A fundamental domain for PGL(2, F q [t])∖PGL(2, F q (( t −1 )))
aa r X i v : . [ m a t h . D S ] S e p A FUNDAMENTAL DOMAIN FOR
P GL p , F q r t sqz P GL ` , F q ` p t ´ q ˘˘ SANGHOON KWON
Abstract.
We give a strong fundamental domain for the quotient of
P GL ` F q pp t ´ qq ˘ by P GL p F q r t sq as a subset of distinct ordered triplepoints of P p F q pp t ´ qqq . Introduction
Given a topological space and a group acting on it, the images of a singlepoint under the group action form an orbit of the action. A strong fundamentaldomain is a subset of the space which contains exactly one point from each ofthese orbits.The action of the modular group
PSL p , Z q on the unit tangent bundle T H of the upper half plane H by Mobius transformation «˜ a bc d ¸ff : p z, v q ÞÑ ˆ az ` bcz ` d , v p cz ` d q ˙ for z P H , v P T z H serves as a key example of arithmetic and geometry. For each p z, v q in T H ,we can find a neighborhood of p z, v q which does not contain any other elementof the PSL p , Z q -orbit of p z, v q .There are various ways of constructing a strong fundamental domain, but acommon choice is the union tp z, v q : z P R, v P T z H u Y ! p w , v q P T H : 0 ď arg p v q ă π ) Y p w , v q P T H : 0 ď arg p v q ă π ( for two ramified points w “ ` ? i and w “ i and the region R “ " z P H : | z | ą , ´ ď Re p z q ă * Y " z P H : | z | “ , ´ ă Re p z q ă * bounded by the vertical lines Re p z q “ ´ and Re p z q “ and the circle | z | “ . Date : January 20, 2019.2000
Mathematics Subject Classification.
Primary 37P20, Secondary 20G25, 20H20.
The boundary at infinity B H may be identified with S and hence with P p R q . Let us say that a mutually distinct ordered triple points p x , x , x q P P p R q is positively ordered if one reaches x before x when starting counter-clockwise from x . We note that there is a bijection between the unit tangentbundle T H and the set of mutually distinct positively ordered triple points p x , x , x q of P p R q .In this article, we construct a fundamental domain for the action of modulargroup on projective general linear group over a field of formal series, namely,the action of P GL p F q r t sq on P GL ` , F q ` p t ´ q ˘˘ .Let K be the field F q pp t ´ qq of Laurent series in t ´ over a finite field F q and Z be the subring F q r t s of polynomials in t over F q of K . We further denote by O the local ring F q rr t ´ ss of K which consists of power series in t ´ over F q .From now on, let G “ P GL ` F q pp t ´ qq ˘ , Γ “ P GL p F q r t sq , h the diagonalelement «˜ t
00 1 ¸ff of G and W “ P GL p O q a maximal compact subgroup of G .Let us denote by P p K q the set of mutually distinct ordered triple points p ω , ω , ω q of P p K q . Since two by two projective general linear group overany field F acts simply transitively on distinct ordered triple points in P p F q byM¨obius transformation, we have a bijection Φ : G Ñ P p K q given by Φ p g q “ g ¨ p , , . We state our main theorem.
Theorem 1.1.
Given p ω , ω , ω q P P p K q , there is a unique γ P Γ suchthat γ ¨ p ω , ω , ω q P S X S X p S Y S q where S “ tp ω , ω , ω q P P p K q : the leading coefficient of ω is 1 u S “ tp ω , ω , ω q P P p K q : deg ω ă ă deg ω u S “ tp ω , ω , ω q P P p K q : deg ω ‰ deg ω ă deg ω u S “ tp ω , ω , ω q P P p K q : deg ω “ deg ω “ deg p ω ´ ω qu . UNDAMENTAL DOMAIN 3
In Section 2, we review the Bruhat-Tits tree of the group
P GL ` , F q ` p t ´ q ˘˘ with an explicit description of vertices and boundary at infinity. We prove thatthe set S X S X p S Y S q in Theorem 1.1 contains at least one point of each Γ -orbit of P p K q in Section 3. In Section 4, we complete the proof of themain theorem and further discuss the diagonal action on Γ z G .2. Tree of G In this section, we review the Bruhat-Tits tree T of G . We give a concreteinterpretation of vertices and edges of T without using Bruhat-Tits theory ofgeneral algebraic groups (see [BT]).Given an element α “ ´8 ÿ i “ n a i t i of K with a n ‰ , let us define r α s “ a ` a t ` ¨ ¨ ¨ ` a n t n t α u “ a ´ t ´ ` a ´ t ´ ` ¨ ¨ ¨r α s L “ a n deg α “ n the polynomial part, fractional part, the leading term and the degree of α ,respectively. Let T be the graph whose vertices are the elements of G { W ,which we can describe as «˜ t n f p t q ¸ff W for some integer n and a rational function f p t q P t n ` Z . Let π n : t n Z Ñ t n ` Z be the projection map which forgets the t n term. Two vertices «˜ t n f p t q ¸ff W and «˜ t n f p t q ¸ff W are adjacent to each other if and only if | n ´ n | “ and f and f satisfy f p t q “ π n p f p t qq , if n “ n ` f p t q “ f p t q ` ct n , if n “ n ´ SANGHOON KWON for some c P F q . It follows that the degree of every vertex of T is equal to q ` .Let us denote by o the standard vertex «˜ ¸ff W. Definition 2.1.
