Fundamental domains in PSL(2,R) for Fuchsian groups
aa r X i v : . [ m a t h . DG ] S e p Fundamental domains in
PSL(2 , R ) forFuchsian groups H UYNH M INH H IEN
Department of Mathematics,Quy Nhon University, Binh Dinh, Vietnam;e-mail: [email protected]
Abstract
In this paper, we provide a necessary and sufficient condition for a setin
PSL(2 , R ) or in T H to be a fundamental domain of a given Fuchsiangroup via its respective fundamental domain in the hyperbolic plane H . Keywords:
Fundamental domain; Hyperbolic plane;
PSL(2 , R ) ; Fuchsian group Fundamental domains arise naturally in the study of group actions on topologicalspaces. The concept fundamental domain is used to describe a set in a topologi-cal space under a group action of which the images under the action tessellate thewhose space. The term fundamental domain is well-known in the model of the hy-perbolic plane H for the action of Fuchsian groups via M ¨obius transformations.If there exists a point in H that is not a fixed point for all elements different from1he unity in a Fuchsian group Γ then there always exist a convex and connectedfundamental domain for Γ named Dirichlet domain (see [1, 3]). Other examplesof fundamental domains are Ford domains [4]. The Poincar´e’s polygon theorem [2, 3, 4] provides a fundamental domain which is a polygon for the Fuchsiangroup generated by the side-pairing transformations. In this case, if the polygonhas finite edges (and hence it is relatively compact), the Fuchsian group is finitelygenerated and the space of Γ -orbits denoted by Γ \ H is compact. Fundamentaldomains have several applications for the study of Γ \ H . If the action of Γ hasno fixed points, the quotient space Γ \ H has a Riemann surface structure that isa closed Riemann surface of genus at least and has the hyperbolic plane H asthe universal covering. Furthermore, it is well-known that any compact orientablesurface with constant negative curvature is isometric to a factor Γ \ H . If the Fuch-sian group has a finite-area fundamental domain then all the fundamental domainshave finite area and have the same area. This area is defined for the measure of thequotient space Γ \ H . In addition, the space Γ \ H is compact if and only if everyfundamental domain in H for Γ is relatively compact [3].There exists a bijection Θ : T H → PSL(2 , R ) . The natural Riemannianmetric on PSL(2 , R ) induces a left-invariant metric function (a metric in usualsense). The topology induced from this metric is the same as the quotient topol-ogy induced from the one in SL(2 , R ) . The Sasaki metric on the unit tangentbundle T H with respect to the hyperbolic metric on H makes Θ an isometry.This induces an isometry from T (Γ \ H ) to the collection of right co-sets Γ g of Γ in PSL(2 , R ) denoted by Γ \ PSL(2 , R ) that is also obtained from a left action ofFuchsian group Γ on PSL(2 , R ) . Furthermore, there is an action of Γ on the unittangent bundle T H by the differential of elements in Γ and this arises fundamen-tal domains for Γ in T H also. However, up to now there have not any resultsabout fundamental domains for Γ in PSL(2 , R ) or in T H . The aim of this paperis to study fundamental domains in PSL(2 , R ) and in T H for Fuchsian groups.A necessary and sufficient condition for a set in PSL(2 , R ) or in T H to be a2undamental domain via its respective fundamental domain in H is provided.The paper is organized as follows. In the next section we present action ofFuchsian groups on the hyperbolic plane H , the unit tangent bundle T H and PSL(2 , R ) and some basic examples of fundamental domain in H . Main resultsare stated and proved in Section 3. In this paper we introduce the necessary background material which is well-known in [2, 3, 4]. The unity of arbitrary group is always denoted by e . Let X be a non-empty set and let G be a group. Let ρ : G × X → X be a (left)group action, that is, ρ ( e, x ) = x and ρ ( g , ρ ( g , x )) = ρ ( g g , x ) for all x ∈ X and g , g ∈ G . For a subset A ⊂ X , denote ρ ( g, A ) = { ρ ( g, x ) , x ∈ A } . Definition 2.1.
