ≥
2, the generalized quaterniongroup Q n +1 is a metacyclic group of order 2 n +1 that admits the presentation h x, y | x n = y = 1 , x n − = y , y − xy = x − i . Remark 4.5.
Let D be a split metacyclic data set of genus g , degree4 · n and twist factor − H, ( G , F )). Suppose that D has the propertythat [( c j , n j ) , ( c j , n j ) , n j ] = [(1 , , (1 , , ≤ j ≤ ℓ . Thenit follows from the proof of Proposition 3.2 that under the order-preservingepimorphism φ H : π orb ( O H ) → H , the tuple [(1 , , (1 , ,
2] would corre-spond to an involution G F n − ∈ H which defines a non-free action on S g .Remark 4.5 motivates the following definition. Definition 4.6. A quaternionic data set is a split metacyclic data set ofdegree 2 n +2 that has the form D = ((4 · n , − , g ; [( c , n ) , ( c , n ) , n ] , . . . , [( c ℓ , n ℓ ) , ( c ℓ , n ℓ ) , n ℓ ]) , such that [( c j , n j ) , ( c j , n j ) , n j ] = [(1 , , (1 , , ≤ j ≤ ℓ. Proposition 4.7.
For g, n ≥ , quaternionic data sets of genus g − correspond to Q n +1 -actions on S g .Proof. Suppose that there exists an action of H = Q n +1 on S g . By Lemma 2.1,there exists an epimorphism π orb ( O H ) → Hξ i φ H y c i mni x c i nni , for 1 ≤ i ≤ ℓ, that is order-preserving on torsion elements. Let H ′ = Z n ⋊ − Z . Since thecanonical projection q : H ′ → H ( ∼ = H ′ / Z ) is order-preserving on H ′ \ ker q ,the map φ H naturally factors via q . Thus, we obtain two possible choicesfor an order-preserving epimorphism π orb ( O H ) → H ′ , of which exactly oneyields an action H ′ on S g ′ (for some g ′ > g ). The weak conjugacy class ofthis action is encoded by a split metacyclic data set of genus g ′ and degree2 n +2 = 4 · n , which has one of the following forms((4 · n , − , g ; [( c , n ) , ( c , n ) , n ] , . . . , [( c ℓ , n ℓ ) , ( c ℓ , n ℓ ) , n ℓ ])or(4 · n , − , g ; [( c , n ) , ( c , n ) , n ] , . . . , [( c ′ ℓ , n ′ ℓ ) , ( c ′ ℓ , n ′ ℓ ) , n ℓ ]) , where c ′ ℓ n ′ ℓ ≡ c ℓ n ℓ + 2 (mod 4) and c ′ ℓ n n ′ ℓ ≡ c ℓ n n ℓ + 2 n − (mod 2 n ) . Further, since ker q ∼ = Z and q preserves the orders of all x ∈ H ′ \ ker q , itfollows that ker q acts freely on S g ′ . Hence, it follows that g ′ = 2 g − , , (1 , , D of genus g ′ = 2 g − φ H ′ : π orb ( O H ′ ) → H ′ , which when composed with canonical projection q : H ′ → H , yields an order-preserving epimorphism φ H : π orb ( O H ) → H . Further,as D does not contain a triple of type [(1 , , (1 , , q acts freely on S g ′ , thereby yielding an action of Q n +1 on S g , where g ′ = 2 g − (cid:3) Example 4.8.
The split metacyclic data set in Example 3.4 is quaternionic.Hence, this represents the weak conjugacy class of an induced Q -action on S .4.3. Lifting cyclic subgroups of mapping classes to split metacyclicgroups.
For n, g ≥
2, let p : S ˜ g → S g be a covering map (that is possiblybranched) with deck transformation group Z n = hF i . Let LMod p ( S g ) (resp.SMod( S g )) denote the liftable (resp. symmetric) mapping class groups of S g under p . Remark 4.9.
The Birman exact sequence [2] in this setting takes the form(B) 1 → h F i → SMod p ( S ˜ g ) → LMod p ( S g ) → . PLIT METACYCLIC ACTIONS ON SURFACES 17
Let G ∈ Mod( S g ) be of finite order. Then G ∈ LMod p ( S g ) if and only if G has a lift ˜ G ∈ SMod p ( S ˜ g ) of finite order so that the sequence ( B ) yields asequence of the form 1 → h F i → h F, ˜ G i → h G i → . Thus, G ∈ LMod p ( S g ) if and only if for any lift ˜ G of G , h G i lifts under p toa metacyclic group h F, ˜ G i .In the following corollary, we characterize the periodic elements in Mod( S , )that generate cyclic groups that lift to finite split metacyclic groups underbranched covers induced by irreducible cyclic actions. Corollary 4.10.
