the discrete subgroups and jÃ¸rgensenâs inequality for sl(m,â p )

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 QIUDepartment of Mathematics, Fudan University, Shanghai 200433, P. R. China

E-mail : [email protected]

Jing Hua YANGInstitute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, P. R. China


Yong Cheng YINDepartment of Mathematics, Zhejiang University, Zhejiang 310027, P. R. China


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 non-

elementary group of Mobius 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 22E35, 22E40, 20H10

1 Introduction

Discreteness criteria for a subgroup of the projective special linear group PSL(2, C) is an im-portant topic in the theory of Riemann surfaces and hyperbolic manifolds. Let Qp be thep-adic number field and Cp be the completion of the algebraic closure of Qp. In order to studythe non-Archimedean orbifolds covered by Mumford curves, Kato [1] discussed the discretenesscriteria of groups of projective general linear group PGL(2, Cp). One of the most importantdiscreteness criteria is Jørgensen’s inequality. Armitage and Parker [2] studied Jørgensen in-equality for discrete subgroups of SL(2, Qp), where SL(2, Cp) is regarded as the covering spaceof SL(2, Qp). In this paper, we improve Jørgensen’s inequality for SL(2, Qp) and generalizediscreteness 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 highdimension special linear group, we classify non-unit elements of SL(m, Cp) for m ≥ 2 intoparabolic elements, elliptic elements and loxodromic elements, and we generalize results ofKato 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)

418 Qiu W. Y., et al.

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 groupSL(2, Cp) is that a non-elementary subgroup G of SL(2, C) is discrete if and only if its subgroupgenerated 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 anycyclic subgroup of G, i.e., a subgroup of G generated by one element, is discrete. This result isneither true for SL(2, C) nor true for SL(m, R). However, we prove that for m ≥ 2, a subgroupG of SL(m, Cp) is discrete if and only if any cyclic subgroup of G is discrete, i.e., any subgroupof G generated by one element is discrete.

Theorem 1.2 The subgroup G of SL(m, Cp) is discrete if and only if any cyclic subgroup ofG is discrete.

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

gi =

⎛⎝ ζi 0

0 ζ−1i

⎞⎠ ,

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 orderonly, 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 andgeometric convergence of subgroups of SL(2, C) and the estimation of the volume of hyperbolicmanifolds. Jørgensen’s inequality was generalized by many authors in various cases. Jørgensen’sinequality also plays an important role in the p-adic analytic space. In [2], Armitage and Parkergave a version of Jørgensen’s inequality of discrete subgroups for SL(2, Qp). In this paper, weimprove 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).

The Discrete Subgroups and Jørgensen’s Inequality for SL(m, Cp) 419

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, thenmin{|tr2(A) − 4|, |tr[A, B] − 2|} ≥ p−

2p−1 .

Note that SL(2, Qp) is a subgroup of SL(2, Cp), and SL(2, Cp) is much more complicatedthan SL(2, Qp), since Qp is locally compact, but Cp is not locally compact.

Theorem 1.7 If a subgroup G of SL(m, Cp) is discrete, then for each g ∈ G\{I}, ‖g − I‖ ≥p−

mp−1 .

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 proveTheorems 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 ≥ 2be a prime number. Let Qp be the algebraic closure of the p-adic number field Qp. Let Cp bethe 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 inequalityarise interesting topological and geometrical properties. If x, y and z are points in Cp satisfying|x−y| < |x−z|, then |x−z| = |y−z|. Let D(a, r) := {z||z−a| ≤ r} and D(a, r) := {z||z−a| < r}denote closed and open disks with center at a ∈ Cp and radius r, respectively. In the topologyof Cp, both D(a, r) and D(a, r) are not only closed but also open. Every point in the diskD(a, r) is the center. This means that if x ∈ D(a, r), then D(a, r) = D(x, r) (D(a, r) = D(x, 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 = d0pt 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 subgroupconsisting of the roots of unity in Kp of order prime to p. If the residue degree f of Kp/Qp isfinite, 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 subgroupconsisting of the p-th roots of unity in Kp. If the ramification index e = e(Kp) is finite, thenthe group μp∞(Kp) is finite. More precisely, �(μp(Kp)) ≤ e

1−1/p .

Lemma 2.5 ([4, p. 151]) Let x ∈ Cp with |x| = 1. Then the sequence {xpn!} converges.

Lemma 2.6 The polynomial (Xpd − 1)/(Xpd−1 − 1) is irreducible over Qp.

Proof Let X = t + 1. Then

(Xpd − 1)/(Xpd−1 − 1) = Xpd−1(p−1) + Xpd−1(p−2) + · · · + Xpd−1+ 1

= (t + 1)pd−1(p−1) + (t + 1)pd−1(p−2) + · · · + (t + 1)pd−1+ 1.


Let g(t) = (t + 1)pd−1(p−1) + (t + 1)pd−1(p−2) + · · ·+ (t + 1)pd−1+ 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

of g(t), where 1 ≤ i ≤ pd−1(p − 1). By Lemma 2.2, we know that |ti| = p− 1

pd−1(p−1) . Supposethat 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 ofg1(t) is equal to |t1 · · · ts| = p

− s

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 contradicts0 < s

pd−1(p−1)< 1. This implies that (Xpd − 1)/(Xpd−1 − 1) is irreducible over Qp. �

Lemma 2.7 Let g(z) = zn + an−1zn−1 + · · · + a0 be a polynomial in Cp[z]. Given a fixed

r > 0, if the coefficients of g(z) satisfy |ai| < rn−i, then all roots of the polynomial g(z) are inthe closed disk D(0, r).