Let G be a group and let X be a topological space. Suppose that ρ : G × X → X is an action. A non-empty open set F ⊂ X is said to be a fundamental domain for G , if(a) S g ∈ G ρ ( g, F ) = X , and(b) ρ ( g, F ) ∩ F = ∅ for all g ∈ G \ { e } .Here e is the unity of G and F denotes the closure of F in X .Due to the fact that G is a group, condition (b) is equivalent to ρ ( g , F ) ∩ ρ ( g , F ) = ∅ for all g , g ∈ G, g = g . We will introduce some examples in the next subsection.3 .2 H and PSL(2 , R ) The hyperbolic plane is the upper half plane H = { ( x, y ) ∈ R : y > } , en-dowed with the hyperbolic ( g z ) z ∈ H , where g z ( ξ, η ) = ξ η + ξ η (Im z ) for ξ = ( ξ , ξ ) , η =( η , η ) ∈ T z H ∼ = C . The group of M ¨obius transformations M¨ob( H ) = { z az + bcz + d : a, b, c, d ∈ R , ad − bc = 1 } can be identified with the projective group PSL(2 , R ) = SL(2 , R ) / {± E } by means of the isomorphism Φ ± (cid:18) a bc d (cid:19)! = z az + bcz + d , where SL(2 , R ) is the group of all real × matrices with unity determinant, and E denotes the unit matrix.Let Γ be a Fuchsian group, which is a discrete subgroup in PSL(2 , R ) . Weconsider the action ρ : Γ × H → H , ρ ( γ, z ) = Φ( γ )( z ) for ( γ, z ) ∈ Γ × H .The action is called free if Φ( γ )( z ) = z for some z ∈ H then γ = e . In this case,there always exist fundamental domains: Proposition 2.1.
Let Γ ⊂ PSL(2 , R ) be a Fuchsian group and take z ∈ H suchthat z = Φ( γ )( z ) holds for all γ ∈ Γ \ { e } . Then the Dirichlet region D z (Γ) = n z ∈ H : d H ( z, z ) < d H ( z, T ( z )) for all T = Φ( γ ) , γ ∈ Γ \ { e } o is a fundamental domain for Γ which contains z . See [1, Lemma 11.5] for a proof. Note that such a z does exist if the actionof Γ on H is free.For g = n ± (cid:0) a bc d (cid:1)o ∈ PSL(2 , R ) , the trace of g is defined by tr( g ) = | a + d | . Element g is called hyperbolic if | tr( g ) | > , elliptic if | tr( g ) | < and parabolicif | tr( g ) | = 2 . Recall that the action of Γ on H is free if and only if Γ does notcontain elliptic elements. 4or any g ∈ PSL(2 , R ) , the cyclic group h g i = { g n : n ∈ Z } ⊂ PSL(2 , R ) is a Fuchsian group. We will consider fundamental domains for h g i with specialclasses of g . For t ∈ R , denote A t = (cid:18) e t/ e − t/ (cid:19) , B t = (cid:18) t (cid:19) , D t = (cid:18) cos t sin t − sin t cos t (cid:19) ∈ SL(2 , R ) and respectively a t = [ A t ] , b t = [ B t ] , d t = [ D t ] ∈ PSL(2 , R ) . Proposition 2.2. (a) For any t > , the set F t = { z = x + iy ∈ H : 1 < y < e t } is a fundamental domain in H for the Fuchsian group h a t i .