For g, n ≥ , let p : S g → S , be a cover with decktransformation group hF i with D F = ( n,
0; ( c , n ) , ( c , n ) , ( c , n )) . Thena G ′ ∈ Mod( S , ) of order m has a conjugate G ∈ LMod p ( S , ) with a lift ˜ G ∈ SMod p ( S g ) such that h F, ˜ G i ∼ = Z n ⋊ k Z m if and only if one of thefollowing conditions hold.(a) D F = ( n,
0; ( c , n ) , ( c , n ) , ( c k, n )) for some k ∈ Z × n such that k ≡ n ) .(b) D F = ( n,
0; ( c , n ) , ( c k, n ) , ( c k , n )) for some k ∈ Z × n such that k ≡ n ) .Proof. Suppose that G ′ ∈ Mod( S , ) has a conjugate G ∈ LMod p ( S , ) witha lift ˜ G ∈ SMod p ( S g ) such that H = h F, ˜ G i ∼ = Z n ⋊ k Z m . First, we claimthat the n i , for 1 ≤ i ≤
3, are not distinct. Suppose that we assume onthe contrary that the n i are indeed distinct. Since G ′ ∈ Aut k ( O hFi ) and |G ′ | >
1, it would have to fix all three cone points of O hFi , which contradictsthe fact that any homeomorphism on a sphere can fix at most two points.Thus, the following two cases arise. Case n = n = n = n . In this case, G ′ fixes the cone point, say oforder n , and should permute the remaining 2 cone points of orders n and n . This implies that D F takes the form in condition (a) in our hypothesis(by Definition 2.8), and hence H = h F, ˜ G i ∼ = Z n ⋊ k Z . Case n i = n, ≤ i ≤
3. In this case, if G ′ permutes all the three conepoints cyclically, then D F takes the form in condition (b) in our hypothesis,and hence H ∼ = Z n ⋊ k Z . Alternatively, G ′ could also fix a cone point oforder n and permute the remaining 2 cone points, in which case, D F willtake the form in condition (a).Conversely, if D F = ( n,
0; ( c , n ) , ( c , n ) , ( c k, n )) for some k ∈ Z × n suchthat k ≡ n ). Up to conjugacy, let G ′ ∈ Aut k ( O hFi ) be an involu-tion so that G ′ maps the cone point represented by ( c , n ) to the cone pointrepresented by ( c · k, n ). To prove our assertion, it would suffice show theexistence of an involution G ∈
Homeo + ( S g ) that induces G ′ . This amountsto showing that there exists a split metacyclic data set D of degree 2 · n with twist factor k encoding the weak conjugacy class ( H, ( G , F )) so that D G has degree 2. Consider the tuple ((2 · n, k ) ,
0; [(1 , , (0 , , , [(1 , , ( n − c , n ) , n ] , [(0 , , ( c , n ) , n ]). By simple computation would reveal that con-ditions (i) - (iv) of Definition 3.1 hold true. Condition (v) is true by taking v = 1, ( p , p , p ) = (1 , ,
0) and ( q , q , q ) = (0 , , w ) such that wc ≡ n ), which proves our claim. For the case when D F = ( n,
0; ( c , n ) , ( c k, n ) , ( c k , n )) for some k ∈ Z × n such that k ≡ n ), let G ′ ∈ Aut k ( O hFi ) be of order 3 so that for1 ≤ i ≤ G ′ i maps the cone point represented by ( c , n ) to the cone pointrepresented by ( c k i , n ). By similar argument as above, we can show thatthe tuple ((3 · n, k ) ,
0; [(1 , , (0 , , , [(2 , , ( n − c , n ) , , [(0 , , ( c , n ) , n ])forms a split metacyclic data set of degree 3 · n with twist factor k uptoequivalence. (cid:3) Example 4.11.
