Acta Mathematica Sinica, English Series Mar., 2013, Vol. 29, No. 3, pp. 417–428 Published online: February 15, 2013 DOI: 10.1007/s10114-013-1606-5 Http://www.ActaMath.com

Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2013

The Discrete Subgroups and Jørgensen’s Inequality for SL(m, Cp) Wei Yuan QIU Department of Mathematics, Fudan University, Shanghai 200433, P. R. China E-mail : [email protected]

Jing Hua YANG Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, P. R. China E-mail : [email protected]

Yong Cheng YIN Department of Mathematics, Zhejiang University, Zhejiang 310027, P. R. China E-mail : [email protected] Abstract In this paper, we give discreteness criteria of subgroups of the special linear group on Qp or Cp in two and higher dimensions. Jørgensen’s inequality gives a necessary condition for a nonelementary group of M¨ obius transformations to be discrete. We give a version of Jørgensen’s inequality for SL(m, Cp ). Keywords

Jørgensen’s inequality, discreteness criteria, non-Archimedean space

MR(2010) Subject Classification

1

22E35, 22E40, 20H10

Introduction

Discreteness criteria for a subgroup of the projective special linear group PSL(2, C) is an important topic in the theory of Riemann surfaces and hyperbolic manifolds. Let Qp be the p-adic number field and Cp be the completion of the algebraic closure of Qp . In order to study the non-Archimedean orbifolds covered by Mumford curves, Kato [1] discussed the discreteness criteria of groups of projective general linear group PGL(2, Cp ). One of the most important discreteness criteria is Jørgensen’s inequality. Armitage and Parker [2] studied Jørgensen inequality for discrete subgroups of SL(2, Qp ), where SL(2, Cp ) is regarded as the covering space of SL(2, Qp ). In this paper, we improve Jørgensen’s inequality for SL(2, Qp ) and generalize discreteness criteria to high dimension SL(m, Cp ), where m ≥ 2. Kato [1] classified non-unit elements of PGL(2, Cp ) into three cases: parabolic elements, elliptic elements and hyperbolic elements. He proved that a discrete subgroup of PGL(2, Cp ) can not contain any parabolic element and any elliptic element of infinite order. For high dimension special linear group, we classify non-unit elements of SL(m, Cp ) for m ≥ 2 into parabolic elements, elliptic elements and loxodromic elements, and we generalize results of Kato to SL(m, Cp ). Received October 21, 2011, accepted May 16, 2012 Supported by National Natural Science Foundation of China (Grants Nos. 10831004 and 11271047) and by Science and Technology Commission of Shanghai Municipality NSF (Grant No. 10ZR1403700)

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Theorem 1.1

Let G be a discrete subgroup of SL(m, Cp ). Then

(1) there is no parabolic element in G; (2) there is no elliptic element of infinite order in G. The basic theorem of discrete subgroups of the complex 2-dimension special linear group SL(2, Cp ) is that a non-elementary subgroup G of SL(2, C) is discrete if and only if its subgroup generated by any two elements in G is discrete. Moreover, for special linear group over R, Jørgensen [3] proved that a non-elementary subgroup G of SL(2, R) is discrete if and only if any cyclic subgroup of G, i.e., a subgroup of G generated by one element, is discrete. This result is neither true for SL(2, C) nor true for SL(m, R). However, we prove that for m ≥ 2, a subgroup G of SL(m, Cp ) is discrete if and only if any cyclic subgroup of G is discrete, i.e., any subgroup of G generated by one element is discrete. Theorem 1.2 G is discrete.

The subgroup G of SL(m, Cp ) is discrete if and only if any cyclic subgroup of

This implies that the inverse of Theorem 3.3 (including Kato’s result) is also true. In the theory of Kleinian groups, if a discrete subgroup G of SL(2, C) contains elliptic elements only, then G is a finite group. However, we construct a subgroup G ⊂ SL(2, Cp ) containing infinitely many elliptic elements of finite order only, but G is discrete. Example 1.3 by

Let ζi be the primitive pi -th root of unity, where i = 1, 2, 3, . . .. If G is generated ⎛ gi = ⎝

⎞ ζi

0

0

ζi−1

⎠,

then G is discrete. However, if Kp is a finite extension of Qp , then we have Theorem 1.4 If a discrete subgroup G of SL(2, Kp ) contains elliptic elements of finite order only, then G is a finite group. Jørgensen’s inequality is a necessary condition for the discreteness of subgroups of SL(2, C). Jørgensen’s inequality has been widely applied in many aspects such as the algebraic and geometric convergence of subgroups of SL(2, C) and the estimation of the volume of hyperbolic manifolds. Jørgensen’s inequality was generalized by many authors in various cases. Jørgensen’s inequality also plays an important role in the p-adic analytic space. In [2], Armitage and Parker gave a version of Jørgensen’s inequality of discrete subgroups for SL(2, Qp ). In this paper, we improve their results as follows. Theorem 1.5 Let A = −I be an element of SL(2, Qp ). Let B be any element of SL(2, Qp ) such that B neither fixes nor interchanges the fixed points of A. If G = A, B is discrete, then 1. if p > 3, then min{|tr2 (A) − 4|, |tr([A, B]) − 2|} ≥ 1; 2. if p = 3, then min{|tr2 (A) − 4|, |tr([A, B]) − 2|} ≥ 13 ;

3. if p = 2, then min{|tr2 (A) − 4|, |tr([A, B]) − 2|} ≥ 14 . We generalize the result of Armitage and Parker to SL(2, Cp ).