An isometry r : Z ě Ñ T is called a parametrized geodesic ray and an isometry ℓ : Z Ñ T is called a parametrized bi-infinite geodesic . Two ge-odesic rays ℓ and ℓ are said to be equivalent if and only if t d T p ℓ p n q , ℓ p n qq : n P Z ą u is bounded above. The Gromov boundary at infinity of T is defined asthe set of equivalence classes r ℓ s of geodesic ray ℓ starting from a fixed vertex v of T .Then, the Gromov boundary B T at infinity of T can be identified with K Yt8u (See Chapter 2 of [Se]). Moreover, given any boundary point ω P B T ,there is a unique geodesic ray from the standard vertex o to ω . We also notethat given any two distinct points ω and ω of B T , there is a bi-infinitegeodesic, which we will denote by p ω ω q , begins at ω and ends at ω .Let B T be the set tp ω , ω , ω q P pB T q : ω i ‰ ω j for 1 ď i ‰ j ď u of distinct ordered triple points in B T . Since two by two projective generallinear group over a field F acts simply transitively on p P p F qq by M¨obiustransformation, we have a bijection Φ : G Ñ B T » P p K q (see also[PS]) given by Φ p g q “ g ¨ p , , . ¨ ¨ ¨ ‚ x ´ ‚ x ´ ‚ x ´ ‚ x ‚ x ‚ x ‚ x ¨ ¨ ¨ ω ω ω x i “ «˜ t i
00 1 ¸ff W Figure 1. deg p ω q “ ´ , deg p ω q “ ´ and deg p ω q “ UNDAMENTAL DOMAIN 5 Regular continued fraction and discrete geodesic flow
In this section, we prove that the set of Theorem 1.1 contains at least onepoint from each Γ -orbit. The main ingredients of the proof are the geometry of T and regular continued fraction expansion of elements in K .Every formal series α can be unqiuely written as α “ a ` a ` a ` . . .for a P F q r t s and non-constant polynomials a i for i ě , which we call a regularcontinued fraction of α . The a i are called the partial quotients of α and we willwrite α “ r a ; a , a , . . . s . Proposition 3.1.
Given p ω , ω , ω q P B T , there is a γ P Γ such that γ ¨ p ω , ω , ω q P S X S X p S Y S q where S “ tp ω , ω , ω q P B T : r ω s L “ u S “ tp ω , ω , ω q P B T : deg ω ă ă deg ω u S “ tp ω , ω , ω q P B T : deg ω ‰ deg ω ă deg ω u S “ tp ω , ω , ω q P B T : deg ω “ deg ω “ deg p ω ´ ω qu . Proof.