(b) For any t > , the set E t = { z = x + iy ∈ H : 0 < x < t } is a fundamental domain in H for the Fuchsian group h b t i . Proof : (a) Obviously F t is open and F t = { z = x + iy ∈ H : 1 ≤ y ≤ e t } . We have Φ( a jt ) = e jt id , with id : H → H denoting the identity. Here Φ( a jt )( F t ) = e jt id( { z = x + iy ∈ H : 1 ≤ y ≤ e t } ) = { z ∈ H : e jt ≤ y ≤ e ( j +1) t } , so that S j ∈ Z Φ( a jt )( F t ) = H and Φ( a jt )( F t ) ∩ Φ( a kt )( F t ) = ∅ for j = k .(b) It is proved analogously to (a). (cid:3) The collection of right co-sets Γ g of Γ in PSL(2 , R ) denoted by Γ \ PSL(2 , R ) can be also obtained by Γ -orbits of the left action ̺ : Γ × PSL(2 , R ) → PSL(2 , R ) , ̺ ( γ, g ) = γg for γ ∈ Γ , g ∈ PSL(2 , R ) . (2.1)This leads to the concept fundamental domain in PSL(2 , R ) .5 emark 2.1. If F ⊂
PSL(2 , R ) is a fundamental domain for Γ and γ ∈ Γ \ { e } ,then γ F is a fundamental domain disjoint from F . For, it is obvious that γ F isopen since F open and γ F = γ F in PSL(2 , R ) . Therefore [ γ ′ ∈ Γ γ ′ γ F = [ γ ′ ∈ Γ γ ′ ( γ F ) = [ γ ′ ∈ Γ ( γ ′ γ )( F ) = [ η ∈ Γ η F = PSL(2 , R ) , and for γ ′ ∈ Γ \ { e } , ( γ F ) ∩ γ ′ ( γ F ) = γ F ∩ ( γ ′ γ ) F = ∅ by (a) in Definition 2.1, due to γ = γ ′ γ . ♦ T H The unit tangent bundle of H is defined by T H = { ( z, ξ ) : z ∈ H , ξ ∈ T z H , k ξ k z = g z ( ξ, ξ ) / = 1 } . (2.2)For g ∈ PSL(2 , R ) we consider the derivative operator D g : T H → T H defined as D g ( z, ξ ) = ( T ( z ) , T ′ ( z ) ξ ) , where T = Φ( g ) . Then D is well-defined. Explicitly, if g = n ± (cid:0) a bc d (cid:1)o , then T ( z ) = az + bcz + d and ad − bc = 1 , whence D g ( z, ξ ) = (cid:16) az + bcz + d , ξ ( cz + d ) (cid:17) . (2.3)Let Γ ⊂ PSL(2 , R ) be a subgroup. Consider the group action κ : Γ × T H → T H , κ ( γ, ( z, ξ )) = D γ ( z, ξ ) for γ ∈ Γ , ( z, ξ ) ∈ T H . If Γ = PSL(2 , R ) then the action is simply transitive (see [1, Lemma 9.2]), thatis, for given ( z, ξ ) , ( w, η ) ∈ T H , there exists a unique g ∈ PSL(2 , R ) such that κ ( g, ( z, ξ )) = D g ( z, ξ ) = ( w, η ) . In particular, we have the following result:6 emma 2.1. For each ( z, ξ ) ∈ T H , there is a unique g ∈ PSL(2 , R ) such that D g ( i, i ) = ( z, ξ ) . Note that if ( z, ξ ) ∈ T H then g = n ± (cid:16) a bc d (cid:17)o is defined by ac + bdc + d = Re z, c + d = Im z, cd ( c + d ) = Re ξ, d − c ( c + d ) = Im ξ. (2.4)We will use these relations afterwards. This section deals with the relation of fundamental domains for a Fuchsian groupin H , PSL(2 , R ) and in T H .The main result of this paper is the following: Theorem 3.1.
Let Γ ⊂ PSL(2 , R ) be a Fuchsian group. For F ⊂ H , denote F = { g = b x a ln y d θ : x + iy ∈ F, θ ∈ [0 , π ) } ⊂ PSL(2 , R ) . Then F is a fundamental domain for Γ in H if and only if F is a fundamentaldomain for Γ in PSL(2 , R ) . Remark 3.1.