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Theorem 1.6 Let A = −I be an element of SL(2, Cp ). Let B be any element of SL(2, Cp ) such that B neither fixes nor interchanges the fixed points of A. If G = A, B is discrete, then 2 min{|tr2 (A) − 4|, |tr[A, B] − 2|} ≥ p− p−1 . Note that SL(2, Qp ) is a subgroup of SL(2, Cp ), and SL(2, Cp ) is much more complicated than SL(2, Qp ), since Qp is locally compact, but Cp is not locally compact. Theorem 1.7 m p− p−1 .

If a subgroup G of SL(m, Cp ) is discrete, then for each g ∈ G\{I}, g − I ≥

This paper is organized as follows. Section 2 consists of definitions and lemmas. In Section 3, we discuss discreteness criteria of subgroups of SL(m, Cp ), and prove Theorems 1.1, 1.2 and 1.4. In Section 4, we give versions of Jørgensen inequality of subgroups of SL(m, Cp ), and prove Theorems 1.5–1.7. 2

Definitions and Lemmas

In this section, we give basic facts and lemmas. All of them can be found in [4, 5]. Let p ≥ 2 be a prime number. Let Qp be the algebraic closure of the p-adic number field Qp . Let Cp be the completion of Qp which is endowed with | · | as the non-Archimedean absolute value on Cp , i.e., for x, y ∈ Cp , we have the ultrametric inequality |x − y| ≤ max{|x|, |y|}. This inequality arise interesting topological and geometrical properties. If x, y and z are points in Cp satisfying ¯ r) := {z||z−a| ≤ r} and D(a, r) := {z||z−a| < r} |x−y| < |x−z|, then |x−z| = |y−z|. Let D(a, denote closed and open disks with center at a ∈ Cp and radius r, respectively. In the topology ¯ r) and D(a, r) are not only closed but also open. Every point in the disk of Cp , both D(a, ¯ r) = D(x, ¯ D(a, r) is the center. This means that if x ∈ D(a, r), then D(a, r) = D(x, r) (D(a, r) resp.). By the ultrametric property, if two disks D1 and D2 in Cp have non-empty intersection, then D1 ⊂ D2 , or D2 ⊂ D1 . Lemma 2.1 ([4, p. 105]) Let d0 > 1 be an integer which is not divisible by p, and d = d0 pt be a natural number. Let ζ be the primitive d-th root of unity. Then |ζ − 1| = 1. Lemma 2.2 ([4, p. 107])

Let ζ be the primitive pd -th root of unity. Then |ζ −1| = p

−

1 pd−1 (p−1)

.

Through this paper, the symbol (A) denotes the cardinality of the set A. Lemma 2.3 ([4, p. 105]) Let Kp be a finite extension of Qp . Let μp (Kp ) be the subgroup consisting of the roots of unity in Kp of order prime to p. If the residue degree f of Kp /Qp is finite, then the group μp (Kp ) is finite. More precisely, (μp (Kp )) ≤ pf − 1. Lemma 2.4 ([4, p. 110]) Let Kp be a finite extension of Qp . Let μp∞ (Kp ) be the subgroup consisting of the p-th roots of unity in Kp . If the ramification index e = e(Kp ) is finite, then e . the group μp∞ (Kp ) is finite. More precisely, (μp (Kp )) ≤ 1−1/p Lemma 2.5 ([4, p. 151]) Lemma 2.6 Proof

n!

Let x ∈ Cp with |x| = 1. Then the sequence {xp } converges. d

d−1

The polynomial (X p − 1)/(X p

− 1) is irreducible over Qp .

Let X = t + 1. Then d

d−1

(X p − 1)/(X p

d−1

− 1) = X p

(p−1) d−1

= (t + 1)p

d−1

+ Xp

(p−1)

(p−2)

d−1

+ · · · + Xp d−1

+ (t + 1)p

(p−2)

+1 d−1

+ · · · + (t + 1)p

+ 1.

