aa r X i v : . [ m a t h . D S ] D ec ON p -ADIC M ¨OBIUS MAPS YUEFEI WANG AND JINGHUA YANG
Abstract.
In this paper, we study three aspects of the p − adic M¨obiusmaps. One is the group PSL(2 , O p ), another is the geometrical charac-terization of the p − adic M¨obius maps and its application, and the otheris different norms of the p − adic M¨obius maps. Firstly, we give a seriesof equations of the p − adic M¨obius maps in PSL(2 , O p ) between matrix,chordal, hyperbolic and unitary aspects. Furthermore, the propertiesof PSL(2 , O p ) can be applied to study the geometrical characterization,the norms, the decomposition theorem of p − adic M¨obius maps, and theconvergence and divergence of p − adic continued fractions. Secondly, weclassify the p − adic M¨obius maps into four types and study the geomet-rical characterization of the p − adic M¨obius maps from the aspects offixed points in P Ber and the invariant axes which yields the decompo-sition theorem of p − adic M¨obius maps. Furthermore, we prove thatif a subgroup of PSL(2 , C p ) containing elliptic elements only, then allelements fix the same point in H Ber without using the famous theorem–Cartan fixed point theorem, and this means that this subgroup haspotentially good reduction. In the last part, we extend the inequali-ties obtained by Gehring and Martin [24,25], Beardon and Short [12] tothe non-archimedean settings. These inequalities of p -adic M¨obius mapsare between the matrix, chordal, three-point and unitary norms. Thispart of work can be applied to study the convergence of the sequence of p − adic M¨obius maps which can be viewed as a special cases of the workin [20] and the discrete criteria of the subgroups of PSL(2 , C p ). Introduction
Statement of results.
We call an element in the projective speciallinear group PSL(2 , C p ) the p -adic M¨obius map, where Q p is the field of p -adic rational numbers and C p is the completion of the algebraic closureof Q p . The projective space P ( C p ) is totally disconnected and not locallycompact, which implies that we can not adopt the method used in dealingwith the complex settings easily. The main tool that we use is the projectiveBerkovich space P Ber (see concrete definitions in section 3), since PSL(2 , C p )acts on the hyperbolic Berkovich space H Ber isometrically and the projectiveBerkovich space is compact with respect to the weak topology.
Date : August 24, 2018.2010
Mathematics Subject Classification.
Key words and phrases. p -adic M¨obius maps, discrete criteria, the convergence theo-rem, three points norms, unitary norms, hyperbolic norms, chordal norms, inequalities. We study the subgroup PSL(2 , O p ), where O p = { z || z | ≤ } firstly,since the study of the unitary groups of the projective special linear groupPSL(2 , C ) is a very important topic in the study of M¨obius maps. Thereexist a lot of equations of M¨obius maps in the unitary group. It is natu-ral to generalize the equations to the non-archimedean settings. We givea series of equations of the p − adic M¨obius maps in PSL(2 , O p ) betweenmatrix, chordal, hyperbolic and unitary aspects. Furthermore, the proper-ties of PSL(2 , O p ) can be applied to study the geometrical characterization,the norms and the decomposition theorem of p − adic M¨obius maps and theconvergence and divergence of p − adic continued fractions.Let g be a p -adic M¨obius map, ρ v ( z, w ) be the chordal metric on theprojective line P ( C p ), ρ ( z, w ) be the hyperbolic metric on the hyperbolicBerkovich space, L ( g ) = p ρ ( g ( ζ Gauss ) ,ζ Gauss ) , and k · k be the matrix norm(see concrete definitions in section 2). Theorem 1.1.
For any p -adic M¨obius map g , the following are equivalent: (1) g ∈ PSL(2 , O p ) ; (2) L ( g ) = 1 ; (3) ρ ( ζ Gauss , g ( ζ Gauss )) = 0 ; (4) g is a chordal isometry; (5) k g k = 1 ; (6) for any h in the Matrix ring M(2 , C p ) , k gh k = k hg k = k ghg − k = k h k . In the second part, we study the characterization of p − adic M¨obius maps.In [26], Kato introduced the idea of the Klenian group to study the p − adicM¨obius maps firstly, and in [6, 30], Vermitage and Parker, Qiu, Yang andYin gave the discrete criteria of subgroups of PSL(2 , C p ). The method thatthey used are also derived from the Kleinian group. We not only follow thephilosophy of the Kleinian group to study the p − adic M¨obius maps, butalso we lay emphasis on the study of the difference between them. First, weclassify the p − adic M¨obius maps into four types which is a bit different thanthose in the Kleinian group, and we study the geometrical characterizationof the p − adic M¨obius maps from the aspects of fixed points in P Ber andthe invariant axes. Furthermore, we can decompose a p − adic M¨obius map g into two involutions α, β (an elliptic element of order 2), namely g = α ◦ β ,and the structure of the fixed points of α, β can reflect the type and otherproperties of g . This method can be used to prove that if a subgroup ofPSL(2 , C p ) contains elliptic elements only, then all elements fix the samepoint in H Ber without using the famous theorem–Cartan fixed point theo-rem, and this means that this subgroup has potentially good reduction(seeconcrete definitions in section 2). In the proof of the results, we should facethree difficulties which do not exist in the archimedean settings. One is that P ( C p ) and P Ber are not locally compact, another is that there exists a newkind of p − adic M¨obius maps–the wild elliptic elements whose geometricalstructure of the fixed points are complicated, and the other is that the primenumber p affects the structure of the fixed points of p − adic M¨obius maps.We give a series of tables to compare the properties of p − adic M¨obius mapsand those of the Kleinian group.the Kleinian group p − adic M¨obius mapstype loxodromicparabolicelliptic loxodromicparabolictame ellipticwild ellipticAssuming g ( z ) = az + bcz + d = α ◦ β with ad − bc = 1, let F g , F α , F β denote thefixed points of g, α, β in P ∪ H respectively.the Kleinian group F g ⊂ P ∪ H F α ∩ F β loxodromicparabolicelliptic two points in P one point in P a geodesic line in P ∪ H ∅ unique point in P unique point in H Let Int( A ) denote the interior of the set A . Assuming g ( z ) = az + bcz + d = α ◦ β with ad − bc = 1, let F g , F α , F β denote the fixed points of g, α, β in P Ber respectively.the p − adic M¨obius map F g ⊂ P ∪ H Ber F α ∩ F β loxodromic two points in P ∅ parabolic F g ∩ H Ber = ∅ Int( F α ∩ F β ) = ∅ tame elliptic a geodesic line in P Ber unique point in H Ber wild elliptic Int( F g ) = ∅ Int( F α ∩ F β ) = ∅ Theorem 1.2.
If the subgroup G ⊂ PSL(2 , C p ) contains elliptic elementsonly, then all the elements of G share at least one fixed point in H Ber .Furthermore, G has potentially good reduction, and G is equicontinuous on P ( C p ) . A point a ∈ P ( C p ) is called the limit point of a subgroup G of PSL(2 , C p )if there exists a point b ∈ P ( C p ) and an infinite sequence { g n | n ≥ } ⊂ G ,where g n = g m if n = m with lim g n ( b ) = a . The set consisting of all limitpoints is called the limit set . G is said to act discontinuously at x ∈ P ( C p )if there is a neighborhood U of x such that g ( U ) ∩ U = ∅ for all but finitelymany g ∈ G . The set of points at which G acts discontinuously is called the discontinuous set . Theorem 1.3.
If the limit set of G is empty, then G has potentially goodreduction. Two points α, β are called antipodal points if there exists an element u ∈ PSL(2 , O p ) such that u (0) = α, u ( ∞ ) = β . Theorem 1.4.
For any p -adic M¨obius map g , there exists an element u ∈ PSL(2 , O p ) such that g = uf , where either f is a loxodromic element withantipodal fixed points, or f = I . In the last part, we extend the inequalities obtained by Gehring andMartin [24, 25], Beardon and Short [12] to the non-archimedean settings.These inequalities of p -adic M¨obius maps are between the matrix, chordal,three-point and unitary norms. This part of work can be applied to studythe convergence of the sequence of p − adic M¨obius maps which can be viewedas a special cases of the work in [20] and the discrete criteria of the subgroupsof PSL(2 , C p ).For any two p -adic M¨obius maps g, h , we define the uniformly conver-gent metric on the PSL(2 , P ( C p )) as follows ρ ( g, h ) = sup z ∈ P ( C p ) ( g ( z ) , h ( z )) . Let M ( g ) = k g − g − k / k g k . Theorem 1.5.
Let g be a p -adic M¨obius map. (1) If p = 3 , then ρ ( g, I ) = M ( g ) . (2) If p = 2 , then − M ( g ) ≤ ρ ( g, I ) ≤ M ( g ) . Let ε ( g ) = max { ρ v ( g ( z ) , z ) , ρ v ( g ( z ) , z ) , ρ v ( g ( z ) , z ) } , where z , z , z are three distinct roots of the equation z = 1. Theorem 1.6.
For any p -adic M¨obius map g , we have − ε ( g ) ≤ M ( g ) ≤ ε ( g ) . Let ε ( g ) = { ρ v ( g (0) , , ρ v ( g (1) , , ρ v ( g ( ∞ ) , ∞ ) } . Theorem 1.7.
For any p -adic M¨obius map, − ε ( g ) ≤ M ( g ) ≤ ε ( g ) . If g is a parabolic element, we can improve the inequality. Let ε ( g ) =max { ρ v ( g (0) , , ρ v ( g ( ∞ ) , ∞ ) } . Corollary 1.8. If g is a parabolic element, then − ε ( g ) ≤ M ( g ) ≤ ε ( g ) . Let U = PSL(2 , O p ). We define d ( g, U ) = inf { ρ ( g, u ) | u ∈ U } . d ( g, U )measures how far an element from the group U . This result is quite differentfrom that in the archimedean setting. Theorem 1.9.
For any p -adic M¨obius map, either d ( g, U ) = 0 , if g ∈ U , or d ( g, U ) = 1 , if g / ∈ U . As an application of this result, we derive a discrete criteria of subgroupsof PSL(2 , C p ). Theorem 1.10. If G is a subgroup of PSL(2 , C p ) with G ∩ U = { I } , then G is a discrete subgroup. Corollary 1.11.
If a subgroup G ⊂ PSL(2 , C p ) contains unit element orloxodromic element only, then G is a discrete subgroup. The other application of these inequalities is to get the convergence the-orem of p -adic M¨obius maps. Theorem 1.12.