Recall that if Γ contains no elliptic elements then there always existfundamental domains in H for Γ and hence fundamental domains in PSL(2 , R ) do always exist. The collection of Γ -orbits of the action ̺ (see (2.1)) denotedby Γ \ PSL(2 , R ) = { Γ g, g ∈ PSL(2 , R ) } is compact if and only if the quotientspace Γ \ H is compact if and only if there is a relatively compact fundamentaldomain (in H or in PSL(2 , R ) ) for Γ . In this case all fundamental domains of Γ is relatively compacts. For proofs of statements in H , see [3, Chapter 3]. ♦ In order to prove Theorem 3.1, we need the following factorization which iscalled NAK decomposition (so-called Iwasawa decomposition).7 emma 3.1 ([5]) . If G = (cid:16) a bc d (cid:17) ∈ SL(2 , R ) then G = B x A ln y d θ with x = ac + bdc + d , y = 1 c + d , θ = − d + ic ) . (3.1) Lemma 3.2. (a) If g = [ G ] ∈ PSL(2 , R ) for G = (cid:16) a bc d (cid:17) ∈ SL(2 , R ) then g = b x a ln y d θ with x = ac + bdc + d , y = 1 c + d , θ = − d + ic ) . (3.2) (b) PSL(2 , R ) = { b x a ln y d θ : x + iy ∈ H , θ ∈ [0 , π ) } = { b x a ln y d θ : x + iy ∈ H , θ ∈ R } . Proof : (a) This follows directly from Lemma 3.1. (b) According to (a), everyelement g = n ± (cid:16) a bc d (cid:17)o ∈ PSL(2 , R ) has the decomposition g = b x a ln y d θ for x + iy ∈ H and θ = − d + ic ) ∈ ( − π, . It remains to verify that wecan find some θ ′ ∈ [0 , π ) such that d θ ′ = d θ and as a consequence, g = b x a ln y d θ ′ .Indeed, the matrix D θ changes by an overall sign if θ changes by π and so doesthe matrix G = B x A ln y D θ . Therefore we can find a unique k ∈ Z such that θ ′ := 2 kπ + θ ∈ [0 , π ) to have d θ ′ = d θ . This implies the first equality in (b).The latter follows from d θ +2 kπ = d θ for all θ ∈ [0 , π ) and k ∈ Z . (cid:3) Proof of Theorem 3.1.
First, denote ˆ F = { G = B x A ln y D θ ∈ SL(2 , R ) : x + iy ∈ F, θ ∈ [0 , π ) } . It is easy to see that ˆ F is open in SL(2 , R ) and since the projection π : SL(2 , R ) → PSL(2 , R ) is an open map and π ( ˆ F ) = F , it follows that F is open in PSL(2 , R ) (note that d θ +2 kπ = d θ for θ ∈ [0 , π ) and k ∈ Z ). To establish part (a) in thedefinition, we first claim that the closure of F in PSL(2 , R ) is F = { g = b x a ln y d θ ∈ PSL(2 , R ) : x + iy ∈ F , θ ∈ [0 , π ) } . ˆ F = { G = B x A ln y D θ : x + iy ∈ F , θ ∈ [0 , π ) } , (3.3)where F denotes the closure of F in H and ˆ F denotes the closure of ˆ F in SL(2 , R ) . The set in the right-hand side of (3.3) is denoted by cl ( ˆ F ) . For ev-ery G ∈ cl ˆ F , we show that G n → G for some sequence ( G n ) n ⊂ ˆ F . Writ-ing G = B x A ln y D θ , we have x + iy ∈ F by the definition of cl ( ˆ F ) . Let ( x n + iy n ) n ⊂ F be such that x n + iy n → x + iy