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Let g(t) = (t + 1)p (p−1) + (t + 1)p (p−2) + · · · + (t + 1)p + 1. If g(t) is reducible over Qp , then g(t) = g1 (t)g2 (t), where g1 (t), g2 (t) are non-constant polynomials in Qp [t]. Let ti be roots d−1

d−1

d−1

−

1

of g(t), where 1 ≤ i ≤ pd−1 (p − 1). By Lemma 2.2, we know that |ti | = p pd−1 (p−1) . Suppose that the degree of g1 (t) is s. Without loss of generality, assume that ti is the root of g1 (t), where 1 ≤ i ≤ s < pd−1 (p − 1). The absolute value of the coefficient on the constant term of s − g1 (t) is equal to |t1 · · · ts | = p pd−1 (p−1) . However the coefficient on the constant term of g1 (t) is in Qp which has the discrete value as the form p−k , where k is an integer. This contradicts d d−1 0 < pd−1 s(p−1) < 1. This implies that (X p − 1)/(X p − 1) is irreducible over Qp .  Lemma 2.7 Let g(z) = z n + an−1 z n−1 + · · · + a0 be a polynomial in Cp [z]. Given a fixed r > 0, if the coefficients of g(z) satisfy |ai | < r n−i , then all roots of the polynomial g(z) are in ¯ r). the closed disk D(0, ¯ r), then |ai αi | < r n−i |αi | < |αn |. By the ultrametric property, |g(α)| = Proof If α ∈ / D(0, n n−1 + · · · + a0 | = |αn | > 0. Then all roots of the polynomial are in the closed disk |α + an−1 α ¯ r). D(0,  Lemma 2.8

Let gn (z) = z m +

m−1 

ain z i

(2.1)

i=0

be a sequence of polynomials in Cp [z]. If all coefficients ain tend to zero as n → ∞, where 0 ≤ i ≤ m − 1, then all roots of gn (z) tend to zero as n → ∞. Proof For any r > 0, we can find a sufficiently large positive integer N such that for any n > N , |ain | < r n−i , since ain tend to zero as n → ∞, where 0 ≤ i ≤ m − 1. By Lemma 2.7, ¯ r). Since r > 0 is arbitrary, all roots of gn (z) tend all roots of gn (z) are in the closed disk D(0, to zero as n → ∞.  3

Discrete Subgroups of SL(m, Cp )

In [1], Kato introduced the method that was used in Kleinian groups to study the discrete subgroups of PGL(2, Cp ). He classified non-unit element g in PGL(2, Cp ) into the following three classes: (a) g is said to be parabolic if it has only one eigenvalue. (b) g is said to be elliptic if it has two distinct eigenvalues λ1 and λ2 with |λ1 | = |λ2 |. (c) g is said to be hyperbolic if it has two eigenvalues λ1 and λ2 with |λ1 | = |λ2 |. Kato pointed out that if g is a parabolic element, then g has only one fixed point in the projective space P1 (Cp ); if g is a hyperbolic element or elliptic element, then g has two distinct fixed point in P1 (Cp ). In the following, we classify non-unit elements in SL(m, Cp ), m ≥ 2. Since the product of all eigenvalues of g ∈ SL(m, Cp ) is one, then either the absolute value of each eigenvalue of g is one or there exists at least one eigenvalue whose absolute value is larger than 1. Thus each non-unit element g ∈ SL(m, Cp ) falls into the following three classes: (a) g is said to be parabolic if 1. the absolute value of any eigenvalue of g is 1, and

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2. g can not be conjugated to a diagonal matrix. (b) g is said to be elliptic if 1. the absolute value of any eigenvalue of g is 1, and 2. g can be conjugated to a diagonal matrix. (c) g is said to be loxodromic if there exists at least eigenvalue of g whose absolute value is larger than 1. For g = (aij ) in the matrix ring M(m, Cp ), the norm of g is defined by g =

max

{|aij |}.

1≤i≤m,1≤j≤m

Obviously, g = 0 implies that each aij = 0. It is easy to verify that αg = |α|g, g + h ≤ max{g, h} and gh ≤ gh. We say that a subgroup G of SL(m, Cp ) is discrete if there exists δ = δ(G) > 0 such that each element g ∈ G\{I} satisfies g − I > δ, where I denotes the unit element. Obviously, a subgroup G of SL(m, Cp ) is discrete if and only if any sequence consisting of distinct elements gn ∈ G is not a Cauchy sequence. Since h−1 gn h − h−1 gh ≤ h−1 gn − gh, we have h−1 gn h − h−1 gh → 0, when gn → g, as n → ∞. This means that conjugation does not change the discreteness. In the following, we introduce some results in [1]. Some of them will be extended to higher dimension. Theorem 3.1 ([1, Lemma 4.2])

Let G be a discrete subgroup of PGL(2, Cp ).