Let ω “ r a ; a , a , a , . . . s ,ω “ r b ; b , b , b , . . . s , p a i , b i , c i P Z Y t8uq ω “ r c ; c , c , c , . . . s be the regular continued fraction expansions of ω i , i “ , , , respectively. Notethat ι “ «˜ ¸ff , σ c “ «˜ c
00 1 ¸ff and u f “ «˜ f p t q ¸ff belong to Γ for every d P F q and f p t q P Z . The actions are given by ι p ω , ω , ω q “ ˆ ω , ω , ω ˙ SANGHOON KWON σ c p ω , ω , ω q “ p cω , cω , cω q and u f p ω , ω , ω q “ p ω ` f, ω ` f, ω ` f q . Given any p ω , ω , ω q P S X p S Y S q , we have σ c p ω , ω , ω q P S X S X p S Y S q for c “ pr ω s L q ´ P F q . Thus, itis enough to find a γ P Γ such that γ ¨ p ω , ω , ω q P S X p S Y S q .Since ω ‰ ω , there exists j ě such that a j ‰ c j and hence we mayassume without loss of generality that a “ and c ‰ by applying ι and u ´ a , . . . , u ´ a j . Let ω “ r a , a , a , . . . s ,ω “ r b ; b , b , b , . . . s , p a i , b i , c i P Z Y t8uq ω “ r c ; c , c , c , . . . s If b ‰ and deg b ă deg c , then p ω , ω , ω q P S X S . If deg b ą deg c ,then ι ˝ u ´ c p ω q “ r ´ c , a , a , . . . s ι ˝ u ´ c p ω q “ r b ´ c , b , b , . . . s ι ˝ u ´ c p ω q “ r c ; c , c . . . s . In this case, ι ˝ u ´ c p ω , ω , ω q P S X S . If deg b “ deg c and b ‰ c ,then ι ˝ u ´ c p ω , ω , ω q belongs to S X S or S X S depending on whether r b s L “ r c s L or not. If b “ c and k ą is the smallest integer such that b k ‰ c k , then we have ι ˝ u ´ c k ˝ ¨ ¨ ¨ ι ˝ u ´ c p ω q “ r ´ c k , . . . , ´ c , a , a , . . . s ι ˝ u ´ c k ˝ ¨ ¨ ¨ ι ˝ u ´ c p ω q “ r b k ´ c k , b k ` , b k ` , . . . s ι ˝ u ´ c k ˝ ¨ ¨ ¨ ι ˝ u ´ c p ω q “ r c k ` ; c k ` , c k ` , . . . s and hence it can be reduced to one of the previous cases. Thus, this proves theexistence of such γ . (cid:3) UNDAMENTAL DOMAIN 7 ¨ ¨ ¨ ‚ x ´ ‚ x ´ ‚ x ´ ‚ x ‚ x ‚ x ‚ x ¨ ¨ ¨ ω ω ω x i “ «˜ t i
00 1 ¸ff W Figure 2. p ω , ω , ω q P S X S X S ¨ ¨ ¨ ‚ x ´ ‚ x ´ ‚ x ´ ‚ x ‚ x ‚ x ‚ x ¨ ¨ ¨ ω ω ω x i “ «˜ t i
00 1 ¸ff W Figure 3. p ω , ω , ω q P S X S X S Now let us introduce some notions for geometry of trees. This is useful whenwe construct a strong fundamental domain of Γ zB T as well as a factor p Γ z GT , φ q of p Γ z G, φ a q which is similar to the geodesic flow system of the unittangent bundle on hyperbolic surfaces. Here, φ h : Γ z G Ñ Γ z G is the righttranslation map given by φ h p x q “ xh .Recall that a parametrized bi-infinite geodesic in T is an isometry ℓ : Z Ñ T .(See the below of Definition 2.1). Note that a bi-infinite geodesic is completelydetermined by two distinct points ℓ p´8q and ℓ p8q of B T and the markedvertex ℓ p q P p ℓ p´8q ℓ p8qq .Let GT be the set of all parametrized bi-infinite geodesics in T . Given atuple p ω , ω , ω q P B T , we have the map p p ω ω q p ω q be the unique vertexprojected from ω to the (unparametrized) bi-infinite geodesic p ω ω q . Forexample, p p ω ω q p ω q “ x ´ for Figure 2 and p p ω ω q p ω q “ x ´ for Figure 3.We define a factor map Θ : B T Ñ GT by Θ p ω , ω , ω q “ ℓ where ℓ isthe unique parametrized bi-infinite geodesic satisfying ℓ p´8q “ ω , ℓ p8q “ ω ,and ℓ p q “ p p ω ω q p ω q . Let φ : GT Ñ GT be the discrete geodesic flow givenby φ p ℓ qp n q “ ℓ p n ` q . Taking the quotient by the discrete subgroup Γ andattaching the factor map Θ , we obtain the following commutative diagram. SANGHOON KWON Γ z G φ h / / Φ (cid:15) (cid:15) Γ z G Φ (cid:15) (cid:15) Γ zB T φ h / / Θ (cid:15) (cid:15) Γ zB T (cid:15) (cid:15) Γ z GT φ / / Γ z GT A fundamental domain for Γ z G In this section, we give a strong fundamental domain for the action of Γ on P p K q . We recall that G may be identified with P p K q via the map Φ inSection 1. Theorem 4.1.
The subset S X S X p S Y S q of B T introduced in Propo-sition 3.1 is a strong fundamental domain of Γ zB T . Proof.