Let { g n } be a sequence of p -adic M¨obius maps, z j , ( j =1 , , be three distinct points and g n ( z j ) → w j , where w j are also threedistinct points. Then the sequence { g n } converges to a p -adic M¨obius map g uniformly, where g ( z j ) = w j , ( j = 1 , , . Motivation.
Firstly the study of the Kleinian group in the archimedeancase has been well developed for a rather long time. It is natural for oneto consider a parallel theory in the non-archimedean settings. Here we wishto give a fairly clear picture of the p − adic M¨obius maps from the point ofview of the Kleinian group. We study three aspects of the p − adic M¨obiusmaps. One is the group PSL(2; O p ), another is the geometrical character-ization of the p − adic M¨obius maps and its application, and the other isdifferent norms of the p − adic M¨obius maps, since PSL(2; O p ) is similar tothe unitary group in the Kleinian group, the geometrical characterization ofthe PSL(2; O p ) maps is useful in the study of the structure and dynamicsof the subgroups of PSL(2; C p ), and the study of norms of M¨obius mapsis a very important topic, and many mathematicians such as Gehring andMartin [24, 25], Beardon and Short [12] do a lot of works in this topic.The other reason is that the three aspects can be viewed as tools tostudy other related topics. The properties of the norms of p − adic M¨obiusmaps have a lot of applications in three topics. One is the discrete crite-ria of subgroups of PSL(2; C p ) and another is the pointwsie convergence of p − adic M¨obius maps(see [26, 32, 34]) and the other is the p − adic contin-ued fraction. The geometrical characterization shows that the subgroup Gcontaining elliptic element only shares one unique point, which means thata non-elementary group should have an loxodromic element which is veryuseful to study the dynamics of the discrete subgroup of PSL(2; C p ). Therapid development of the Berkovich space and the arithmetical dynamicalsystem (see [3], [4], [11], [13], [14], [15], [16], [19], [21], [31])also promotestudying the p − adic Mobius maps. Especially, the study of p − adic Mobiusmaps can be applied in studying the quantum mechanics and quantum cos-mology [22, 23]). In this paper we give the affirrmative answers to all thesethree questions in the non-archimedean settings.1.3. Outline of the paper.
Outline of the paper. In section 1, we presentour main results of the paper. In section 2, the basic theories of the p − adicanalysis and the Berkovich space are briefly reviewed. In section 3, we obtaina few preliminary results of p − adic M¨obius maps. Section 4 contains proofsof the equations of p − adic M¨obius maps in PSL(2; O p ). In section 5, wegive the results of the geometrical characterization of the p − adic M¨obiusmaps. In section 6, the inequalities of p − adic M¨obius maps between matrix,chordal, three points, hyperbolic, and unitary norms are derived. In section7, we prove the decomposition theorem of the p − adic M¨obius maps anddiscuss its application.2. Some Preliminary Results
The non-archimedean space P ( C p ) . Let p ≥ Q p be the field of p -adic numbers and C p be the completion of thealgebraic closure of Q p . Denote | C ∗ p | the valuation group of C p . Then everyelement r ∈ | K ∗ | can be expressed as r = p s with s ∈ Q . We have the strong triangle inequality | x − y | ≤ max {| x | , | y |} for x, y ∈ C p . If x, y and z are points of C p with | x − y | < | x − z | , then | x − z | = | y − z | .For any a ∈ C p , and r >
0, we define D ( a, r ) − = { z ∈ C p | | z − a | < r } to be the open disk of radius r and centered at a . Similarly, D ( a, r ) = { z ∈ C p | | z − a | ≤ r } denote the closed disk of radius r and centered at a . Both D ( a, r ) − and D ( a, r ) are closed and open topologically, and every point in disk D ( a, r ) − is the center. This denotes that if x ∈ D ( a, r ) − , then D ( a, r ) − = D ( x, r ) − (resp. D ( a, r ) = D ( x, r )). By the strong triangle inequality, if two disks D and D in C p have non-empty intersection, then either D ⊂ D , or D ⊂ D .For any z, w ∈ P ( C p ), we define the chordal distance ρ v ( z, w ) = | z − w | max { , | z |} max { , | w |} for z, w ∈ C p , ρ v ( z, w ) = 1max { , | w |} for w ∈ C p and z = ∞ , and ρ v ( z, w ) = 0for z = w = ∞ .It follows the definition of the chordal distance that if | z | ≤ , | w | ≤ ρ v ( z, w ) = | z − w | , and if | z | > , | w | ≤
1, then by the strong triangleinequality, we have | z − w | = | z | , and hence ρ v ( z, w ) = | z − w || z | = 1, and if | z | > , | w | >
1, then ρ v ( z, w ) = | z − w || z || w | = | z − w | . Lemma 2.1 ( [2]) . Let d > be an integer which is not divisible by p ,and d = d p t be a natural number. Let ζ be the primitive d -th root of unity.Then | ζ − | = 1 . Lemma 2.2 ( [2]) . Let ζ be the primitive p d -th root of unity. Then | ζ − | = p − pd − p − . The Berckovich space.
We shall use a few properties of about thestructure of the Berkovich affine line and its topology. Here we give a briefintroduction to the Berkovich space. More details can be found in [7].The underlying point set for the
Berkovich affine line A Ber is the collec-tion of all multiplicative seminorms [ · ] x on the polynomial ring C p [ z ] whichextend the absolute value on C p . Recall that a multiplicative seminorms on the ring C p [ z ] is a function [ · ] x : C p [ z ] → [0 , + ∞ ). • [0] x = 0 , [1] x = 1; • [ f g ] x = [ f ] x [ g ] x for all f, g ∈ C p [ z ]; • [ f + g ] x ≤ max { [ f ] x , [ g ] x } for all f, g ∈ C p [ z ].It is a norm provided that [ f ] x = 0 if and only if f = 0. The topologyon A Ber is the weakest one for which the mapping x [ f ] x is continuousfor all f ∈ C p [ z ].Recall the Berkovich’s classification Theorem: Every point x ∈ A Ber can be viewed as a nested sequence of disks D ( a , r ) ⊃ D ( a , r ) ⊃ · · · .Moreover, points in A Ber can be divided into four types: • A point in A Ber corresponding to a nested sequence { D ( a i , r i ) } of diskswith lim r i = 0 is said to be of type I . • A point in A Ber corresponding to a nested sequence { D ( a i , r i ) } of diskswith nonempty intersection, for which r = lim r i > | C ∗ | of C , is said to be of type II . • A point in A Ber corresponding to a nested sequence { D ( a i , r i ) } of diskswith nonempty intersection, for which r = lim r i > | C ∗ p | of C p , is said to be of type III . • A point in A Ber corresponding to a nested sequence { D ( a i , r i ) } of diskswith empty intersection is said to be of type IV .We call points of type I, II, III the nonsingular points , and pointsof type IV the singular points . Every nonsingular point in A Ber has arepresentation which is the intersection of the corresponding nested sequenceof disks. So a nonsingular point in A Ber can be identifying with a point a (type I) or a disk D ( a, r ) (type II, III).We define a partial order on A Ber as follows. For x, y ∈ A Ber , define x (cid:22) y if and only if [ f ] x ≤ [ f ] y for all f ∈ C p [ z ]. If x, y are two points in A Ber identifying with disks D ( a, r ) and D ( a ′ , r ′ ) respectively, then x (cid:22) y ifand only if D ( a, r ) ⊂ D ( a ′ , r ′ ).For a point x ∈ A Ber , we denote the set of elements larger than x by[ x, ∞ [ = { w ∈ A Ber | x (cid:22) w } . Observe that [ x, ∞ [ is isomorphic, as an ordered set, to [0 , + ∞ [ ⊂ R .Given two points x, y in A Ber , we have that[ x, ∞ [ ∩ [ y, ∞ [ = [ x ∨ y, ∞ [ , where x ∨ y is the smallest element larger than x and y . If x is different from y , then the element x ∨ y is a point of type II. We also denote[ x, y ] = { w ∈ A Ber | x (cid:22) w (cid:22) x ∨ y } ∪ { w ∈ A Ber | y (cid:22) w (cid:22) x ∨ y } . The sets ] x, y ] , [ x, y [ and ] x, y [ are defined in the obvious way.For a set E ⊂ K , denote diam( E ) = sup z,w ∈ E | z − w | the diameter of E inthe non-Archimedean metric. For x ∈ A Ber , which is corresponding to thenested sequence { D ( a i , r i ) } of disks, the diameter of x is given bydiam( x ) = lim i →∞ diam( D ( a i , r i )) . For a nonsingular element x ∈ A Ber identifying with the disk D ( a, r ), thediameter of x coincides with the diameter (radius r ) of D ( a, r ).In order to endow the Berkovich affine line with a topology, we define anopen disk of A Ber by D ( a, r ) − = { x ∈ A Ber | diam( a ∨ x ) < r } , for a ∈ K and r >
0. Similaryly, a closed disk of A Ber is defined by D ( a, r ) = { x ∈ A Ber | diam( a ∨ x ) ≤ r } . Let P ( C p ) be the projective line over C p , which can be viewed as P ( C p ) = C p ∪ {∞} . We can also introduce the Berkovich projective line P Ber over P ( K ) similarly. In [7], Baker and Rumely pointed out that P Ber can be de-fined as A Ber ∪ {∞} , where ∞ ∈ P ( K ) is regarded as a point of type I. P Ber can be also identifying with the disjoint union of a closed set X whichis homeomorphic to D (0 ,
1) and an open set Y which is homeomorphic to D (0 , − . This provides a useful way to visualize P Ber . Lemma 2.3 ( [7]) . The space P Ber is uniquely path-connected. More pre-cisely, given any two distinct points x, y ∈ P Ber , there is a unique arc [ x, y ] in P Ber from x to y , and if x ∈ X and y ∈ Y , then the arc contains ζ Gauss ∈ A Ber , where ζ Gauss is identifying with the disk D (0 , . We say that a metric space (
X, d ) is an R -tree , if for any two points x, y ∈ X , there is a unique arc from x to y and this arc is the geodesicsegment. Lemma 2.4 ( [7]) . Let x, y ∈ D (0 , . Then the metric d ( x, y ) = 2diam( x ∨ y ) − diam( x ) − diam( y ) makes D (0 , into an R -tree. Therefore, we can define a metric on P Ber as d p ( x, y ) = d ( x, y ) if x, y ∈ X which can be identified with D (0 , d p ( x, y ) = d ( x, y ) if x, y ∈ Y whichcan be identified with D (0 , − , and d p ( x, y ) = d ( x, ζ Gauss ) + d ( ζ Gauss , y ) if x ∈ X and y ∈ Y .The Berkovich hyperbolic space H Ber is defined by H Ber = P Ber \ P ( C ) . Since ∞ ∈ P Ber is of type I, H Ber can be also viewed as A Ber \ C . Thus H Ber has a tree structure induced by A Ber . Over H Ber we can define the hyperbolic distance , ρ ( x, y ) = 2 log diam( x ∨ y ) − log diam( x ) − log diam( y ) , x, y ∈ H Ber . Lemma 2.5 ( [7]) . H Ber is a complete metric space under ρ ( x, y ) . Lemma 2.6 ( [7]) . Suppose that w, y, x ∈ H Ber . Then ρ ( x, y ) = ρ ( x, w ) + ρ ( w, y ) if and only if w belongs to [ x, y ] . The action of a rational map φ over P Ber . Let φ ∈ C p ( T ) be anonconstant rational function of degree d ≥
1. Since type I points are densein P Ber , for any x ∈ H Ber , there exists a sequence x n tending to x withrespect to the Berkovich topology. We can define φ ( z ) = lim n −→∞ φ ( x n ).If d = 1, φ has an algebraic inverse and thus induces an automorphismof P Ber . Define Aut( P Ber ) to be the group of automorphisms of P Ber . Thefollowing lemmas can be found in [7].