(1) There exists no parabolic element in G. (2) An element g ∈ G is elliptic if and only if it is of finite order. In this paper, we show that if G is a discrete subgroup of SL(m, Cp ), then the elliptic element in G is of finite order, and G does not contain any parabolic element. Lemma 3.2 Let I denote the unit matrix and J denote a nilpotent matrix in M(m, Cp ). Let n! λ ∈ Cp with |λ| = 1. If f = λI + J, then the sequence f p  converges. Proof Since J is a nilpotent matrix, there exists a positive integer N such that J N = 0. Thus for any positive integer k > N , we have     k k−1 k f k = (λI + J)k = λk I + λ J + ···+ λk−N J N . 1 N pn!





pn! pn! −1 i−1 i

, where 1 ≤ i ≤ N . Taking some fixed positive

n!

n! n! −1 integer t such that N < p , then for any 1 ≤ i ≤ N , |i| > p−t , we have | pi | = | pi || pi−1 |≤ pn! pn! −i i pn! t−n! | i |≤p for sufficiently large n. Hence i λ J → 0, as n → ∞, since |λ| = 1. By n! pn!  Lemma 2.5, {λ } converges. Therefore, the sequence f p  converges. Choose k = pn! . Then

i

=

t

Theorem 3.3

Let G be a discrete subgroup of SL(m, Cp ). Then

(1) there is no parabolic element in G; (2) there is no elliptic element of infinite order in G. Proof

(1) Suppose that there is a parabolic element g ∈ SL(m, Cp ). Since conjugation does

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not change the discreteness, we can assume that g has the Jordan standard form as ⎛ ⎞ A1 ⎜ ⎟ ⎜ ⎟ .. g=⎜ ⎟, . ⎝ ⎠ As where each Jordan block Ai has the form λi Ii + Ji , where λi is the eigenvalue of g, and Ii denotes the unit matrix, and Ji is the nilpotent matrix. n!

n!

By Lemma 3.2, the {Api } converges. This implies that {g p } converges which is a contradiction. Hence there is no parabolic element in G. (2) Suppose that g is an elliptic element of infinite order in SL(m, Cp ). We can assume that ⎛ ⎞ λ1 ⎜ ⎟ ⎜ ⎟ .. g=⎜ ⎟, . ⎝ ⎠ λm where λi ∈ Cp are eigenvalues of g with |λi | = 1, 1 ≤ i ≤ m. n!

Therefore, λsi = λti for any positive integers s, t. By Lemma 2.5, the sequence {λpi } n! converges. Thus {g p } is the sequence consisting of distinct elements and a convergent sequence. This contradicts the hypothesis. Thus there is no elliptic element of infinite order in G.  Lemma 3.4 n → ∞.

If gn ∈ SL(m, Cp ) → I, as n → ∞, then all eigenvalues of gn tend to 1, as

Proof The eigenpolynomial fn (λ) = |λI − gn | tends to polynomial (λ − 1)m , since gn tends to I. By Lemma 2.8, all eigenvalues λn tends to 1.  Corollary 3.5 If there exists a positive number δ = δ(G) such that for any g ∈ G, max{|λ1 − 1|, |λ2 − 1|, . . . , |λm − 1|} ≥ δ, where λ1 , λ2 , . . . , λm are eigenvalues of g, then G is discrete. Proof If G is not discrete, then there exists a sequence gn tending to I, as n → ∞. By Lemma 3.4, we know that eigenvalues λ1,n , λ2,n , . . . , λm,n of gn tend to 1 which contradicts the hypothesis. Thus G is discrete.  Theorem 3.6 G is discrete. Proof

The subgroup G of SL(m, Cp ) is discrete if and only if any cyclic subgroup of

Obviously, if G is discrete, then each subgroup of G is discrete.

By Theorem 3.3, the cyclic subgroup generated by an elliptic element of infinite order or a parabolic element is not discrete. Hence the subgroup G does not contain any elliptic element of infinite order and any parabolic element, which yields that there only exist loxodromic elements or elliptic elements of finite order. If g is a loxodromic element, then g has at least one eigenvalue whose absolute value is larger than 1. Let λ be the eigenvalue of g with |λ| > 1. By the ultrametric property, we have |λ − 1| > 1. If g is an elliptic element of the order n, namely, g n = I, where n is the smallest