For i = 1 ,
2, consider the branched cover p : S → O hF i i ( ≈ S , ), where D F = (8 ,
0; (1 , , (1 , , (5 , D F = (8 ,
0; (3 , , (3 , , (7 , G ∈ LMod p ( S , ) repre-sented by an automorphism G ∈
Aut ( S , ), that permutes two cone pointsof order 8 and fixes order 4 cone point, lifts to a ˜ G ∈ SMod p ( S ) with D ˜ G = (2 ,
1; ((1 , , h F i , ˜ G i ∼ = Z ⋊ Z . Moreover, the weakconjugacy class of ( hF i , ˜ Gi , ( G , F i )), for i = 1 ,
2, is encoded by((2 · , ,
0; [(1 , , (0 , , , [(1 , , (7 , , , [(0 , , (1 , , · , ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(0 , , (7 , , , respectively. The geometric realization of these actions is illustrated inFigure 5 below, where for each i , the action F i is realized by the rotation ofa polygon of type P F i described in Theorem 2.5. G π (1 ,
8) (1 , ,
4) (5 , G π (3 ,
8) (3 , ,
4) (7 , Figure 5.
The realizations of two distinct Z ⋊ Z -actionson S . Proposition 4.12.
For g, n ≥ , let p : S n ( g − → S g be a regularcover with deck transformation group Z n = hF i . Then any involution G ′ ∈ Mod( S g ) has a conjugate G ∈ LMod( S g ) with a lift ˜ G ∈ SMod( S n ( g − ) such that h F, ˜ G i ∼ = D n .Proof. Let G ′ ∈ Mod( S g ) be an involution with D G ′ = (2 , g ; ((1 , , t )). ByTheorem 3.3 and Remark 4.9, it suffices to show that there exists a dihedraldata set D degree 2 · n and genus n ( g − h F, ˜ G i , ˜ G, F ). When g ≥
1, we take D to be the tuple((2 n, − , g ; [(1 , , (0 , , , . . . , [(1 , , (0 , , | {z } t times , PLIT METACYCLIC ACTIONS ON SURFACES 19 and when g = 0, t ≥
4, and so we take D to be the tuple((2 n, − , g ; [(1 , , (0 , , , . . . , [(1 , , (0 , , | {z } t − , [(1 , , (1 , n ) , , [(1 , , (1 , n ) , . It is an easy computation to check that D satisfies conditions (i)-(iv) ofDefinition 3.1 in both cases. When g = 0, taking v = 1,( p , . . . , p ℓ ) = (1 , , . . . , , and ( q , . . . , q ℓ ) = (0 , . . . , , , , g = 1, we take v = 1,( p , . . . , p ℓ ) = (1 , , . . . , , and ( q , . . . , q ℓ ) = (0 , . . . , , thereby verifying condition (vi). Thus, we have shown that D is a dihedraldata set as desired. Finally, it follows from Theorem 3.3 that D encodes theweak conjugacy class of ( h F, ˜ G i , ˜ G, F ). (cid:3) Note that the same Z -action can lift to multiple non-isomorphic groupsunder a regular cyclic cover. We illustrate this phenomenon in the followingexample. Example 4.13.
Let p : S → S be a regular 4-sheeted cover with decktransformation group Z = hF i as illustrated in Figure 6 below. F ˜ G π π ˜ G π F ˜ G π Figure 6.
Two distinct lifts ˜ G , ˜ G ∈ SMod( S ) of an invo-lution in G ∈ Mod( S ).The involution G ∈ Mod( S ) with D G = (2 ,
1; (1 , , (1 , G , ˜ G ∈ SMod( S ) (as indicated) such that h F, ˜ G i ∼ = D and h F, ˜ G i ∼ = Z × Z . Note that the weak conjugacy classes of ( hF , ˜ G i , ˜ G , F )is represented by ((2 · , − ,
1; [(1 , , (0 , , , [(1 , , (0 , , Z m -actions whose corresponding orbifolds are spheres with a cone point oforder m . Proposition 4.14.
For g, n ≥ , let p : S n ( g − → S g be a regular n -sheeted cover with deck transformation group Z n = hF i . Let G ′ ∈ Mod( S g ) of be order m such that D G ′ = ( m,
0; ( c , m ) , . . . , ( c ℓ , m ℓ )) with m ℓ = m (say). Then G ′ has a conjugate G ∈ LMod p ( S g ) with a lift ˜ G ∈ SMod p ( S g ) such that h F, ˜ G i ∼ = Z n ⋊ k Z m if the following conditions hold. (a) There exists a , . . . , a ℓ − ∈ Z , and k ∈ Z × n , k m ≡ n ) such that ℓ − X i =1 a i ( k c i mmi − ℓ − Y s = i +1 k c s mms ≡ n ) . (b) Let d i nn i ≡ a i ( k c i mmi −
1) (mod n ) , for ≤ i ≤ ℓ − , where gcd( d i , n i ) = 1 and n i | n . Then we have lcm( n , n , . . . , n ℓ − ) = n. Proof.