Lemma 2.7 ( [7]) . If φ ( z ) ∈ C p ( z ) is nonconstant, then φ : P Ber −→ P Ber takes points of each type ( I, II, III, IV ) to points of the same type. Thus φ ( z ) has a given type if and only if z does. Lemma 2.8 ( [7]) . Let f ( z ) ∈ C p ( z ) be a nonconstant rational function, andsuppose that x ∈ A Ber is a point of type II, corresponding to a disc D ( a, r ) in C p under Berkovich’s classification. Then f ( x ) corresponds to the disc D ( b, R ) if and only if there exist a , a , . . . , a m , b , b , . . . , b n ∈ C p for which D ( b, R ) \ ∪ ni =1 D ( b i , R ) − is the image under f ( z ) of D ( a, r ) \ ∪ mi =1 D ( a i , r ) − . Reduction on rational function over C p . Let O p = { z ∈ C p || z | ≤ } , O ∗ = { z ∈ C p || z | = 1 } , M = { z || z | < } and k = O p / M . We also call k the residue field of C p . If x ∈ O p , we denote the reduction of x modulo M by¯ x . For any z ∈ C p , there exists a homogeneous coordinate [ x, y ] for z , where x, y ∈ O with at least one in O ∗ . Reduction induces a well-defined map P ( C p ) → P ( k ) by [ x, y ] = [¯ x, ¯ y ]. Any rational function f ( z ) ∈ C p ( z ) canbe written in homogeneous coordinates as f ([ x, y ]) = [ g ( x, y ) , h ( x, y )] where g, h ∈ O p [ x, y ] are relatively prime homogeneous polynomials of degree d =deg( f ). We can ensure that at least one coefficient of either g or h hasvaluation zero (i.e., absolute value 1). The reduction map induces a map O p [ x, y ] → k [ x, y ]. Definition 2.9.
Let f ( z ) ∈ C p ( z ) be a map with homogenous presentation f ([ x, y ]) = [ g ( x, y ) , h ( x, y )] , where g, h ∈ O p [ x, y ] are relatively prime homo-geneous polynomials of degree d = deg( f ) , and at least one coefficient of g or h has absolute value . We say that f has good reduction if ¯ g and ¯ h haveno common zeros in k × k besides ( x, y ) = (0 , . If there is some linear fractional transformation h ∈ PSL(2 , C p ) suchthat h − ◦ f ◦ h has good reduction, we say that f has potentially goodreduction .0 Lemma 2.10 ( [33]) . Let f ∈ PSL(2 , C p ) be a rational function of degreeone. Then f has good reduction if and only if f ∈ PSL(2 , O ) . Lemma 2.11 ( [7]) . Let f ∈ C p ( z ) be a non-constant rational function.Then f has good reduction if and only if f − ( ζ Gauss ) = ζ Gauss . The p-adic M¨obius maps
We classify non-unit elements in PSL(2 , C p ) = SL(2 , C p ) / {± I } . Since theproduct of all eigenvalues of g ∈ PSL(2 , C p ) is one, either the absolute valueof each eigenvalue of g is one or there exists at least one eigenvalue whoseabsolute value is larger than 1. Thus each non-unit element g ∈ PSL(2 , C p )falls into the following four classes:(a) g is said to be parabolic if the absolute value of any eigenvalue of g is 1, and g can not be conjugated to a diagonal matrix.(b) g is said to be loxodromic if there exists at least eigenvalue of g whose absolute value is larger than 1.(c) g = I is said to be elliptic if the absolute value of any eigenvalue of g is 1, and g can be conjugated to a diagonal matrix.In this paper, we classifies the elliptic elements more precisely.(d) g is said to be tame elliptic if the two eigenvalues λ , λ of g satisfy | λ − | = | λ − | = 1.(e) g is said to be wild elliptic if one of the eigenvalues of g lies in thedisc D (1 , − .For g = ( a ij ) in the matrix ring M( m, C p ), the norm of g is defined by k g k = max ≤ i ≤ m, ≤ j ≤ m {| a ij |} . Obviously, k g k = 0 implies that each a ij = 0.It is easy to verify that k αg k = | α | k g k , k g + h k≤ max {k g k , k h k} and k gh k≤k g kk h k .For any element g ∈ PSL(2 , C p ), there exist two lifts g , g in SL(2 , C p )with k g k = k g k . We define k g k = k g k = k g k . If g, h ∈ PSL(2 , C p )correspond the lifts g , g ∈ SL(2 , C p ) and h , h ∈ SL(2 , C p ) respectively,then we define k g − h k = inf ≤ i ≤ , ≤ j ≤ k g i − h j k .If d = 1, φ has an algebraic inverse and thus induces an automorphismof P Ber . Define Aut( P Ber ) to be the group of automorphisms of P Ber . Thefollowing lemmas can be found in [7].
Lemma 3.1 ( [7]) . The path distance metric ρ ( x, y ) on H Ber is independentof the choice of homogenous coordinates on P Ber , in the sense that if h ( z ) ∈ C p ( z ) is a p -adic M¨obius map, then ρ ( h ( x ) , h ( y )) = ρ ( x, y ) for all x, y ∈ H Ber . Lemma 3.2 ( [30]) . Let w satisfy | w | = λ | w − a | , where λ ∈ | C ∗ p | , and w, a ∈ C p . (1) If λ > , then w ∈ D ( a, | a | λ ) \ D ( a, | a | λ ) − . (2) If λ = 1 , then w ∈ P ( C p ) \ D (0 , | a | ) − ∪ D ( a, | a | ) − . (3) If < λ < , then w ∈ D (0 , λ | a | ) \ D (0 , λ | a | ) − . Proposition 3.3. A p -adic M¨obius map g is (1) parabolic if it is conjugate to z → z + 1 ; (2) elliptic if it is conjugate to z → kz for some k with | k | = 1 , k = 1 ; (3) loxodromic if it is conjugate to z → kz for some k with | k | > . Proposition 3.4.
Let f and g be two p -adic M¨obius maps neither of which isthe identity. Then f and g are conjugate if and only if trace ( f ) = trace ( g ) . Proposition 3.5.
Let g = I be a p -adic M¨obius map. Then (1) g is parabolic if and only if ( a + d ) − ; (2) g is elliptic if and only if < | ( a + d ) − | ≤ ; (3) g is loxodromic if and only if | ( a + d ) − | > .Proof. Since the trace of the matrix is invariant under the conjugation, with-out lose of generality, we can assume that g ( z ) = ad z , if g is elliptic or loxo-dromic which yields that tr ( g ) = a + d = λ + 1 /λ , where λ is the eigenvalueof g . Thus ( a + d ) − λ − /λ ) . By the non-archimedean property ofthe metric, we have | λ − /λ | >
1, if g is loxodromic, and 0 < | λ − /λ | < g is elliptic. If g is parabolic, we can assume that g ( z ) = z + 1 which yieldsthat a + d = 2.Conversely, if ( a + d ) − λ − /λ = 0 which denotes λ = 1.Since g ∈ PSL(2 , C p ), we have λ = 1 which yields that g is parabolic. If | ( a + d ) − | >
1, then | λ − /λ | > | λ | > | /λ | > g is loxodromic. If 0 < | ( a + d ) − | <
1, then 0 < | λ − /λ | < | λ | <
1. Thus g is elliptic. (cid:3) The properties of
PSL(2 , O p ) Lemma 4.1 ( [33]) . If f ∈ PSL(2 , O p ) , we have ρ v ( f ( z ) , f ( w )) = ρ v ( z, w ) . Lemma 4.2 ( [30]) . Let g be any p -adic M¨obius map. Then ρ ( g ( ζ Gauss ) , ζ Gauss ) = 2 log p k g k . Lemma 4.3.