in H as n → ∞ . Then x n → x as well as y n → y in R . Taking G n = B x n A ln y n D θ ∈ cl ( ˆ F ) , we obtain G n → G in SL(2 , R ) after a short check.Next, for any g = b x a ln y d θ ∈ PSL(2 , R ) we have z := x + iy ∈ Φ( γ )( F ) for some γ ∈ Γ as F ⊂ H is a fundamental domain for Γ . Take ˜ z = ˜ x + i ˜ y =Φ( γ − )( z ) ∈ F and write γ = [ T ] with T = (cid:16) t t t t (cid:17) ∈ SL(2 , R ) . Let ˜ θ = θ + 2 arg( t ˜ z + t ) + 2 kπ ∈ [0 , π ) for a unique k ∈ Z to obtain h := b ˜ x a ln ˜ y d ˜ θ ∈ F . Thus x + iy = t ˜ z + t t ˜ z + t and θ = ˜ θ − t ˜ z + t ) − kπ implies g = γh ∈ γ F after a short calculation. This completes the proof for (a)in the definition.For part (b), suppose on the contrary that there exists g ∈ F ∩ γ F for some γ ∈ Γ \ { e } . Then g = b x a ln y d θ and g = γb x ′ a ln y ′ d θ ′ for x + iy ∈ F and x ′ + iy ′ ∈ F . A short calculation shows that x + iy = Φ( γ )( x ′ + iy ′ ) ∈ F ∩ Φ( γ )( F ) that contradicts to the fact that F is a fundamental domain. Thus F ∩ γ F = ∅ forall γ ∈ Γ \ { e } .Conversely, assume that F is a fundamental domain for Γ . Then F ⊂ H isopen since F is open. For any z = x + iy ∈ H , then g = b x a ln y ∈ PSL(2 , R ) = ∪ γ ∈ Γ γ F implies that g = γh for some γ ∈ Γ and h ∈ F = { b x a ln y d θ , x + iy ∈ , ≤ θ ≤ π } . Write h = b ˜ x a ln ˜ y d ˜ θ . Then ˜ z = ˜ x + i ˜ y ∈ F and z = Φ( γ )(˜ z ) yield z ∈ Φ( γ )( F ) . This proves (a) in Definition 2.1. Finally, assume that z = x + iy ∈ F and z = Φ( γ )( z ′ ) for some γ ∈ Γ \ { e } and z ′ = x ′ + iy ′ ∈ F .Then take g = b x a ln y d π and h = b x ′ a ln y ′ d θ with θ = 2 arg( h z + h ) + 2 kπ for a unique k ∈ Z such that θ ∈ [0 , π ) ; here h = π ( H ) , H = (cid:16) h h h h (cid:17) .Then g = γh after a short calculation. This means that F ∩ γ F 6 = ∅ which isimpossible since F is a fundamental domain. (cid:3) The next result follows directly from Proposition 2.2 and Lemma 3.1.
Corollary 3.1. (a) For t > , the set F t = { g = b x a ln y d θ ∈ PSL(2 , R ) : x ∈ R , < y < e t , θ ∈ [0 , π ) } (3.4) is a fundamental domain in PSL(2 , R ) for the Fuchsian group h a t i . (b) For t > , the set E t = { g = b x a ln y d θ ∈ PSL(2 , R ) : 0 < x < t, y > , θ ∈ [0 , π ) } (3.5) is a fundamental domain in PSL(2 , R ) for the Fuchsian group h b t i . It is well-known that
PSL(2 , Z ) is a Fuchsian group and the set F = n z ∈ H : | z | > , | Re z | < o is a fundamental domain of PSL(2 , Z ) in H (see [1, Proposition 9.18]). It followsfrom Lemma 3.2 and Theorem 3.1 that Corollary 3.2.
The set F = n g = [ G ] ∈ PSL(2 , R ) , G = (cid:16) a bc d (cid:17) ∈ SL(2 , R ) :2 | ac + bd | < c + d , ( ac + bd ) + 1 > ( c + d ) o is a fundamental domain in PSL(2 , R ) for PSL(2 , Z ) . Lemma 3.3.