By Theorem 3.3 and Remark 4.9, it suffices to show that the tuple D = (( m · n, k ) ,
0; [( c , m ) , ( d , n ) , m ] , . . . , [( c ℓ − , m ℓ − ) , ( d ℓ − , n ℓ − ) , m ℓ − ] , [( c ℓ , m ℓ ) , (0 , , m ℓ ])forms a split metacyclic data set of genus n ( g −
1) + 1 that represents theweak conjugacy class of ( h F, ˜ G i , ˜ G, F ) for some lift ˜ G of G under p . It canbe verified easily that D satisfies conditions (i)-(iii) of Definition 3.1, andfurther, condition (iv) follows from condition (a) in our hypothesis. Bytaking v = 1, ( p , . . . , p ℓ ) = (0 , . . . , , w ) such that wc ℓ ≡ m ), we seethat condition (v)(a) holds. Finally, condition (v)(b) follows from condition(b) in our hypothesis, and our assertion follows. (cid:3) Using similar arguments, we can show the following.
Proposition 4.15.
For g, n ≥ , let p : S n ( g − → S g be a regular n -sheeted cover with deck transformation group Z n = hF i . Let G ′ ∈ Mod( S g ) of be order m such that D G ′ = ( m,
0; ( c , m ) , . . . , ( c ℓ , m ℓ )) with m i = m ,for ≤ i ≤ ℓ . Then G ′ has a conjugate G ∈ LMod p ( S g ) with a lift ˜ G ∈ SMod p ( S g ) such that h F, ˜ G i ∼ = Z n ⋊ k Z m if following conditions hold.(i) There exists a , . . . , a ℓ ∈ Z , and k ∈ Z × n , k m ≡ n ) such that ℓ X i =1 a i ( k c i mmi − ℓ Y s = i +1 k c s mms ≡ n ) . (ii) There exists ( p , . . . , p ℓv ) , ( q , . . . , q ℓv ) ∈ Z ℓv and v ∈ N such that con-dition (v)(b) of Definition 3.1 holds, where for ≤ i ≤ ℓ , we have c i mn i ≡ c i mm i (mod m ) and c i nn i ≡ a i ( k c i mmi −
1) (mod n ) . A consequence of Propositions 4.14-4.15 is the following.
Corollary 4.16.
For g ≥ and prime n , let p : S n ( g − → S g be a regular n -sheeted with deck transformation group Z n = hF i . Let G ′ ∈ Mod( S g ) ofbe order m such that the genus of O hG ′ i is zero. Then G ′ has a conjugate G ∈ LMod p ( S g ) with a lift ˜ G ∈ SMod p ( S g ) such that h F, ˜ G i ∼ = Z n ⋊ k Z m ifthere exists k ∈ Z × n such that | k | = m .Proof. Let D G ′ = ( m,
0; ( c , m ) , . . . , ( c ℓ , m ℓ )). First, let us assume (withoutloss of generality) that m ℓ = m . By choosing( a , . . . , a ℓ − ) = (0 , . . . , , , − ( k c ℓ − mmℓ − − · k c ℓ − mmℓ − · ( k c ℓ − mmℓ − − − ) , PLIT METACYCLIC ACTIONS ON SURFACES 21 we see that condition (i) of Proposition 4.14 holds true. Moreover, since | k | = m , we have gcd(( k c ℓ − mmℓ − − , n ) = 1, and so condition (ii) alsoholds, and our assertion follows.Similarly, for the case when each m ℓ < m , the result follows by taking( a , . . . , a ℓ ) = (0 , . . . , , , − ( k c ℓ − mmℓ − − · k c ℓ mmℓ · ( k c ℓ mmℓ − − ) , and applying Proposition 4.15. (cid:3) Infinite split dihedral group.
By an infinite dihedral group , we meana group that is isomorphic to
Z ⋊ − Z and admits a presentation of theform h x, y | y = 1 , y − xy = x − i . In this subsection, we give an explicit construction of an infinite dihedralsubgroup of Mod( S g ) when g ≥
5. Let T c ∈ Mod( S g ) denote the left-handedDehn twist about a simple closed curve c in S g . A root of T c of degree s isa G ∈ Mod( S g ) such that G s = T c . In the following lemma, by using somebasic properties of Dehn twists [9, Chapter 3], we show that a root of Dehntwist cannot generate an infinite dihedral group. Lemma 4.17.