Let g be any p -adic M¨obius map. Then the best Lipschitz con-stant (relative the chordal metric) for g is given by L ( g ) = p ρ ( ζ Gauss ,g ( ζ Gauss )) ,namely ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ) . Furthermore, there exist at least twopoints z, w ∈ P ( C p ) such that ρ v ( g ( z ) , g ( w )) = L ( g ) ρ v ( z, w ) .Proof. We can assume that g ( z ) = az + bcz + d . The element g has at least onefixed point a g . If | a g | ≤
1, let h ( z ) = z − a g and ι ( z ) = 1 /z , and then ιhgh − ι − fixes ∞ . Since h and ι fix the point ζ Gauss , we have ρ ( ιhgh − ι − ( ζ Gauss ) , ζ Gauss ) = ρ ( ιhg ( ζ Gauss ) , ζ Gauss ) = ρ ( g ( ζ Gauss ) , ζ Gauss )which yields that L ( g ) = L ( ιhgh − ι − ). If | a g | >
1, let h ( z ) = z − /a g and ι ( z ) = 1 /z . Thus ιhιgιh − ι − fixes ∞ . Similarly, since h and ι fix the point ζ Gauss , we have ρ ( ιhιgιh − ι − ( ζ Gauss ) , ζ Gauss ) = ρ ( ιhιg ( ζ Gauss ) , ζ Gauss ) = ρ ( g ( ζ Gauss ) , ζ Gauss )2which yields that L ( g ) = L ( ιhιgιh − ι − ). Therefore, without loss of gener-ality, we can assume that g ( z ) = az + bd with ad = 1. If | a | >
1, we considerthe inverse g − ( z ) = da z − ba , since L ( g ) = L ( g − ). Thus we can assume that | a | ≤ z, w ∈ P ( C p ), we have ρ v ( g ( z ) , g ( w )) = | g ( z ) − g ( w ) | max { , | g ( z ) |} max { , | g ( w ) |} = | z − w | max {| az + b | , | d |} max {| aw + b | , | d |} = | z − w | max { , | z |} max { , | w |} max { , | z |} max { , | w |} max {| az + b | , | d |} max {| aw + b | , | d |} = ρ v ( z, w ) max { , | z |} max { , | w |} max {| az + b | , | d |} max {| aw + b | , | d |} = ρ v ( z, w ) | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } . When | a | = 1, let D = D ( − ba ,
1) and D = D (0 , D = D , then | ba | ≤ | b | ≤ | a | = 1, namely L ( g ) = 1. If z, w ∈ D = D ,then | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = 1 , namely ρ v ( g ( z ) , g ( w )) = L ( g ) ρ v ( z, w ). If z ∈ D = D and w / ∈ D = D , | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = | w || w + b/a | . Since | ba | ≤
1, we have | w | = | w + ba | which yields that | w || w + b/a | = 1. Thus ρ v ( g ( z ) , g ( w )) = L ( g ) ρ v ( z, w ). If z, w / ∈ D = D , we have | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = | w || z || z + b/a || w + b/a | . Since | ba | ≤
1, we have | w | = | w + ba | and | z | = | z + ba | which yields that | z || w || z + b/a || w + b/a | = 1 . Thus ρ v ( g ( z ) , g ( w )) = L ( g ) ρ v ( z, w ). If D ∩ D = ∅ , then | ba | > | b | > | a | = 1, namely L ( g ) = | b | . If z, w ∈ D , then | z | > , | w | > , | z + b/a | ≤ , | w + b/a | ≤ | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = | ba | = | b | . ρ v ( g ( z ) , g ( w )) = L ( g ) ρ v ( z, w ). If z, w ∈ D , then | z | ≤ , | w | ≤ , | z + b/a | < , | w + b/a | < | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = 1 | z + b/a || w + b/a | = | ab | = | b | − ≤ L ( g ) . Thus ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ). If z ∈ D and w ∈ D , then | z | > , | w | ≤ , | z + b/a | ≤ , | w + b/a | > | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = | z || w + b/a | = 1 ≤ L ( g ) . If z / ∈ D ∪ D , then | z | > , | z + b/a | > { , | z |} max {| a z + ab | , } = | z || z + b/a | = 1 . Thus if z, w / ∈ D ∪ D , then ρ v ( g ( z ) , g ( w )) = ρ v ( z, w ) ≤ L ( g ) ρ v ( z, w ). If z / ∈ D ∪ D , w ∈ D , then ρ v ( g ( z ) , g ( w )) = ρ v ( z, w ) | w | = | b | ρ v ( z, w ) ≤ L ( g ) ρ v ( z, w ). If z / ∈ D ∪ D , w ∈ D , then ρ v ( g ( z ) , g ( w )) = ρ v ( z, w ) / | w + b/a | = | a/b | ρ v ( z, w ) ≤ L ( g ) ρ v ( z, w ). Therefore, ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ).When | a | <
1, let D = D ( − ba , | a | ) and D = D (0 , D ⊂ D , then | ba | ≤ | a | which yields that L ( g ) = | d | = | a | − . If z, w ∈ D , then | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = | a | max { , | z |} max { , | w |} ≤ | a | = L ( g ) , namely ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ). If z, w / ∈ D , then | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = | a | | z || w || a | | z || w | = | a | − which yields that ρ v ( g ( z ) , g ( w )) = L ( g ) ρ v ( z, w ). If z ∈ D , w / ∈ D , then | a | max { , | z |} max { , | w |} max {| a z + ab | , } max {| a w + ab | , } = | a | | w | max { , | z |}| a | | w | = max { , | z |} ≤ | a | = L ( g ) . Thus ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ) . If D ∩ D = ∅ , then | ba | > | a | whichyields that | b | > | a | . Thus L ( g ) = | b | . If z ∈ D , thenmax { , | z |} max { , | a z + ab |} ≤ . If z ∈ D , then max { , | z |} max { , | a z + ab |} = | b || a | . If z / ∈ D ∪ D , then max { , | z |} max { , | a z + ab |} = | z || a z + ab | . z, w ∈ D , then | a | max { , | z |} max { , | w |} max { , | a z + ab |} max { , | a w + ab |} = | b | which yields that ρ v ( g ( z ) , g ( w )) = L ( g ) ρ v ( z, w ). If z, w ∈ D , then | a | max { , | z |} max { , | w |} max { , | a z + ab |} max { , | a w + ab |} ≤ | a | ≤ ≤ L ( g ) . Thus ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ). If z ∈ D , w ∈ D , then | a | max { , | z |} max { , | w |} max { , | a z + ab |} max { , | a w + ab |} ≤ | a || b | ≤ L ( g ) . Thus ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ) . If z, w / ∈ D ∪ D , then | a | max { , | z |} max { , | w |} max { , | a z + ab |} max { , | a w + ab |} = | a | | z || w || a z + ab || a w + ab | . If | z | = | z + ba | , then | z || z + b/a | = 1. If | z | < | z + b/a | , then | z || z + b/a | <
1. If | z | > | z + b/a | , then | z | = | b/a | . Therefore, we have | a | | z || w || a z + ab || a w + ab | ≤ | b | . Thus ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w ). (cid:3) Remark 4.4.
In [33], the Lipschitz constant of the rational map with re-spect to the chordal metric is derived by the resultant(see definition below).Let O ∗ p = { α ∈ C p : | α | = 1 } . Let g ( z ) = a z n + a z n − + ... + a n − z + a n and h ( z ) = b z m + b z n − + ... + b m − z + b m . The rational function φ ( z ) = g ( z ) h ( z ) ∈ C p ( z ) is called a normalized form if all the coefficients of g ( z ) and h ( z ) are in O p and at least one coefficient of g ( z ) and h ( z ) are in O ∗ p . Let α i , ≤ i ≤ n be the roots of g ( z ), and β i , ≤ i ≤ m be the roots of h ( z ). LetRes( g ( z ) , h ( z )) = a n b m Q ni =1 Q mj =1 ( α i − β j ) be the resultant of two polyno-mials, and Res( φ ) = Res( g ( z ) , h ( z )) be the resultant of the rational function φ . The absolute value of the resultant Res( φ ) depends only on the map φ . Lemma 4.5 ( [33]) . Let φ : P ( C p ) −→ P ( C p ) be a rational map. Then ρ v ( φ ( z ) , φ ( w )) ≤ | Res( φ ) | − ρ v ( z, w ) for all v, w ∈ P ( C p ) . Let g ( z ) = az + bcz + d ∈ PSL(2 , C p ). By Lemma 4.5, we haveRes( g ( z )) = Res(( a/t ) z + b/t, ( c/t ) z + d/t ) = 1 / k g k = L ( g ) , where t = max {| a | , | b | , | c | , | d |} = k g k .In [33], it is shown that there are two points x, y ∈ P ( C p ) such thatsup x = y ρ v ( φ ( x ) , φ ( y )) /ρ v ( x, y ) = | Res( φ ) | − . Since P ( C p ) is not compact, thesupreme can not be omitted in general cases. However, we get the Lipschitz5constant by other method and show that we can get the supreme when it isthe p -adic M¨obius map.Proof of Theorem 1.1 Proof.
Since ρ ( g ( ζ Gauss ) , ζ Gauss ) = 2 log p k g k , k g k = 1 is equivalent to ρ ( g ( ζ Gauss ) , ζ Gauss ) = 0 which yields that L ( g ) = 1. The converse is alsotrue. This means that (1), (2), (3) are equivalent.(2) ⇒ (4) If L ( g ) = 1, L ( g − ) = 1. Hence by Lemma 4.3, we have ρ v ( z, w ) = ρ v ( g − ( g ( z )) , g − ( g ( w ))) ≤ L ( g − ) ρ v ( g ( z ) , g ( w )) ≤ L ( g ) ρ v ( z, w )which yields that ρ v ( g ( z ) , g ( w )) = ρ v ( z, w ).(4) ⇒ (2) If g is a chordal isometry, then ρ v ( g ( z ) , g ( w )) = ρ v ( z, w ). ByLemma 4.3, we have L ( g ) = 1.(1) ⇒ (5) Since k g k = 1, let g = az + bcz + d , we have max {| a | , | b | , | c | , | d |} = 1which yields that g ∈ PSL(2 , O p ).(5) ⇒ (1) If g ∈ PSL(2 , O p ), let g = az + bcz + d , then max {| a | , | b | , | c | , | d |} ≤ ad − bc = 1. If max {| a | , | b | , | c | , | d |} <
1, then | ad − bc | < max {| ad | , | bc |} <
1. This is a contradiction. Hence max {| a | , | b | , | c | , | d |} = 1 which yields that k g k = 1.(1) ⇒ (6) Since k g k = 1, k g − k = 1. Since k gh k≤k g kk h k≤k h k ,and k h k = k g − gh k≤k g − kk gh k≤k gh k , we have k gh k = k h k .Similarly, k hg k = k h k . We can rewrite h as hg − , and then we have k ghg − k = k hg − k = k h k , since g ∈ PSL(2 , O p ).(6) ⇒ (1) Let h ∈ PSL(2 , O p ). Then k h k = k gh k = k g k which yieldsthat k g k = 1. (cid:3) The metric properties of PSL(2 , O p ) can be used to study the p − adiccontinued fractions. An infinite p − adic continued fraction is a formal ex-pression a b + a b + a b + · · · , where a i , b i ∈ C p and a i = 0. We denote this continued fraction by K ( a n | b n ).Let t n = a n z + b n and T n = t ◦ t ◦ t ◦ · · · for n = 1 , , , . . . . The continuedfraction is said to be convergent classically if the sequence { T n } converges,else it is said to diverge classically. In the following part, we study thesimplest case K (1 | b i ) with | b i | ≤
1. Since T n +1 ( ∞ ) = T n (0), by Theorem1.1, if the sequence { T n (0) } converges, then 0 = lim n →∞ ρ v ( T n (0) , T n ( ∞ )) = ρ v (0 , ∞ ) = 1. This is a contradiction. This implies that the continuedfraction K (1 | b i ) with | b i | ≤ Example 4.6.