Let g and g be conjugate in PSL(2 , R ) and g = hg h − for h ∈ PSL(2 , R ) . Then if F ⊂ PSL(2 , R ) is a fundamental domain for h g i then F = h F is a fundamental domain for h g i . Proof.
Obviously F = h F . Since F is a fundamental domain for h g i , we have [ j ∈ Z g j F = [ j ∈ Z hg j h − h F = [ j ∈ Z hg j F = h ( [ j ∈ Z g j F ) = h PSL(2 , R ) = PSL(2 , R ) , and if j ∈ Z , g j = e , then g j = e yields F ∩ g j F = h F ∩ hg j h − h F = h F ∩ hg j F = h ( F ∩ g j F ) = ∅ . Also both F ⊂ PSL(2 , R ) and F ⊂ PSL(2 , R ) are open. (cid:3) Recall that every hyperbolic (resp. parabolic) element is conjugate with a t (resp. b t ) for some t ∈ R . Note that h a t i = h a − t i and h b t i = h b − t i . The nextresult follows from the preceding lemma. Proposition 3.1.
Let g ∈ PSL(2 , R ) be a hyperbolic element (resp. parabolicelement). If h ∈ PSL(2 , R ) and t ∈ R are such that g = h − a t h (resp. g = h − b t h ) then F = h F | t | (resp. E = h E | t | ) is a fundamental domain for Γ = h g i ,where F | t | (resp. E | t | ) is a fundamental domain for h a t i (resp. h b t i ) given by (3.4) (resp. (3.5) ). Next we define
Θ : T H → PSL(2 , R ) by Θ( z, ξ ) = g for ( z, ξ ) ∈ T H ,where g ∈ PSL(2 , R ) satisfies D g ( i, i ) = ( z, ξ ) . Then Θ is well-defined andbijective owing to Lemma 2.1. Recall that there exist metrics on PSL(2 , R ) and T H such that Θ is an isometry. 11 emma 3.4. Let F ⊂ H and denote T F = { ( z, ξ ) ∈ T H : z ∈ F } . Then Θ( T F ) = { g ∈ PSL(2 , R ) : g = b x a ln y d θ : x + iy ∈ F, θ ∈ [0 , π ) } . Proof :
For any g = b x a ln y d θ ∈ PSL(2 , R ) with x + iy ∈ F , if g = n ± (cid:16) a bc d (cid:17)o then we take z = x + iy ∈ F and ξ = Re ξ + i Im ξ satisfying x = ac + bdc + d , y = 1 c + d , Re ξ = 2 cd ( c + d ) , Im ξ = d − c ( c + d ) . Then k ξ k z = | ξ | y = 1 means that ( z, ξ ) ∈ T F and (2.4) shows Θ( z, ξ ) = g. On the other hand, for ( z, ξ ) ∈ T F and Θ( z, ξ ) = g ∈ PSL(2 , R ) . If g = n ± (cid:16) a bc d (cid:17)o = b x a ln y d θ then x = ac + bdc + d , y = c + d by Lemma 3.2 (a). Onceagain (2.4) implies that z = x + iy ∈ F . This completes the proof. (cid:3) The relation of fundamental domains in H and in T H is the following: Theorem 3.2.
Let Γ be a Fuchsian group. A set F ⊂ H is a fundamental domainfor Γ if and only if T F ⊂ T H is a fundamental domain for Γ . Proof :
Let F ⊂ H and F = { g = b x a ln y d θ , x + iy ∈ F, θ ∈ [0 , π ) } ⊂ PSL(2 , R ) . Then Θ − ( F ) = T F by Lemma 3.4 and this follows from Theorem3.1 and the fact that Θ is an isomorphism. (cid:3) See also [3, Exercise 24] for the sufficiency.
Acknowledgments:
This work was supported by the Research Project of VietnamMinistry of Education and Training, Grant No. B2019-DQN-10.
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