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positive integer, then we can assume that g has the form as ⎛ ⎞ λ1 ⎜ ⎟ ⎜ ⎟ .. g=⎜ ⎟, . ⎝ ⎠ λm where λi is an eigenvalue of g, 1 ≤ i ≤ m. Since g n = I, each eigenvalue λi of g satisfies λni = 1, where 1 ≤ i ≤ m. Let n = d0 pt ≥ 2, where d0 and t are non-negative integers, and d0 is prime to p. At least one of the eigenvalues of g is n-th primitive root of unity, since n is the smallest positive integer. Take λ the n-th primitive root of unity. If d0 = 1, namely, n = pt , then λ is a primitive pt -th root of unity. By 1 − Lemma 2.2, |λ − 1| = p pt−1 (p−1) ≥ p−1 . If d0 > 1, then by Lemma 2.1, |λ − 1| = 1. Thus max{|λ1 − 1|, |λ2 − 1|, . . . , |λm − 1|} ≥ p−1 . By Corollary 3.5, the subgroup G is discrete.  In the theory of Kleinian groups, there is a classical result that if a subgroup G is discrete and contains elliptic elements only, then G contains finitely many elements. This is also true in the isometric group of complex hyperbolic manifold in 2 or higher dimensions. In the nonArchimedean setting, we prove that this result is true for the finite extension of Qp , but this is not true for Cp . The following lemma shows that for some fixed positive integer n, there only exist finitely many extensions of Qp of degree n. Lemma 3.7 ([4]) degree n of Qp .

For a given integer n ≥ 1, there are only finitely many extensions Kp of

Lemma 3.8 Let Kp be a finite extension of degree t of Qp . If H be the union of all finite

p : Kp ] ≤ 2, then the number of primite roots of unity in H is finite.

p with [K extensions K Proof Since [K˜p : Kp ] ≤ 2 and [Kp : Qp ] ≤ t, we have [K˜p : Qp ] ≤ 2t. By Lemmas 2.4 and 2.5, we know that the number of primitive roots of unity in each field of finite extension of Qp is finite. By Lemma 3.7, the number of fields of finite extension of Qp of degree less than 2t is finite. Hence the number of primitive roots of unity in H is finite.  2

Lemma 3.9 If λ is the eigenvalue of the elliptic element g of finite order, then p− p−1 ≤ |tr(g) − 2| ≤ 1, where tr(g) denotes the trace of g. Proof Since λ is the eigenvalue of the elliptic element g of finite order, λ is the primitive root 1 of unity. By Lemmas 2.1 and 2.2, p− p−1 ≤ |λ − 1| ≤ 1. Since trace is invariant by conjugation, we have tr(g) = a + d = λ + λ−1 which implies that |tr(g) − 2| = |λ + λ−1 − 2| = |λ − 1|2 /|λ|. 2 Hence p− p−1 ≤ |tr(g) − 2| ≤ 1.  If g has an eigenvalue is −1, then the other eigenvalue should also be −1, since the determinant is 1. Hence g can be conjugated to the diagonal matrix −I, and thus g = h(−I)h−1 = −hh−1 = −I. Theorem 3.10 Let Kp be a finite extension of Qp . If a discrete subgroup G of SL(2, Kp ) contains elliptic elements of finite order only, then G is a finite group. Proof

p : Kp ] ≤ 2. Then K

p is a finite extension of ˜ p be a finite extension of Kp with [K Let K

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p is locally compact. By Lemma 3.7, there are only finitely Qp . Since Qp is locally compact, K

p . Let H be union of all such K

p . This implies that H is locally compact. many such K Take a fixed element g ∈ G\{I}. Suppose that there exists an element h ∈ G which can not commutate with g. Then g can not be −I or I. So the eigenvalue λ of g satisfies λ2 = 1. Thus g, h can be conjugated to ⎛ ⎞ ⎛ ⎞ λ 0 a b ¯=⎝ ⎠, h ⎠ ∈ SL(2, Kp (λ)). g¯ = ⎝ 0 λ−1 c d Then the commutator

⎛

¯ −1 = ⎝ ¯ = g¯h¯ ¯ g −1 h [¯ g , h]

ad − bcλ−2

−abλ2 + ab

cd − cdλ−2

−bcλ2 + ad

⎞ ⎠.

¯ = 2ad−bc(λ2 + 12 ) = 2−bc[(λ+ 1 )2 −4] = 2−bc(λ− 1 )2 . By Lemma 3.9, Therefore, tr[¯ g , h] λ λ λ 2 p− p−1 ≤ |bc(λ − λ1 )2 | ≤ 1. Since λ2 is also a primitive root of unity and λ2 = 1, by Lemmas 2.1 1 2 1 and 2.2, we have p− p−1 ≤ |λ2 − 1|/|λ| = |λ2 − 1| ≤ 1. Therefore, p− p−1 ≤ |bc| ≤ p p−1 . Suppose that there exist infinitely many distinct elements hn which can not commutate with g, and let hn have the following form ⎛ ⎞ an bn ⎠. hn = ⎝ cn dn Since an dn − bn cn = 1 and bn cn are bounded, an dn is also bounded. We also have an , dn are bounded, since an + dn is bounded. Suppose that bn , cn are bounded, then an , dn , bn , cn are all bounded. Since Qp is locally compact, an , dn , bn , cn have convergent subsequences. Then hn has the convergent subsequence which contradicts the discreteness of G. Suppose that {bn } or {cn } is unbounded, without loss of generality, we suppose that bn → ∞, 2 1 as n → ∞. Since p− p−1 ≤ |bn cn | ≤ p p−1 , cn → 0, as n → ∞. Consider the sequence {h1 hn }. Since ⎛ ⎞ h1 h n = ⎝