11 + 11 + 11 + · · · does not converge classically.
This example shows that the convergence and divergence of p − adic con-tinued fractions are different from those in complex settings.5. Reduction and p − adic M¨obius maps We call an elliptic element f ∈ PSL(2 , C p ) of order 2 an involution . If g ∈ PSL(2 , C p ) is a loxodromic element or an elliptic element, g have twofixed points a g and r g in P ( C p ). We call the geodesic line A g which connects a g and r g the axis of g . Let A be a geodesic line in P Ber . A geodesic line B is orthogonal to A if there exists a p -adic M¨obius transformation f whichis an involution such that endpoints of B are two fixed points of f , and f interchanges endpoints of A .We introduce a new conception, a tailed geodesic line , in order toanalyze the geometrical characterization of 2 − adic M¨obius maps. Let A be a geodesic line and x be the point satisfying inf y ∈ A ρ ( x, y ) = log p y ∈ A , A x = A ∪ [ x, y ]. Since the Berkovich space is a R − tree, it is obviously that A x is independent of the chosen point y , A isthe geodesic line associated with A x , and x is called a tail . If A is the axisof an involution f , then there exists a unique point x fixed by f such that A x is a tailed geodesic line. Hence we call A x the tailed axis of f . Lemma 5.1.
Let g be a p − adic M¨obius map. Then there exist two involu-tion f, h ∈ PSL(2 , C p ) such that g = f ◦ h . Furhtermore (1) The axes A of h and B of f are orthogonal to the axis A g of g . (2) The endpoints of A are different from the endpoints of B . (3) The element g is parabolic if and only if A and B share the uniqueendpoint.When p ≥ , (4) the element g is elliptic if and only if A ∩ B = ∅ ; (5) the element g is loxodromic if and only if A ∩ B = ∅ .When p = 2 , (6) the element g is elliptic if and only if the two tailed geodesic lines A x ∩ B y = ∅ ; (7) the element g is loxodromic if and only if the two tailed axes A x ∩ B y = ∅ .Proof. If g is loxodromic or elliptic, without loss of generality, let g = λ z .The axis A g is the geodesic line connecting 0 and ∞ . Let h ( z ) = − b z and f ( z ) = g ◦ h − ( z ) = − λ b z . Thus h has two fixed points bk, − bk and f hastwo fixed points λbk, − λbk , where k = −
1. It is easy to see that f and7 h interchange 0 and ∞ , and f ◦ f = h ◦ h = z , namely f and h are twoinvolutions. Hence A, B are orthogonal to A g respectively. We prove (1),(2).In case of p ≥
3, if A ∩ B = ∅ , since the P Ber is a R − tree, there existsa point x ∈ A ∩ B corresponding the disc D , which contains endpoints bk, − bk, λbk, − λbk and | bk | = | λbk | which implies that | λ | = 1, i.e. g iselliptic element. If A ∩ B = ∅ , either bk, − bk / ∈ D ( λbk, | λbk | ) or λbk, − λbk / ∈ D ( bk, | bk | ) which implies that | λbk − bk | > | bk | or | λbk − bk | > | λbk | , i.e. | λ − | > | − λ | >
1. Hence g is loxodromic. We prove (4),(5).In case of p = 2, if the tailed axes A x ∩ B y = ∅ , we denote the tails of thetailed axes by D ( bk, | bk | ) and D ( λbk, | λbk | ), the two tailed axes A x and B y do not intersect, since D ( bk, | bk | ) − ∩ D ( λbk, | λbk | ) − = ∅ , and D ( λbk, | λbk | ) = D ( λbk, | λbk | ). Conversely, if two tailed axes do not intersect, then | λbk | 6 = | bk | , namely | λ | 6 = 1. This implies that g is a loxodromic element. We prove(6),(7)If g is parabolic, without loss of generality, we can assume that g ( z ) = z +1. Let f ( z ) = − z and h ( z ) = − z −
1. Then g ( z ) = f ◦ h ( z ) = − ( − z −
1) = z + 1. Since the axis A of f is the geodesic line connecting { , ∞} and theaxis B of h is the geodesic line connecting {− , ∞} , we have ∞ ∈ A ∩ B .Conversely, if A and B share only one endpoint, without loss of generality,we can assume that A is the geodesic line connecting { , ∞} , and B is thegeodesic line connecting { , } . Then f ( z ) = − z and h ( z ) = z z − whichimplies that g ( z ) = f ◦ h ( z ) = − z z − is a parabolic element, since g hasa unique fixed point in P ( C p ). We know that if A and B have differentendpoints, then g is either loxodromic or elliptic. (cid:3) By the proof of Lemma 5.1, we know that for any p -adic M¨obius map g ,there exist two half turns f and h such that g = f h . Furthermore, followingthe proof, since f and h are not unique, we can make the axis of f containthe Gauss point ζ Gauss .We denote the set of fixed points of an element g by F g = { x ∈ P Ber | g ( x ) = x } . Let a, b ∈ P ( C p ), and x ∈ H Ber . Proposition 5.2.
Let g ∈ PSL(2 , C p ) . (1) If g is a loxodromic element, then the set of fixed points of g containstwo points in P ( C p ) . (2) If g is a tame elliptic element, then the set of fixed points of g is ageodesic line in P Bek , and F g ∩ P ( C p ) contains two points. (3) Let g be a wild elliptic element. Then the interior of the set of thefixed points of g contains a geodesic line in P Ber . (4) If g is a parabolic element, then the fixed points of g is an open discwith its boundary with respect to the Berkovich topology.Proof. If g is loxodromic or elliptic, we can assume that g ( z ) = λ z with fixedpoints 0 , ∞ . If | λ | > | λ | <
1, then by Lemma 4.2, we know that g can8not fix any point in H Ber . This proves (1). If | λ | = 1, then g ∈ PSL(2 , O p )which implies that g fixes every point on the geodesic line connecting 0 , ∞ .Furthermore, in case of | λ − | = | λ + 1 | = 1, for any x ∈ H Ber correspondingto the disc D ( a, r ), we have g ( D ( a, r )) = D ( λ a, | λ | r ) = D ( λ a, | λ | r ), if gx = x . Then | λ − || a | ≤ r which implies that | a | ≤ r , i.e. 0 ∈ D ( a, r ).Hence x is on the geodesic line connecting 0 and ∞ . This proves (2). When | λ − | <
1, for any x ∈ H Ber corresponding to the disc D ( a, r ), we have g ( D ( a, r )) = D ( λ a, | λ | r ) = D ( λ a, | λ | r ), if gx = x . This implies that | a | ≤ r | λ − | . Hence g fixes any point in the hyperbolic disc B ( x, | λ − | ), i.e.the interior of the set of the fixed points of g contains a geodesic line in P Ber .This proves (3).If g is parabolic, we can assume that g = z + 1. For any x ∈ H Ber corresponding to the disc D ( a, r ), if g ( x ) = x , then D ( a, r ) = D ( a + 1 , r )which implies that r ≥ (cid:3) Lemma 5.3.
If a geodesic line A is orthogonal to the other geodesic line B ,then B is also orthogonal to A .Proof. Without loss of generality, we can assume that A is a geodesic linewith endpoints 0 and ∞ and B is a geodesic line with endpoints − f ( A ) = B, f ( B ) = A , if f ( z ) = z +1 z − . (cid:3) Lemma 5.4.
Let a geodesic line A be orthogonal to the other geodesic line B . (1) If p ≥ , then A intersects B at one unique point. (2) If p = 2 , then A ∩ B = ∅ .Proof. Without loss of generality, we may assume that A is the geodesicconnecting 0 and ∞ and B is a geodesic line with endpoints − α, α . Let thepoint x correspond to the disc D ( α, | α | ) which lies on the geodesic line B and contains the points − α, α .If p ≥
3, then | α | = | α | which implies that D ( α, | α | ) contains 0. Hence B intersects A . Conversely, if B intersects A , then there exists a point x corresponding to a disc D (0 , r ) containing α or − α which implies that | α | ≤ r . Hence D (0 , r ) contains both α and − α . If there exist two points x , x ∈ A ∩ B corresponding to two discs D (0 , r ) and D (0 , r ) respectively,then either D (0 , r ) ⊂ D (0 , r ) or D (0 , r ) ⊂ D (0 , r ). Without loss ofgenerality, we can assume that D (0 , r ) ⊂ D (0 , r ). Let l be the geodesicsegment connecting α and x , and l be the geodesic segment connecting − α and x . Hence l ∪ l ⊂ B , but l ∩ l contains a segment containing x and x . This is a contradiction. Hence A intersects B at a uniquely point.If p = 2, then for any x lying on the geodesic line connecting α, − α ,we have that the disc corresponding to x must contain α or − α . Withoutloss of generality, let x correspond to the disc D ( α, r ). If x lies on A , then D ( α, r ) contains 0 which implies that | α | ≤ r . Hence − α ∈ D (0 , r ). Since ζ corresponds to the disc D ( α, | α | ) containing both α and − α , the geodesic9line contains the segment which connecting ζ and x . This is a contradiction.Hence A ∩ B = ∅ . (cid:3) We say that g keeps a set A invariant if g ( A ) = g − ( A ) = A . Lemma 5.5.