For g ≥ , no root of T c is a generator of any infinite dihedralsubgroup of Mod( S g ) .Proof. Let G be a root of T c of degree s . Suppose we assume on the contrarythat for some g ≥
2, there exists an infinite dihedral subgroup H ∼ = Z ⋊ − Z of Mod( S g ) that admits the presentation H = h F, G | F = 1 , F − GF = G − i . First, we consider the case when s = 1, that is, G = T c . Then we havethat F − T c F = T − c = ⇒ T F − ( c ) = T − c , which is impossible. Thus, we have that H = h F, T c i , which contradicts ourassumption.For s >
1, suppose that H = h F, G i . Then the subgroup h F, G s i of H would also be a split metacyclic group. Since G s = T c , this would contradictour conclusion in the previous case, and so our assertion follows. (cid:3) Considering Lemma 4.17, a natural question is whether product of commut-ing Dehn twists can generate an infinite dihedral group. In the followingexample, we answer this question in the affirmative.
Example 4.18.
Let F ′ ∈ Mod( S ) be of order 3 with D F ′ = (3 ,
0; ((1 , , , ((2 , , . First, we note that F has four fixed points on S . Further, it induces a lo-cal rotation angle of 2 π/ ,
3) pairs in D F ′ ) and rotation angle of 4 π/ ,
3) pairs in D F ′ ), as indicated inFigure 7. Considering this action on two distinct copies of S , we removeinvariant disks around a distinguished (1 , , two annuli connecting the resulting boundary components across the twosurfaces so that:(a) each annulus connects a pair of boundary components where the inducedrotation angle is the same, as shown in Figure 7 below, and further,(b) the annulus connecting the boundary components with rotation 4 π/ c ) has a 1 / rd twist, while the other(with the nonseparating curve d ) has a − / rd twist. (1 , cd (2 ,
3) (1 , , ,
3) (1 , ,
3) (2 , π G Figure 7.
Realization of an infinite dihedral subgroup of Mod( S ).Thus, by applying the theory developed in [19], we obtain an F ∈ Mod( S ),which is a root of the bounding pair map T c T − d of degree 3. Now, we con-sider the hyperelliptic involution G ∈ Mod( S ) with D G = (2 ,
0; ((1 , , GF G − = F − , and so we have h F, G i ∼ = Z ⋊ − Z .Generalizing the construction in Example 4.18, we have the following. Proposition 4.19.
For g ≥ , there exists an infinite dihedral subgroupof Mod( S g ) that is generated by the hyperelliptic involution and a root of abounding pair map of degree . Hyperbolic structures realizing split metacyclic actions
We begin this section by providing an algorithm for obtaining the hy-perbolic structures that realize finite split metacyclic subgroups of Mod( S g )(up to weak conjugacy) as groups of isometries. Step
1. Consider a weak conjugacy class represented by ( H, ( G , F )). Step
2. Use Theorem 3.3 to determine the conjugacy classes D F (resp. D G )of the generators F (resp. G ). Step
3. We apply Lemma 2.7, and Theorems 2.5-2.6, to obtain the hyper-bolic structures that realize H as a group of isometries.We now describe the geometric realizations of some split metacyclic actionson S and S represented by the split metacyclic data sets listed in Tables 1and 2 in Section 6. PLIT METACYCLIC ACTIONS ON SURFACES 23 G π (1 ,
3) (1 , ,
3) (2 , , , Figure 8.
A realization of a D -action hF , Gi on S , where D G =(2 , ,
1; ) and D F = (3 ,
1; (1 , , (2 , F is realizedthrough two 1-compatibilities between two actions F ′ and F ′′ on S with D F ′ = (3 ,
0; ((1 , , D F ′′ = (3 ,
0; ((2 , , hF , Gi , G , F ) is encoded by the first splitmetacyclic data set in Table 1. G π (1 , ,
2) (1 , ,
4) (1 , , , , G π Figure 9.