a1 an + b1 cn

a1 bn + b1 dn

an c1 + d1 cn

bn c1 + d1 dn

⎠,

then tr[h1 hn ] = a1 an + b1 cn + bn c1 + d1 dn . Since b1 , c1 are nonzero and an , dn are bounded, then a1 an + b1 cn + bn c1 + d1 dn → ∞, as n → ∞. Therefore, when n is sufficiently large, h1 hn is a loxodromic element which contradicts that G has elliptic elements only. Hence there do not exist infinitely many elements which can not commutate with g. Suppose that h ∈ G can commutate with g. Then h and g can be conjugated to diagonal matrices simultaneously. Since eigenvalues of h ∈ G are primitive roots of unity in K˜p , then by Lemma 3.8, there exist finitely many such h. To sum up, there are only finitely many elements in G.  But the result we proved above is not true for Cp , even for Qp , since Qp and Cp are infinite extensions of Qp . We can find infinitely many primitive roots, but the distance between all the primitive roots and unity is larger than a fixed positive number.

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Example 3.11 generated by

425

Let ζi be the primitive pi -th root of unity, where i = 1, 2, 3, . . .. If G is ⎛ gi = ⎝

⎞ ζi

0

0

ζi−1

⎠,

then G is discrete. Proof Since G is generated by gi , then for each g ∈ G, g = giε11 giε22 · · · giεnn , where εi ∈ {−1, 1}. Then eigenvalues of g are ζiε11 ζiε22 · · · ζiεnn and the reciprocal which are roots of unity. Hence each element in G is an elliptic element of finite order. By Theorem 3.6, we know the group G is discrete. However, G contains infinitely many elliptic elements, since each generator gi is different from each other.  4

Jørgensen’s Inequality for SL(m, Cp )

In [2], Armitage and Parker gave a version of Jørgensen’s inequality in the non-Archimedean metric space, especially for SL(2, Qp ). Theorem 4.1 ([2, Theorem 4.2]) Let A be an element of SL(2, Qp ) conjugate to a diagonal obius matrix. Let B be any element of SL(2, Qp ) so that, when acting on Qp ∪ {∞} via M¨ transformations, B neither fixes nor interchanges the fixed points of A. If G = A, B is discrete, then max{|tr2 (A) − 4|, |tr([A, B]) − 2|} ≥ 1. In our paper, we improve Jørgensen’s inequality for SL(2, Qp ), and construct a version of Jørgensen’s inequality for SL(m, Cp ) by algebraic method. Lemma 4.2 Then

Let λ be the n-th primitive root of unity with [Qp (λ) : Qp ] ≤ 2, where n ≥ 2.

(1) If p > 3, then |λ − 1| = 1; (2) If p = 3, then |λ − 1| ≥ (3) If p = 2, then |λ − 1| ≥

√1 ; 3 1 . 2

Proof Let n = d0 pd , where d0 is a positive integer which is prime to p and d is a non-negative integer. If the positive integer d0 > 1, then by Lemma 2.1, we have |λ − 1| = 1. When d0 = 1, λ is the pd -th primitive root of unity, where d is a positive integer and λ is the root of the pd irreducible polynomial Xpd−1−1 = 0. Hence [Qp (λ) : Qp ] = pd−1 (p − 1). It is divided into X −1 three different cases. If p > 3, then [Qp (λ) : Qp ] = pd−1 (p − 1) > 2, which is a contradiction. Therefore, there does not exist such kind of λ. If p = 3, then [Qp (λ) : Qp ] = 3d−1 (3 − 1) ≤ 2 implies that d = 1. Therefore, λ is the 3-th primitive root of unity, and then |λ − 1| = √13 . If p = 2, then [Qp (λ) : Qp ] = 2d−1 (2 − 1) ≤ 2 deduces that d = 1, or d = 2. If λ is the 2-th primitive root of unity, then |λ − 1| = 12 . If λ is the 22 -th primitive root of unity, then 1  |λ − 1| = 2− 2 . We draw our conclusion. According to our results, the discrete subgroup does not contain any parabolic element which yields that a generator A ∈ SL(2, Cp ) can be conjugated to a diagonal matrix. If the subgroup G is generated by −I and B ∈ SL(2, Cp ), then the group G = {(−1)i B j } is very trivial. Hence we do not consider −I as the generator.

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Theorem 4.3 Let A = −I be an element of SL(2, Qp ). Let B be any element of SL(2, Qp ) such that B neither fixes nor interchanges the fixed points of A. If G = A, B is discrete, then 1. if p > 3, then min{|tr2 (A) − 4|, |tr([A, B]) − 2|} ≥ 1; 2. if p = 3, then min{|tr2 (A) − 4|, |tr([A, B]) − 2|} ≥ 13 ;

3. if p = 2, then min{|tr2 (A) − 4|, |tr([A, B]) − 2|} ≥ 14 .