Let A g be the axis of g . If g is a loxodromic element or anelliptic element, then g keeps the axis A g invariant. Furthermore, g fixesevery point of the axis A g if and only if g is an elliptic element.Proof. Without loss of generality, let g = λz . Hence A g is the geodesic lineconnecting 0 , ∞ . If | λ | >
1, then g maps each disk D (0 , r ) to D (0 , | λ | r )which is also on the geodesic line. If | λ | = 1, then g maps each disk D (0 , r )to D (0 , | λ | r ) which is the disk D (0 , r ), namely g fixes the point ζ ,r . (cid:3) Lemma 5.6. If p ≥ , and g = h ◦ f is a tame elliptic element, where h, f are two involutions, then two axes of h and f only intersect at a uniquepoint.Proof. Without loss of generality, we can assume that g ( z ) = λ ( z ). Thus f ( z ) = − b z and h ( z ) = − λ b z . By Lemma 2.1, we have | λb − b | = | λ − || b | = | b | . This implies that two axes of h and f only intersect at the point ζ , | b | which corresponds to the disc D (0 , | b | ). (cid:3) Lemma 5.7. If p ≥ , and g = h ◦ f is a wild elliptic element, where h, f are two involutions, then two axes of h and f only intersect on a segment,and this segment belongs to the fixed points of g .Proof. Without loss of generality, we can assume that g ( z ) = λ z , f ( z ) = − z and h ( z ) = − λz . Let A be the axis of f ( z ) and B be the axis of h ( z ). Hencethe endpoints of A are {− , } and the endpoints of B are {−√ λ, √ λ } .Since p ≥
3, then 1 = | − − | = | − √ λ − √ λ | which implies that ζ Gauss lies on both the axes A and B . Since | λ − | < | λ + 1 | <
1, wehave min {|√ λ − | , |√ λ + 1 |} < x ∈ A ∩ B ∩ H Ber which corresponds to the disc D (1 , |√ λ − | ) or the disc D (1 , |√ λ + 1 | ). Hence A ∩ B contains either the segment connecting ζ , |√ λ − | and ζ , or the segment connecting ζ , |√ λ +1 | and ζ , . This segment belongsto the fixed points of fixed points of f . (cid:3) Lemma 5.8. If p ≥ , and A and B are two geodesic lines with four distinctendpoints, then there exists a unique geodesic line which is orthogonal to A and B simultaneously.Proof. Without loss of generality, we can assume that A is the geodesic withendpoints 0 and ∞ , and B is the other geodesic line with endpoints a and b . If A does not intersect B , then we have | a − b | < max {| a | , | b |} . Bythe ultrametric property, we have | a | = | b | > | a − b | . Let C be a geodesic0line with endpoints ζ, − ζ . By Theorem 5.1, we have that the geodesic line C is orthogonal to the geodesic line A . Let g = z − az − b . Then the geodesicline B is mapped to the geodesic line A by g . If the geodesic g ( C ) is alsoorthogonal to A , then g ( − ζ ) + g ( ζ ) = 0 and g ( C ) ∩ A = ∅ . This impliesthat ζ − aζ − b + − ζ − a − ζ − b = 0, namely ζ = √ ab . Then the geodesic line C connecting −√ ab, √ ab is orthogonal to both A and B simultaneously. (cid:3) Lemma 5.9. If p = 2 , and A x and B y are two tailed geodesic lines withfour distinct endpoints, then there exists a unique geodesic line which isorthogonal to A and B simultaneously.Proof. Without loss of generality, we can assume that the endpoints of A are − ,
1, and the endpoints of B are t, s . We claim that we can find a p -adicM¨obius map f = az + bcz + d such that f ( −
1) = − f ( −
1) = − f ( t ) + f ( s ) = 0.Since f ( −
1) = − f ( −
1) = −
1, we have a = d, b = c , and at + bct + d + as + bcs + d = 0which yields that 2 abst + ( a + b )( s + t ) + 2 ba = 0. We can lift the solutionto the projective space, namely 2 ABst + ( A + B )( s + t ) + AB = 0, and A − B = C . Since any two curves in the projective space intersect, wehave solutions in the projective space. If the solution is ( A : B : 0), namely C = 0, then A = B or A = − B . This implies that st + s + t + 1 = 0 or st − ( s + t ) + 1 = 0, namely s = − t = − s = 1 or t = 1. Thiscontradicts that two tailed geodesic line have no common endpoints. Hence C = 0, namely there exists p -adic M¨obius map f such that f ( −
1) = − f ( −
1) = − f ( t ) + f ( s ) = 0.Hence we can assume that the tailed geodesic line A has the endpoints − , − λ, λ . Then the twotailed geodesic line are orthogonal to the line connecting 0 , ∞ simultane-ously. (cid:3) Lemma 5.10. If p = 2 , and A x is a tailed geodesic line with the tail x ∈ B ,and A ∩ B = ∅ , then there exists a tailed geodesic line B y such that B y isorthogonal to A , y ∈ A and B ⊂ B y .Proof. Let y be the point on the geodesic line A satisfying ρ ( x, y ) = log 2,and l be the segment connecting x and y such that B y = l ∪ B is the tailedgeodesic line satisfying the condition. (cid:3) Lemma 5.11. If p = 2 , and g = h ◦ f is a tame elliptic element, where h, f are two involutions, then two tailed geodesic lines of h and f only intersectat a unique point.Proof. By Lemma 5.9, we can assume that the fixed points of f are − , h are − λ, λ . Since g is a tame elliptic element, wehave | λ − | = 1. Hence the tailed point of the tailed geodesic line of h is1 ζ , , and the tailed point of the tailed geodesic line of f is also ζ , . Since D (1 , − ∩ D ( λ, − = ∅ , then the two tailed geodesic lines intersect theunique point ζ , . (cid:3) We give the following lemma without proof, which follows from Lemma5.7and Lemma 5.9 directly.
Lemma 5.12. If p = 2 , and g = h ◦ f is a wild elliptic element, where h, f are two involutions, then two tailed axes A x , B y of h and f only intersecton a segment, and this segment belongs to the fixed points of g . Proof of Theorem 1.2
Proof. If T g ∈ G F g = ∅ , then there exist finitely many elements g , . . . , g n such that T ni =1 F g i = ∅ , since the Berkovich space is compact with respectto the weak topology. Hence if we can show that T ni =1 F g i = ∅ for anypositive integer n , then we prove the theorem.Let f, g be two elliptic elements, and denote the axes of f, g by A f , A g respectively. When p ≥
3, by Lemma 5.8, we have that there exists ainvolution a whose axis A is orthogonal to A f and A g simultaneously. ByLemma 5.1, there exist two involutions b, c such that f = a ◦ b and g = a ◦ c .We denote the set of the fixed points of f, g by F f , F g respectively, and theaxes of a, b, c by A, B, C respectively. By Lemma 5.7, we know that F f ⊃ A ∩ B = ∅ , and F g ⊃ A ∩ C = ∅ , and B ∩ C = ∅ , since h = f − ◦ g is elliptic.Choosing x ∈ A ∩ B, y ∈ A ∩ C, z ∈ B ∩ C , there exists w ∈ [ x, z ] ∩ [ y, z ] ∩ [ x, y ],since P Ber is an R − tree . This means that F f ∩ F g = ∅ .By induction, T ni =1 ,i = k F g i = ∅ for k = 1 , . . . , n , and then we want to show T ni =1 F g i = ∅ . Since T ni =1 ,i = n − F g i = ∅ , T ni =1 ,i = n F g i = ∅ , F g n − ∩ F g n = ∅ ,choosing x ∈ T ni =1 ,i = n − F g i , y ∈ T ni =1 ,i = n F g i , z ∈ F g n − ∩ F g n , there exists w ∈ [ x, z ] ∩ [ y, z ] ∩ [ x, y ] such that w ∈ T ni =1 F g i . This implies that eachelement in G share at least one fixed point.When p = 2, by Lemma 5.8, we have that there exists a involution a whose axis A is orthogonal to A f and A g simultaneously. By Lemma 5.1,there exist two involutions b, c such that f = a ◦ b and g = a ◦ c . Wedenote the set of the fixed points of f, g by F f , F g . Thanks to Lemma 5.9and Lemma 5.10, there exist two tailed geodesic line A x , A y who share thesame associated geodesic line A which are orthogonal to two axes A f and A g . We denote the tailed axes of b, c by B x , C y . By Lemma 5.12, we knowthat F f ⊃ A x ∩ B x = ∅ , and F g ⊃ A y ∩ C y = ∅ , and B x ∩ C y = ∅ , since h = f − ◦ g is elliptic. Choosing u ∈ A x ∩ B x , v ∈ A y ∩ C y , w ∈ B y ∩ C y ,there exists ω ∈ [ u, v ] ∩ [ u, w ] ∩ [ v, w ], since P Ber is an R − tree . This meansthat F f ∩ F g = ∅ .Following the proof of the case p = 2, it is obviously that when p = 2,each element in G share at least one fixed point.By conjugation, we can assume that each element in G shares the uniquefixed point ζ Gauss . By Lemma 2.11, we know that each element in G has2good reduction which yields that G has a potentially good reduction. ByLemma 2.10, we know that if g has good reduction, then g ∈ PSL(2 , O ).Since each element f ∈ G can be written as φf ′ φ − , where f ′ has goodreduction and φ ∈ PSL(2 , C p ), ρ v ( f ( x ) , f ( y )) = ρ v ( φf ′ φ − ( x ) , φf ′ φ − ( y )) ≤ L ρ v ( f ′ φ − ( x ) , f ′ φ − ( y )) ≤ L ρ v ( φ − ( x ) , φ − ( y )) ≤ L L ρ v ( x, y ), where L , L depending only on φ . Hence G is equicontinuous on P ( C p ). (cid:3) Theorem 5.13. If G is a discrete subgroup of PSL(2 , C p ) and the limit setof G is empty, then G has potentially good reduction.Proof. We have that G contains no loxodromic element g , since the fixedpoints of g are in the limit set of G which yields that G contains para-bolic elements and elliptic elements only. Since G is a discrete subgroup ofPSL(2 , C p ), G contains no parabolic elements. By Theorem 1.2, we knowthat G has potentially good reduction. (cid:3) Example 5.14.
Let f n ( z ) = z + p − n ( n ≥ , and the group G is generatedby each f n . Then G contains parabolic elements only and does not havepotentially good reduction.Proof. For any disc D ( a, r ) which is fixed by f n , we have r ≥ p n . Since n is arbitrary, the only point fixed by G is the ∞ . Since each generator cancommutate with each other, we know that each element in G can only fixedthe unique point ∞ in P Ber . (cid:3) Example 5.15.
Let G ⊂ PSL(2 , C p ) , and ζ i be the p i -th primitive root ofunity. Suppose that G is generated by g i = (cid:18) ζ i ζ − i (cid:19) , for all the positive integer i ≥ . Then G is discrete, and the limit set Λ( G ) of G is { , ∞} is a compact set.Proof. In [30], we have proved that G is a discrete subgroup of PSL(2 , C p ).Furthermore, the points { , ∞} are the fixed points of all the elements g i , i ≥
1, namely 0 , ∞ are the limit sets of G (cid:3) Norms of p -adic M¨obius maps and its applications Proposition 6.1.