Let us assume that γ ¨ p ω , ω , ω q “ p η , η , η q and p ω , ω , ω q , p η , η , η q P S X S X p S Y S q . Then, γ maps p p ω ω q p ω q to p p η η q p η q . Since p ω , ω , ω q and p η , η , η q arecontained in S X p S Y S q , it follows that p p ω ω q p ω q and p p η η q p η q are ofthe form «˜ t deg ω
00 1 ¸ff W and «˜ t deg η
00 1 ¸ff W, respectively. Since G is the disjoint union G “ \ i “ Γ «˜ t i
00 1 ¸ff W of double cosets with respect to Γ and W , this implies that deg p ω q “ ˘ deg p η q .If deg p ω q “ deg p η q ě , then γ “ «˜ a b d ¸ff for some a, d P F ˆ q and b P Z with deg p b q ď deg p ω q . Since deg p η q “ deg p γ ¨ ω q “ deg p ad ´ ω ` d ´ b q ă , UNDAMENTAL DOMAIN 9 we must have b “ . Moreover, r ω s L “ r η s L “ r ad ´ ω s L “ implies that γ “ «˜ ¸ff . If deg p ω q “ deg p η q ă , then γ “ «˜ a c d ¸ff for some a, d P F ˆ q and c P Z with deg p c q ď ´ deg p ω q . The positivity of deg p η q and the similar argument implies that γ is the identity.If deg p ω q “ ´ deg p η q , then either γ or γ ´ is of the form «˜ da b ¸ff for some a, d P F ˆ q and deg p b q ď | deg p ω q| . Then deg p η q “ deg p d p aω ` b q ´ q ă which is impossible. Therefore, γ must be an identity. Together withProposition 3.1, we can conclude that S X S X p S Y S q is a fundamentaldomain. (cid:3) Recall that φ h is the right translation map x ÞÑ xh from Γ z G to itself. Wediscuss the system p Γ z G, φ h q in the remaining part. Let τ h : G Ñ G be theright translation map τ h p g q “ gh . Proposition 4.2.
Let ϕ h : B T Ñ B T be the map given by ϕ h p ω , ω , ω q “ ˆ ω , p ω ´ ω q ω t ` p ω ´ ω q ω p ω ´ ω q t ` ω ´ ω , ω ˙ , then the following commutative diagram holds. G τ h / / Φ (cid:15) (cid:15) G Φ (cid:15) (cid:15) B T ϕ h / / B T Proof.
Suppose that p ω , ω , ω q “ g ¨ p , , with g “ ˜ a bc d ¸ . Then, Φ ˝ φ h ˝ Φ ´ p ω , ω , ω q “ Φ ˝ φ h p g q “ Φ p gh q “ g ¨ p , t,
8q “ p ω , ω , ω q where ω “ at ` bct ` d and ω “ a ` bc ` d . If c “ , then b “ ω d , a “ p ω ´ ω q d and ω “ 8 . Thus, we have ω “ p ω ´ ω q t ` ω . If c ‰ , then a “ ω c and b “ ω d . In this case, we have dc “ ω ´ ω ω ´ ω which yields ω “ p ω ´ ω q ω t ` p ω ´ ω q ω p ω ´ ω q t ` ω ´ ω . This completes the proof of the proposition. (cid:3)
According to the Theorem 4.1, there is a unique point in S X S X p S Y S q X Γ ¨ p ω , ω , ω q which we will denote by rp ω , ω , ω qs Γ . Corollary 4.3.
Let S “ S X S X p S Y S q and ψ h : S Ñ S be the map givenby ψ h p ω , ω , ω q “ „ˆ ω , p ω ´ ω q ω t ` p ω ´ ω q ω p ω ´ ω q t ` ω ´ ω , ω ˙ Γ . Then, the diagram Γ z G φ h / / Φ (cid:15) (cid:15) Γ z G Φ (cid:15) (cid:15) S ψ h / / S commutes. References [BPP] A. Broise-Alamichel, J. Parkkonen and F. Paulin,
Equidistribution and counting underequilibrium states in negatively curved spaces and graphs of groups , Progress in Mathe-matics , Birkhuser (2019)[BT] F. Bruhat and J. Tits,
Groupes rductifs sur un corps local, I. Donnes radicielles values ,Inst. Hautes tudes Sci. Publ. Math. (1972), 5-251[PS] F. Paulin and U. Shapira, On continued fraction expansions of quadratic irrationals inpositive characteristic , Groups, Geometry and Dynamics (2020), 81-105[Se] J.-P. Serre, Trees , Springer Monographs in Mathematics (2003)
UNDAMENTAL DOMAIN 11
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