The realizations of two distinct D -actions hF , G i and hF , G i on S , where D F = (4 ,
1; ((1 , , D G = (2 , ,
1; ),and D G = (2 ,
1; ((1 , , F is realized via two1-compatibilities between two actions F ′ and F ′′ on S , where D F ′ = (4 ,
0; ((1 , , , (1 , D F ′′ = (4 ,
0; ((3 , , , (1 , hF , G i , G , F ) and ( hF , G i , G , F )are encoded by split metacyclic data sets nos 3 and 6, respectively,in Table 1. (1 , , G π (2 ,
3) (2 , , ,
3) (1 , , ,
3) (2 , ,
3) (2 , Figure 10.
A realization of a Z ⋊ − Z -action hF , Gi on S ,where D G = (4 , ,
1; ) and D F = (3 ,
1; ((1 , , , ((2 , , F is realized via two 1-compatibilities between the action F ′ on two copies of S with D F ′ = (4 ,
0; ((1 , , , (1 , , (3 , F ′ is realized by a 1-compatibility between theaction F ′′ on two copies of S with D F ′′ = (4 ,
0; (1 , , (1 , , (3 , hF , Gi , G , F ) is encoded by the splitmetacyclic data set no 13 in Table 2. G π G π G π (1 , ,
2) (1 , ,
2) (1 , ,
2) (1 , ,
2) (3 , ,
8) (5 , , Figure 11.
Realization of Z ⋊ − Z -action hF , G i , Z ⋊ Z -action hF , G i and Z ⋊ Z -action hF , G i on S , where D G = D G = (2 ,
2; ((1 , , D G = (2 , ,
1; ) and D F = (8 ,
1; ((1 , , F is realized viatwo 1-compatibilities between two actions F ′ and F ′′ on S where D F ′ = (8 ,
0; (1 , , (1 , , (3 , D F ′′ =(8 ,
0; (1 , , (5 , , (7 , hF , G i i , G i , F ) 1 ≤ i ≤ Classification of the weak conjugacy classes in
Mod( S ) and Mod( S )In this section, we will use Theorem 3.3 to classify the weak conjugacyclasses in Mod( S ) and Mod( S ). For brevity, we will further assume the fol-lowing equivalence of the split metacyclic data sets (i.e. the weak conjugacyclasses). Definition 6.1.
Two split metacyclic data sets D = (( m · n, k ) , g ; [( c , n ) , ( c , n ) , n ] , · · · , [( c ℓ , n ℓ ) , ( c ℓ , n ℓ ) , n ℓ ])and D ′ = (( m · n, k ) , g ; [( c ′ , n ′ ) , ( c ′ , n ′ ) , n ′ ] , · · · , [( c ′ ℓ , n ′ ℓ ) , ( c ′ ℓ , n ′ ℓ ) , n ′ ℓ ])are said to be equivalent if for each tuple [( c ′ i , n ′ i ) , ( c ′ i , n ′ i ) , n ′ i ], there existsa unique tuple [( c j , n j ) , ( c j , n j ) , n j ] satisfying the following conditions:(i) ( c ′ i , n ′ i ) = ( c j , n j ),(ii) n ′ i = n j , and(iii) c ′ i nn ′ i ≡ c j nn j k a i + b i ( k c j mnj −
1) (mod n ) for some a i , b i ∈ Z . Note that equivalent data sets D and D ′ as in Definition 6.1 satisfy D ′ i = D i ,for i = 1 ,
2. We will now provide a classification of the weak conjugacyclasses of finite split metacyclic subgroups of Mod( S ) and Mod( S ) (up tothis equivalence) in Tables 1 and 2, respectively, P L I T M E T A C Y C L I C A CT I O N S O N S U R F A C E S Group Weak conjucacy classes in
Mod( S ) Cyclic factors [ D G ; D F ] Z ⋊ − Z ((2 · , − ,
1; [(0 , , (1 , , , ,
1; ); (3 ,
1; (1 , , (2 , · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(0 , , (2 , , ∗ [(2 ,
1; ((1 , , ,
1; (1 , , (2 , Z ⋊ − Z ((2 · , − ,
1; [(0 , , (1 , , , ,
1; ); (4 ,
1; ((1 , , · , − ,
0; [(1 , , (0 , , , [(0 , , (1 , , , [(0 , , (3 , , ,
1; ((1 , , ,
0; ((1 , , , ((3 , , · , − ,
0; [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (3 , , , ,