Suppose that f, g, h ∈ PSL(2 , C p ) . Then (1) ρ ( f h, gh ) = ρ ( f, g ) , and ρ ( hf, hg ) ≤ L ( h ) ρ ( f, g ) . (2) If h ∈ PSL(2 , O p ) , ρ ( h − f h, h − gh ) = ρ ( f, g ) .Proof. Since h is an automorphism on P ( C p ), we have ρ ( f h, gh ) = sup z ∈ P ( C p ) ρ v ( f h ( z ) , gh ( z )) = sup w = h ( z ) ∈ P ( C p ) ρ v ( f ( w ) , g ( w )) = ρ ( f, g ) . ρ v ( h ( z ) , h ( w )) ≤ L ( h ) ρ v ( z, w ), we have ρ v ( hf ( z ) , hg ( z )) ≤ L ( h ) ρ v ( f ( z ) , g ( z ))which yields that ρ ( hf, hg ) ≤ L ( h ) ρ ( f, g ).Since ρ v ( h − f h, h − gh ) = ρ v ( h − f, h − g ) ≤ L ( h ) ρ v ( f, g ) ≤ L ( h ) L ( h − ) ρ v ( h − f, h − g ) , we have ρ v ( h − f, h − g ) ≤ L ( h ) ρ v ( f, g ) ≤ ρ v ( h − f, h − g ) which yields that ρ ( h − f h, h − gh ) = ρ ( f, g ). (cid:3) Let m ( g ) = k g − g − k and M ( g ) = k g − g − kk g k . Proposition 6.2.
Let p be a prime number. Then p − p − ≥ − .Proof. Let f ( x ) = x − x − . Then f ′ ( x ) = f ( x )(ln x − (1 − x )) / ( x − whichyields that f ′ ( x ) > x ≥
3. Since f (3) = 3 − ≥ − = f (2), we have p − p − ≥ − if p is a prime number. (cid:3) Theorem 6.3.
Let p ≥ , and g be a p -adic M¨obius map. Then ρ ( g, I ) = M ( g ) .Proof. By Lemma 4.3, there exists an element h ∈ PSL(2 , O p ) such that hgh − = az + bd with ad = 1. By Proposition 6.1, we have ρ ( hgh − , I ) = ρ ( g, I ). By Theorem 1.1, we know that k g − g − k = k h ( g − g − ) h − k = k hgh − − hg − h − k . Thus we can rewrite hgh − as g . Hence M ( g ) = max {| a − d | , | b |} max {| a | , | d | , | b |} . Moreover ρ v ( g ( z ) , z ) = | a z + ab − z | max { , | z |} max { , | a z + ab |} . If g is parabolic, then ρ v ( g ( z ) , z ) = | b | max { , | z |} max { , | z + b |} ≤ M ( g ) , and ρ ( g, I ) ≥ ρ v ( g (0) ,
0) = M ( g ) . Thus ρ ( g, I ) = M ( g ).If g is loxodromic, we can assume that | a | >
1. Since ρ v ( g (0) ,
0) = | ab | max { , | ab |} , ρ v ( g (1) ,
1) = | a + ab − | max { , | a + ab |} = | a + ab | max { , | a + ab |} , max {| ab | , | a + ab |} ≥ , we have M ( g ) = 1 ≥ ρ ( g, I ) ≥ max { ρ v ( g (0) , , ρ v ( g (1) , } = 1 = M ( g ) . g is elliptic, we have | a − | ≤
1. If | ab | > | a − | , then M ( g ) = | ab | ≥ ρ ( g, I ) ≥ ρ v ( g (0) ,
0) = | ab | = M ( g ). If | ab | ≤ | a − | , there exists a number ω ∈ O p such that | ( a − ω + ab | = | a − | which yields that M ( g ) = | a − | ≥ ρ ( g, I ) ≥ ρ v ( g ( ω ) , ω ) = | a − | = M ( g ) . (cid:3) If p = 2, then 2 − M ( g ) ≤ max {| a − d | , | b |} max {| a | , | d | , | b |} ≤ M ( g ). Thus we give thefollowing theorem without proof. Theorem 6.4.
Let p = 2 , and g be a p -adic M¨obius map. Then − M ( g ) ≤ ρ ( g, I ) ≤ M ( g ) . Theorem 6.5.
For any p -adic M¨obius map g , ρ ( g, I ) ≤k g − I k .Proof. Following Theorem 6.3, we can assume that g ( z ) = az + bd with ad = 1.Thus k g − I k = max {| a − | , | b | , | d − |} . Moreover ρ v ( g ( z ) , z ) = | a z + ab − z | max { , | z |} max { , | az + b |} ≤ . If g is loxodromic, we can assume | a | >
1. Hence max {| a − | , | d − |} > ρ ( g, I ) ≤k g − I k .If g is parabolic, then a = 1. We have ρ v ( g ( z ) , z ) = | b | max { , | z |} max { , | z + b |} ≤ | b | = k g − I k . If g is elliptic, then | a | = 1. We have ρ v ( g ( z ) , z ) = | a z + ab − z | max { , | z |} max { , | az + b |} ≤ | a z + ab − z | max { , | z |}≤ max {| ( a − z | , | ab |} max { , | z |} ≤ max {| a − | , | ab |} ≤k g − I k . (cid:3) Let ε ( g ) = max { ρ v ( g ( z ) , z ) , ρ v ( g ( z ) , z ) , ρ v ( g ( z ) , z ) } , where z , z , z are three distinct roots of the equation z = 1. Theorem 6.6.
For any p -adic M¨obius map g , we have − ε ( g ) ≤ M ( g ) ≤ ε ( g ) .Proof. Since ε ( g ) = max { ρ v ( g ( z ) , z ) , ρ v ( g ( z ) , z ) , ρ v ( g ( z ) , z ) } ≤ ρ ( g, I ),by Theorem 6.3 and 6.4, we have ε ( g ) ≤ ρ ( g, I ) ≤ M ( g ).Since z , z , z are three distinct roots of the equation z = 1, and let z =1, we have z + z + 1 = 0, and by Lemma2.1, Lemma 2.2 and Proposition6.2, we have 2 − ≤ | z − z | ≤
1. This implies that | b | = | cz + ( d − a ) z − b + cz + ( d − a ) z − b + cz + ( d − a ) z − b |≤ max {| cz + ( d − a ) z − b | , | cz + ( d − a ) z − b | , | cz + ( d − a ) z − b |} . ε ′ ( g ) by max {| cz + ( d − a ) z − b | , | cz + ( d − a ) z − b | , | cz + ( d − a ) z − b |} . Hence | b | ≤ ε ′ ( g ) which yields thatmax {| cz + ( d − a ) z | , | cz + ( d − a ) z | , | cz + ( d − a ) z |} ≤ ε ′ ( g ) . Thusmax {| cz +( d − a ) z + cz +( d − a ) z | , | cz +( d − a ) z − cz − ( d − a ) z |} ≤ ε ′ ( g ) , namely | c + ( d − a ) | ≤ ε ′ ( g ) and | c ( z z ) + ( d − a ) | ≤ ε ′ ( g ) which yields that | c ( z z − | ≤ ε ′ ( g ). This implies that | c | ≤ ε ′ ( g ) and | ( d − a ) | ≤ ε ′ ( g ),namely max {| a − d | , | b | , | c |} ≤ ε ′ ( g ). For any z with | z | = 1, we havemax {| az + b | , | cz + d |} ≤ max {| az | , | b | , | cz | , | d |} = k g k which yields thatmax {| a − d | , | b | , | c |} ≤ max {| a − d | , | b | , | c |} ≤ ε ′ ( g ). This implies thatmax {| a − d | , | b | , | c |} / k g k≤ max {| a − d | , | b | , | c |} / k g k≤ ε ′ ( g ) / max {| az + b | , | cz + d |} max { , | z |} for any z with | z | = 1. Since ε ′ ( g )max {| az + b | , | cz + d |} max { , | z |} ≤ max { | cz + ( d − a ) z − b | max {| az + b | , | cz + d |} max { , | z |} , | cz + ( d − a ) z − b | max {| az + b | , | cz + d |} max { , | z |} , | cz + ( d − a ) z − b | max {| az + b | , | cz + d |} max { , | z |} }≤ ε ( g ) , we have M ( g ) ≤ ε ( g ). (cid:3) Corollary 6.7.
For any p -adic M¨obius map g , ε ( g ) ≤ ρ ( g, I ) ≤ ε ( g ) .Proof. By Theorem6.3 and Theorem 6.6, we get the inequality directly. (cid:3)
Let ε ( g ) = { ρ v ( g (0) , , ρ v ( g (1) , , ρ v ( g ( ∞ ) , ∞ ) } . Theorem 6.8.
For any p -adic M¨obius map, − ε ( g ) ≤ M ( g ) ≤ ε ( g ) .Proof. Since ε ( g ) = max { ρ v ( g (0) , , ρ v ( g (1) , , ρ v ( g ( ∞ ) , ∞ ) } ≤ ρ ( g, I ),by Theorem 6.3, we have ε ( g ) ≤ ρ ( g, I ) ≤ M ( g ).Since ρ v ( g ( z ) , z ) = | cz + ( d − a ) z − b | max {| az + b | , | cz + d |} max { , | z |} , we have ρ v ( g (0) ,
0) = | b | max {| b | , | d |} , ρ v ( g ( ∞ ) , ∞ ) = | c | max {| a | , | c |} , and ρ v ( g (1) ,
1) = | c + ( d − a ) − b | max {| a + b | , | c + d |} . Since max {| a − d | , | b | , | c |} ≤ max {| b | , | c | , | c + ( d − a ) − b |} and max {| a + b | , | c + d |} ≤ max {| a | , | c | , | b | , | d |} = k g k , we have M ( g ) = max {| a − d | , | b | , | c |}k g k ≤ max {| b | , | c | , | c + ( d − a ) − b |}k g k ≤ max { | b | max {| b | , | d |} , | c | max {| a | , | c |} , | c + ( d − a ) − b | max {| a + b | , | c + d |} } = ε ( g ). (cid:3) Let ε ( g ) = max { ρ v ( g (0) , , ρ v ( g ( ∞ ) , ∞ ) } . Corollary 6.9. If g is a parabolic element, then − ε ( g ) ≤ M ( g ) ≤ ε ( g ) .Proof. Since ε ( g ) < ε ( g ), we have 2 − ε ( g ) ≤ − ε ( g ) ≤ M ( g ).By Proposition 3.5, if g is a parabolic element, then ( a + d ) = 4 whichyields that 4 bc = − ( a − d ) . Thus it implies that | a − d | = p | || bc | ≤ p | bc | ≤ max {| b | , | c |} which yields that max {| a − d | , | b | , | c |} ≤ max {| b | , | c |} . Hence M ( g ) = max {| a − d | , | b | , | c |}k g k ≤ max {| a − d | , | b | , | c |}k g k≤ max { | b | max {| b | , | d |} , | c | max {| a | , | c |} } = ε ( g ) = max { ρ v ( g (0) , , ρ v ( g ( ∞ ) , ∞ ) } . (cid:3) As applications of these inequalities, we can get the convergence theoremof p -adic M¨obius maps.Let { f n } be a sequence of p -adic M¨obius maps, and U be the set of pointsat which the sequence { f n } converges pointwisely, and f = lim n →∞ f n on U .Write f n ֒ → ( U , f ) to mean that U is the set of convergence of f n and that f n → f on (and only on) U . In [34], we proved the following theorem. Theorem 6.10 ( [34]) . Suppose that there exists a sequence p -adic M¨obiusmaps f n such that f n ֒ → ( U , f ) with U = ∅ . Then one of the followingpossibilities occurs: ( a ) U = P ( C p ) , and f is a p -adic M¨obius map; ( b ) U = P ( C p ) , and f is constant on the complement of one point on U ;( c ) U = { z , z } and f ( z ) = f ( z ) ; or ( d ) f is constant on U . We can reprove this theorem by the use of the three-point norms.If U contains only one point, it is the case ( d ), and if U contains twopoints only, it is the case ( c ) or ( d ). Hence we only need to consider the casewhen U contains at least three points.We prove the following theorem by using different norms of p − adic M¨obiusmaps without using the cross ratios of p − adic M¨obius maps. Theorem 6.11.