Proof If [A, B] = I, then ABA−1 B −1 = I. This implies that AB = BA which means that B can fix or interchange the fixed point of A which contradicts the hypothesis. Hence [A, B] = I. By Theorem 3.3, we know that A and [A, B] are either loxodromic elements or elliptic elements of finite order. Let λ and λ1 be eigenvalues of A. If A is a loxodromic element, then the absolute value of one of the eigenvalues is larger than 1. Without loss of generality, we can assume that |λ| > 1. Since |λ − λ1 | = |λ| > 1, we have |tr2 (A) − 4| = |(λ + λ1 )2 − 4| = |λ − λ1 |2 = |λ| > 1. If A is an elliptic element of finite order, then there exist three different cases. Since λ2 is an n-th primitive root of unity and λ2 = 1, when p > 3, by Lemma 4.2, |λ2 − 1| ≥ 1 implies that |tr2 (A) − 4| = |(λ + λ1 )2 − 4| = |λ2 − 1|2 /|λ|2 ≥ 1; when p = 3, by Lemma 4.2, |λ − 1| ≥ √13 implies that |tr2 (A) − 4| = |(λ + λ1 )2 − 4| = |λ2 − 1|2 /|λ|2 ≥ 13 ; when p = 2, by Lemma 4.2, |λ2 − 1| ≥ 12 implies that |tr2 (A) − 4| = |(λ + λ1 )2 − 4| = |λ2 − 1|2 /|λ|2 ≥ 14 . Let μ and μ−1 be eigenvalues of [A, B]. Similar to the analysis above, when [A, B] is a loxodromic element, |tr[A, B]−2| = |μ−1|2 /|μ| > 1; when p > 3, |tr[A, B]−2| = |μ−1|2 /|μ| ≥ 1; when p = 3, |tr[A, B] − 2| = |μ − 1|2 /|μ| ≥ 13 ; when p = 2, |tr[A, B] − 2| = |μ − 1|2 /|μ| ≥ 14 .  We can also build a Jørgensen’s inequality for SL(2, Cp ). Theorem 4.4 Let A = −I be an element of SL(2, Cp ). Let B be any element in SL(2, Cp ) such that B neither fixes nor interchanges the fixed points of A. If G = A, B is discrete, then 2 min{|tr2 (A) − 4|, |tr[A, B] − 2|} ≥ p− p−1 . Proof If [A, B] = I, then ABA−1 B −1 = I. This implies that AB = BA which means B can fix or interchange the fixed point of A which contradicts the hypothesis. Hence [A, B] = I. Let λ and λ1 be eigenvalues of A. Since A is not −I, λ2 = 1. If A is loxodromic element, the absolute value of one of the eigenvalues is larger than 1. Without loss of generality, we can assume that |λ| > 1. Hence |λ − λ1 | = |λ| > 1, and then |tr2 (A) − 4| = |(λ + λ1 )2 − 4| = |λ − λ1 |2 = |λ| > 1. If A is an elliptic element of finite order, then λ is the n-th primitive root of unity. Let n = d0 pd , where d0 is a positive integer which is prime to p, and d is a non-negative integer. There are two different cases. If d0 = 1, namely, λ2 is the pd -th root of 1 1 − unity, where d is a positive integer, then by Lemma 2.2, |λ2 − 1| = p pd−1 (p−1) ≥ p− p−1 . Hence 2 |tr2 (A) − 4| = |(λ + λ1 )2 − 4| = |λ2 − 1|2 /|λ|2 ≥ p− p−1 . When d0 > 1, since λ2 is the d0 pd -th root 2 of unity, then by Lemma 2.1, |λ2 − 1| = 1. Hence |tr2 (A) − 4| = |λ2 − 1|2 /|λ|2 = 1 ≥ p− p−1 . Let ζ and ζ −1 be the eigenvalues of the [A, B]. Since [A, B] = I, ζ = 1. If [A, B] is a loxodromic element, then the absolute value of one of the eigenvalues is larger than 1. Without loss of generality, we can assume that |ζ| > 1. Hence |ζ − ζ1 | = |ζ| > 1 which implies that |tr[A, B] − 2| = |(ζ + 1ζ ) − 2| = |ζ − 1|2 /|ζ| = |ζ| > 1. If [A, B] is an elliptic element of finite order, then ζ is an n-th primitive root of unity. Let n = d0 pd , where d0 is a positive integer which is prime to p, and d is a non-negative integer. There are two different cases. If d0 = 1,

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1

1

namely, ζ is the pd -th root of unity, then by Lemma 2.2, |ζ − 1| = p pd−1 (p−1) ≥ p− p−1 . Hence 2 |tr[A, B] − 2| = |ζ + ζ1 − 2| = |ζ − 1|2 /|ζ| ≥ p− p−1 . If d0 > 1, namely, ζ is a d0 pd -th primitive root of unity, then by Lemma 2.1, |ζ − 1| = 1. Therefore, 2

|tr[A, B] − 2| = |ζ − 1|2 /|ζ| = 1 ≥ p− p−1 .