1; ); (4 ,
0; ((1 , , , ((3 , , · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(1 , , (3 , , , [(0 , , (1 , , ,
1; ((1 , , ,
1; ((1 , , Z ⋊ − Z ((4 · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(1 , , (2 , , ,
0; ((1 , , , ((1 , , ,
1; (1 , , (2 , · , − ,
0; [(3 , , (0 , , , [(3 , , (1 , , , [(1 , , (2 , , ,
0; ((3 , , , ((1 , , ,
1; (1 , , (2 , Z ⋊ − Z ((2 · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (1 , , ,
1; ((1 , , ,
0; ((1 , , , (1 , , (5 , · , − ,
0; [(1 , , (1 , , , [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (1 , , , ,
1; ); (6 ,
0; ((1 , , , (1 , , (5 , Z ⋊ − Z ((4 · , − ,
0; [(1 , , (0 , , , [(0 , , (1 , , , [(3 , , (1 , , † [(4 ,
0; ((1 , , , ((1 , , ,
0; ((1 , , , ((3 , , · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(1 , , (3 , , † [(4 ,
0; ((1 , , , ((1 , , ,
1; ((1 , , · , − ,
0; [(3 , , (0 , , , [(0 , , (1 , , , [(1 , , (1 , , † [(4 ,
0; ((3 , , , ((1 , , ,
0; ((1 , , , ((3 , , · , − ,
0; [(3 , , (0 , , , [(3 , , (1 , , , [(1 , , (3 , , † [(4 ,
0; ((3 , , , ((1 , , ,
1; ((1 , , Z ⋊ Z ((2 · , ,
0; [(1 , , (0 , , , [(1 , , (7 , , , [(0 , , (1 , , ,
1; ((1 , , ,
0; (1 , , (1 , , (5 , · , ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(0 , , (7 , , ,
1; ((1 , , ,
0; (3 , , (3 , , (7 , Z ⋊ Z ((3 · , ,
0; [(1 , , (0 , , , [(2 , , (6 , , , [(0 , , (1 , , ,
1; (1 , , (2 , ,
0; (1 , , (2 , , (4 , · , ,
0; [(1 , , (0 , , , [(2 , , (1 , , , [(0 , , (6 , , ,
1; (1 , , (2 , ,
0; (3 , , (6 , , (5 , Table 1.
The weak conjugacy classes of finite non-abelian split metacyclic subgroups of Mod( S ). Note that eachdata set of type † is quaternionic, and therefore corresponds to the weak conjugacy action of a Q -action on S .(*Thesuffix refers to the multiplicity of the tuple in the split metacyclic data set.) N . K . D HAN W AN I , K . R A J EE V S A R A T HY , AN D A . S AN G H I Group Weak conjucacy classes in
Mod( S ) Cyclic factors [ D G ; D F ] Z ⋊ − Z ((2 · , − ,
1; [(0 , , (1 , , ) ∗ [(2 , ,
1; ); (3 ,
1; ((1 , , , ((2 , , · , − ,
0; [(1 , , (0 , , , [(0 , , (1 , , ) [(2 ,
2; ((1 , , ,
1; ((1 , , , ((2 , , Z ⋊ − Z ((2 · , − ,
1; [(1 , , (0 , , ) [(2 ,
2; ((1 , , , ,
1; )]((2 · , − ,
1; [(1 , , (1 , , ) [(2 , ,
1; ); (4 , ,
1; )]((2 · , − ,
0; [(1 , , (0 , , , [(0 , , (1 , , , [(0 , , (1 , , ) [(2 ,
2; ((1 , , ,
0; ((1 , , , ((1 , , , ((3 , , · , − ,
0; [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (1 , , ) [(2 , ,
1; ); (4 ,
0; ((1 , , , ((1 , , , ((3 , , · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , ) [(2 ,
1; ((1 , , , ,
1; )]((2 · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , ) [(2 ,
2; ((1 , , , ,
1; )] Z ⋊ − Z ((2 · , − ,
1; [(0 , , (1 , , , ,
1; ); (5 ,
1; (1 , , (4 , · , − ,
1; [(0 , , (2 , , , ,
1; ); (5 ,
1; (2 , , (3 , · , − ,
0; [(1 , , (0 , , , [(1 , , (4 , , , [(0 , , (1 , , ,
2; ((1 , , ,
1; (1 , , (4 , · , − ,
0; [(1 , , (0 , , , [(1 , , (3 , , , [(0 , , (2 , , ,
2; ((1 , , ,
1; (2 , , (3 , Z ⋊ − Z ((4 · , − ,
1; [(0 , , (1 , , , ,
1; ); (3 ,
1; ((1 , , , ((2 , , · , − ,
0; [(1 , , (0 , , , [(0 , , (1 , , , [(1 , , (0 , , , [(1 , , (2 , , ,