Let { f n } be a sequence of p -adic M¨obius maps and z j , j =1 , , be three distinct points with f n ( z j ) → w j , where w j are also threedistinct points. Then a sequence { f n } converge to a p -adic M¨obius map f uniformly, where f ( z j ) = w j , j = 1 , , . Proof.
We can find a p -adic M¨obius map h such that h ( z ) = u , h ( z ) = u , h ( z ) = u , where u i are the three distinct roots of z = 1. Then hf n h − ( u i ) → w i . By Corollary 6.7, we know that ε ( hf − f n h − ) ≤ ρ ( hf − f n h − , I ) ≤ ε ( hf − f n h − ). This yields that hg − f n h − convergesto I uniformly, namely f n converges to f uniformly. (cid:3) Proposition 6.12 ( [34]) . Let f ∈ PSL(2 , C p ) . Then f preserves the chordalcross ratio, namely ρ v ( f ( x ) , f ( y )) ρ v ( f ( z ) , f ( w )) ρ v ( f ( x ) , f ( z )) ρ v ( f ( y ) , f ( w )) = ρ v ( x, y ) ρ v ( z, w ) ρ v ( x, z ) ρ v ( y, w ) . Theorem 6.13.
Let { f n } be a sequence of p-adic maps, and suppose thatthere exist three distinct points x , x , x in P ( C p ) such that lim n →∞ f n ( x ) =lim n →∞ f n ( x ) = α , lim n →∞ f n ( x ) = β , where α = β . Then f n → α on P ( C p ) \ x , namely U = P ( C p ) , and f is constant on the complement of one pointon U .Proof. Without loss of generality, we can assume that x = 0 , x = 1 , x = ∞ and α = 0 , β = ∞ . Assuming x ∈ P ( C p ) \ { , , ∞} , if the sequence { f n ( x ) } does not converge to 0, there exists a subsequence { f n j ( x ) } anda fixed positive number δ such that | f n j ( x ) | > δ . ρ v ( f n j (0) , f n j (1)) ρ v ( f n j ( ∞ ) , f n j ( x )) ρ v ( f n j (0) , f n j ( ∞ )) ρ v ( f n j (1) , f n j ( x )) = ρ v (0 , ρ v ( ∞ , x ) ρ v (0 , ∞ ) ρ v (1 , x ) = 0 . Letting n j tend to ∞ ,0 = lim n j →∞ ρ v ( f n j (0) , f n j (1)) ρ v ( f n j ( ∞ ) , f n j ( x )) ρ v ( f n j (0) , f n j ( ∞ )) ρ v ( f n j (1) , f n j ( x )) = ρ v (0 , ρ v ( ∞ , x ) ρ v (0 , ∞ ) ρ v (1 , x ) = 0 . This is a contradiction. Hence lim n →∞ f n ( x ) = 0. (cid:3) Combing Theorem 6.11 and Theorem 6.13, we prove Theorem 6.10.7.
The decomposition theorem of p − adic M¨obius maps Two points α, β are called antipodal points if there exists an element u ∈ PSL(2 , O p ) such that u (0) = α, u ( ∞ ) = β . Theorem 7.1.
For any p -adic M¨obius map g , there exists an element u ∈ PSL(2 , O p ) such that g = uf , where either f is a loxodromic element withantipodal fixed points, or f = I .Proof. If g is a loxodromic element or elliptic element, by Lemma 5.1, thereexist two involutions a, b such that g = ab , the (tailed) axes of a and b areorthogonal to the axis of g , and the (tailed) axis of a containing ζ Gauss .Let α , β be the fixed points of b . We claim that there exists an element in h ∈ PSL(2 , O p ) such that h ( α ) + h ( β ) = 0.8 Claim : Without loss of generality, we can assume that | α | ≤
1, otherwisewe can consider 1 /α , since 1 /z ∈ PSL(2 , O p ). Let u ( z ) = z − α ∈ PSL(2 , O p )which yields that u ( α ) = 0. Hence we can assume that α = 0. If | β | ≤ h ( z ) = z − β which implies that h (0) + h ( β ) = 0. If | β | >
1, thenlet h ( z ) = az + bcz + d with h (0) + h ( β ) = 0. Hence bd + aβ + bcβ + d = 0which yields that β ( ad + bc ) + 2 bd = 0. Let X = ad, X = bc, X = bd .Hence we have three equations:(1) β ( X + X ) + 2 X = 0(2) X − X = 1(3) X + X = λ Thus we have X = λ +12 , X = λ − , and X = − λβ . Since ab = X X and cd = X X , let a = λ +12 t , b = − λβ t , c = λ − s , and d = − λβ s which yields that − λβ ( λ +1)4 ts + λβ ( λ − ts = 1, namely − λβ ts = 1. Obviously, λ = 1 /β , t = 2,and s = 1 is one of the solution of the equations with max {| a | , | b | , | c | , | d |} ≤
1, namely h ∈ PSL(2 , O p ). Hence we prove the claim.Since h ( α ) + h ( β ) = 0, the geodesic line connecting 0 , ∞ is orthog-onal to the geodesic line connecting h ( α ) , h ( β ) , and this line contains ζ Gauss . Let l be the geodesic line which connects the endpoints of invo-lution a . Since all the elements in PSL(2 , O p ) fix the point ζ Gauss , the point ζ Gauss ∈ h ( l ). By Lemma 5.1, we know that there exists an involution c which fixes the point ζ Gauss such that chbh − is a loxodromic element whosefixed points are 0 , ∞ , and hah − c is an elliptic element in PSL(2 , O p ), since hah − c fixes the point ζ Gauss . Thus hgh − = hah − cchbh − which yieldsthat g = h − hah − chh − ( chbh − ) h , where h − hah − ch ∈ PSL(2 , O p ) and h − ( chbh − ) h is a loxodromic element with antipodal fixed points.If g is a parabolic element, then we can assume the fixed point of g is ∞ after conjugating by an element in PSL(2 , O p ) by the Claim. Thus g ( z ) = ab , where a ( z ) = − z, b ( z ) = − z + b . Since a ( z ) = − z contains thepoint ζ Gauss , by similar discussion above, we can find a involution c whichfixes the point ζ Gauss such that az ∈ PSL(2 , O p ), and cb is a loxodromicelement with antipodal fixed points. (cid:3) Let U = PSL(2 , O p ). We define d ( g, U ) = inf { ρ ( g, u ) | u ∈ U } . Theorem 7.2.
For any p -adic M¨obius map, either d ( g, U ) = 0 , if g ∈ U , or d ( g, U ) = 1 , if g / ∈ U . Proof. If g ∈ U , then d ( g, U ) = 0. If g / ∈ U , then by Theorem 7.1, thereexist u ∈ PSL(2 , O p ) and a loxodromic element f with antipodal fixed pointssuch that g = uf . Hence there exists an element h ∈ PSL(2 , O p ) such that v = hf h − = λz . Thus d ( g, U ) = d ( uf, U ) = d ( v, U ).For any s ( z ) = az + bcz + d ∈ PSL(2 , O p ), since ad − bc = 1 and max {| a | , | b | , | c | , | d |} ≤
1, we have that | bd | = 1, or | ac | = 1, or | bd | > , | ac | <
1, otherwise, if | bd | > | ac | >
1, then | d | < | b | ≤ | c | < | a | ≤ | ad − bc | ≤ max {| ad | , | bc |} <
1. This contradicts ad − bc = 1. Othercases are similar. If | bd | = 1, then ρ v ( v (0) , s (0)) = 1. If | ac | = 1, then ρ v ( v ( ∞ ) , s ( ∞ )) = 1. If | bd | > , | ac | <
1, then 1 ≥ | b | > | d | and | a | < | c | ≤ ad − bc = 1, we have | b | = | c | = 1 > max {| a | , | d |} . This implies that | dc | < ρ v ( v ( − dc ) , s ( − dc )) = | λc ( − dc ) + ( λd − a )( − dc ) − b | max { , | − dc |} max {| a ( − dc ) + b | , | c ( − dc ) + d |} = | /c || /c | = 1 . This yields that for any p -adic M¨obius map g / ∈ U , and any s ∈ U , ρ ( g, s ) =1, namely d ( g, U ) = 1. (cid:3) Let G be a subgroup of PSL(2 , C p ). We say that G a discrete subgroup ifthere exists a positive number ε such that for any non-unit element g with k g − I k > ε . Theorem 7.3. If G is a subgroup of PSL(2 , C p ) and G ∩ U = I , then G isa discrete subgroup.Proof. By Theorem 6.5 and Theorem 7.2, we have d ( f, U ) ≤ ρ ( f, I ) ≤k f − I k . Since G ∩ U = I , for any nonunit element f , k f − I k≥ (cid:3) Corollary 7.4.
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E-mail address : [email protected] Jinghua Yang, Shanghai Univ., Shanghai, 200444, P.R.China
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