Furthermore, we give a Jørgensen’s inequality of discrete subgroup of SL(m, Cp ) as an isolation of identity. Although we can not control the matrix by the trace in higher dimensions, we can control the eigenvalues of matrix to give the discreteness criteria of subgroups of SL(m, Cp ). In [6], Martin gave a version of Jørgensen’s inequality for the real M¨ obius transform in higher dimensions. Theorem 4.5 ([6, Theorem 4.5]) Let f and g be M¨ obius transformations of S n . If f and g together generate a discrete non-elementary group, then max{g i f g −i −I : i = 0, 1, 2, . . . , n} > √ 2 − 3. In order to give the Jøgensen inequality in all dimensions, we also show that if an element is too close to the identity, then the eigenvalue is very close to 1. In [6], Martin discussed the group generated by finitely many elements, and estimated the maximum distance between the generator and the identity. m ¯ r), Lemma 4.6 If g ∈ SL(m, Cp ) and g − I < p− p−1 , then all eigenvalues of g are in D(1, 1 − p−1 . where r < p

Proof Let g = (bij ) ∈ SL(m, Cp ). Since g − I < p− p−1 , then |bij − δij | < p− p−1 , where δij = 1, if i = j; otherwise δij = 0, if i = j. m

Then eigenpolynomial      λ − b11 −b12 ... −b1m      λ − b22 ··· −b2m   −b21   |λI − g| =  .. ..  .. ..   . . . .      ... ... −bm(m−1) λ − bmm     (λ − 1) + 1 − b11 −b12 ...   −b21 (λ − 1) + 1 − b22 ···  =  .. .. ..  . . .    ... ... −bm(m−1)

m

−b1m −b2m .. . (λ − 1) + 1 − bmm

      .     

Hence the eigenpolynomial can be rewritten as G(λ−1) = (λ−1)m +am−1 (λ−1)m−1 +· · ·+ a0 , where the coefficient ai of eigenpolynomial G(λ − 1) is a combination of the cij = δij − bij m by product or addition. By the ultrametric property, we have |ai | ≤ max{|cij |} < p− p−1 . Since m 1 1 1 |ai | m−i ≤ p− (m−i)(p−1) ≤ p− p−1 , there exists a positive number r satisfying 0 < r < p− p−1 such ¯ r).  that |ai | < r m−i . By Lemma 2.7, each eigenvalue of g is in D(1, Theorem 4.7 m p− p−1 .

If a subgroup G of SL(m, Cp ) is discrete, then for each g ∈ G\{I}, g − I ≥

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Proof By Theorem 3.3, we know that each element in G is either a loxodromic element or an elliptic element of finite order. If g in G is a loxodromic element, then there exists at least one eigenvalue λ whose absolute value is larger than 1. Hence |λ − 1| = |λ| > 1. If g is an elliptic element of finite order, then there exists at least one eigenvalue λ which is an n-th primitive root of unity. Let n = d0 pd , where d0 is a positive integer which is prime to p, and d is a 1 1 − non-negative integer. If d0 = 1, then by Lemma 2.2, |λ − 1| = p pd−1 (p−1) ≥ p− (p−1) . If d0 > 1, 1 then by Lemma 2.1, |λ − 1| = 1. Therefore, |λ − 1| ≥ p− p−1 . This shows that λ, which is 1 one of the eigenvalues of g, satisfies |λ − 1| ≥ p− (p−1) . Hence by Lemma 4.6, we know that m  g − I ≥ p− p−1 . Acknowledgements

We would like to thank the referee for his/her valuable comments.

References [1] Kato, F.: Non-archimedean orbifolds covered by Mumford curves. J. Algebraic Geom., 14, 1–34 (2005) [2] Armitage, J. V., Parker, J. R.: Jørgensen’s inequality for non-Archimedean metric spaces. Geometry and Dynamics of Groups and Spaces, 265, 97–111 (2008) [3] Jørgensen, T.: A note on the subgroups of SL(2, C). Quart. J. Math., 28(2), 209–211 (1977) [4] Alain, M. R.: A Course in p-adic Analysis, Springer-Verlag, New York, 2000 [5] Artin, E.: Algebraic Numbers and Algebraic Function, Nelson, 1968 [6] Martin, G. J.: On discrete Mobius groups in all dimensions: A generalization of Jørgensen’s inequality. Acta Math., 163, 253–289 (1989)