0; ((1 , , , ((1 , , ,
1; ((1 , , , ((2 , , · , − ,
0; [(1 , , (0 , , , [(0 , , (1 , , , [(3 , , (0 , , , [(3 , , (2 , , ,
0; ((3 , , , ((1 , , ,
1; ((1 , , , ((2 , , Z ⋊ − Z ((2 · , − ,
1; [(0 , , (1 , , , ,
1; ); (6 ,
1; (1 , , (2 , · , − ,
0; [(1 , , (0 , , , [(0 , , (1 , , , [(0 , , (5 , , ,
2; ((1 , , ,
0; ((1 , , , ((5 , , · , − ,
0; [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (5 , , , ,
1; ); (6 ,
0; ((1 , , , ((5 , , · , − ,
0; [(1 , , (0 , , , [(1 , , (2 , , , [(1 , , (1 , , , [(0 , , (1 , , ,
2; ((1 , , ,
1; (1 , , (2 , Z ⋊ − Z ((4 · , − ,
1; [(0 , , (1 , , , ,
1; ); (4 ,
1; ((1 , , Z ⋊ Z ((2 · , ,
1; [(0 , , (1 , , , ,
1; ); (8 ,
1; ((1 , , · , ,
0; [(1 , , (1 , , , [(0 , , (1 , , , [(1 , , (1 , , , ,
1; ); (8 ,
0; (1 , , (3 , , (1 , , (5 , · , ,
0; [(1 , , (1 , , , [(0 , , (3 , , , [(1 , , (3 , , , ,
1; ); (8 ,
0; (1 , , (1 , , (3 , , (7 , Z ⋊ Z ((2 · , ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(1 , , (1 , , , [(1 , , (3 , , ,
2; ((1 , , ,
1; ((1 , , Z ⋊ − Z ((2 · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(1 , , (5 , , , [(0 , , (1 , , ,
2; ((1 , , ,
1; ((1 , , Z ⋊ − Z ((4 · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(1 , , (4 , , ,
0; ((1 , , , ((1 , , ,
1; (2 , , (3 , · , − ,
0; [(1 , , (0 , , , [(1 , , (2 , , , [(1 , , (3 , , ,
0; ((1 , , , ((1 , , ,
1; (1 , , (4 , · , − ,
0; [(3 , , (0 , , , [(3 , , (1 , , , [(1 , , (4 , , ,
0; ((3 , , , ((1 , , ,
1; (2 , , (3 , · , − ,
0; [(3 , , (0 , , , [(3 , , (2 , , , [(1 , , (3 , , ,
0; ((3 , , , ((1 , , ,
1; (1 , , (4 , Z ⋊ − Z ((2 · , − ,
0; [(1 , , (0 , , , [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (3 , , ,
2; ((1 , , ,
0; ((1 , , , (3 , , (7 , · , − ,
0; [(1 , , (0 , , , [(1 , , (2 , , , [(0 , , (1 , , , [(0 , , (1 , , ,
2; ((1 , , ,
0; ((1 , , , (1 , , (9 , · , − ,
0; [(1 , , (1 , , , [(1 , , (3 , , , [(0 , , (1 , , , [(0 , , (3 , , , ,
1; ); (10 ,
0; ((1 , , , (3 , , (7 , · , − ,
0; [(1 , , (1 , , , [(1 , , (1 , , , [(0 , , (1 , , , [(0 , , (1 , , , ,
1; ); (10 ,
0; ((1 , , , (1 , , (9 , Table 2.
The weak conjugacy classes of finite non-abelian split metacyclic subgroups of Mod( S ).(*The suffix refersto the multiplicity of the tuple in the split metacyclic data set.) PLIT METACYCLIC ACTIONS ON SURFACES 27
Acknowledgements
The first and third authors were supported by the UGC-JRF fellowship.The authors would also like to thank Dr. Siddhartha Sarkar for some helpfuldiscussions.
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Department of Mathematics, Indian Institute of Science Education and Re-search Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh,India
E-mail address : [email protected] Department of Mathematics, Indian Institute of Science Education and Re-search Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh,India
E-mail address : [email protected] URL : https://home.iiserb.ac.in/ e kashyap/ Department of Mathematics, Indian Institute of Science Education and Re-search Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh,India
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