Proof If α /∈ D(0, r), then |aiαi| < rn−i|αi| < |αn|. By the ultrametric property, |g(α)| =

|αn + an−1αn−1 + · · · + a0| = |αn| > 0. Then all roots of the polynomial are in the closed disk

D(0, r). �

Lemma 2.8 Let

gn(z) = zm +m−1∑i=0

ainzi (2.1)

be a sequence of polynomials in Cp[z]. If all coefficients ain tend to zero as n → ∞, where0 ≤ 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 anyn > N , |ain| < rn−i, since ain tend to zero as n → ∞, where 0 ≤ i ≤ m − 1. By Lemma 2.7,all roots of gn(z) are in the closed disk D(0, r). Since r > 0 is arbitrary, all roots of gn(z) tendto 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 discretesubgroups of PGL(2, Cp). He classified non-unit element g in PGL(2, Cp) into the followingthree 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 distinctfixed point in P1(Cp).

In the following, we classify non-unit elements in SL(m, Cp), m ≥ 2. Since the product ofall 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 eachnon-unit element g ∈ SL(m, Cp) falls into the following three classes:

(a) g is said to be parabolic if1. the absolute value of any eigenvalue of g is 1, and


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 islarger than 1.

For g = (aij) in the matrix ring M(m, Cp), the norm of g is defined by

‖g‖ = max1≤i≤m,1≤j≤m

{|aij |}.

Obviously, ‖g‖ = 0 implies that each aij = 0. It is easy to verify that ‖αg‖ = |α|‖g‖, ‖g +h‖ ≤max{‖g‖, ‖h‖} and ‖gh‖ ≤ ‖g‖‖h‖.

We say that a subgroup G of SL(m, Cp) is discrete if there exists δ = δ(G) > 0 such thateach 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 ofdistinct elements gn ∈ G is not a Cauchy sequence. Since ‖h−1gnh − h−1gh‖ ≤ ‖h−1‖‖gn −g‖‖h‖, we have ‖h−1gnh−h−1gh‖ → 0, when gn → g, as n → ∞. This means that conjugationdoes not change the discreteness.

In the following, we introduce some results in [1]. Some of them will be extended to higherdimension.

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 ellipticelement 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). Letλ ∈ Cp with |λ| = 1. If f = λI + J , then the sequence 〈fpn!〉 converges.

Proof Since J is a nilpotent matrix, there exists a positive integer N such that JN = 0. Thusfor any positive integer k > N , we have

fk = (λI + J)k = λkI +(

k

1

)λk−1J + · · · +

(k

N

)λk−NJN .

Choose k = pn!. Then(pn!

i

)= pn!

i

(pn!−1i−1

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

integer t such that N < pt, then for any 1 ≤ i ≤ N , |i| > p−t, we have |(pn!

i

)| = |pn!

i ||(pn!−1i−1

)| ≤|pn!

i | ≤ pt−n! for sufficiently large n. Hence(pn!

i

)λpn!−iJ i → 0, as n → ∞, since |λ| = 1. By

Lemma 2.5, {λpn!} converges. Therefore, the sequence 〈fpn!〉 converges. �

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


not change the discreteness, we can assume that g has the Jordan standard form as

g =

⎛⎜⎜⎜⎝

A1

. . .

As

⎞⎟⎟⎟⎠ ,

where each Jordan block Ai has the form λiIi + Ji, where λi is the eigenvalue of g, and Ii

denotes the unit matrix, and Ji is the nilpotent matrix.

By Lemma 3.2, the {Apn!

i } converges. This implies that {gpn!} converges which is a contra-diction. 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

g =

⎛⎜⎜⎜⎝

λ1

. . .

λm

⎞⎟⎟⎟⎠ ,

where λi ∈ Cp are eigenvalues of g with |λi| = 1, 1 ≤ i ≤ m.

Therefore, λsi �= λt

i for any positive integers s, t. By Lemma 2.5, the sequence {λpn!

i }converges. Thus {gpn!} 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 If gn ∈ SL(m, Cp) → I, as n → ∞, then all eigenvalues of gn tend to 1, asn → ∞.

Proof The eigenpolynomial fn(λ) = |λI − gn| tends to polynomial (λ− 1)m, since gn tends toI. 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 → ∞. ByLemma 3.4, we know that eigenvalues λ1,n, λ2,n, . . . , λm,n of gn tend to 1 which contradicts thehypothesis. Thus G is discrete. �

Theorem 3.6 The subgroup G of SL(m, Cp) is discrete if and only if any cyclic subgroup ofG is discrete.

Proof 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 aparabolic element is not discrete. Hence the subgroup G does not contain any elliptic element ofinfinite order and any parabolic element, which yields that there only exist loxodromic elementsor elliptic elements of finite order.

If g is a loxodromic element, then g has at least one eigenvalue whose absolute value islarger 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, gn = I, where n is the smallest


positive integer, then we can assume that g has the form as

g =

⎛⎜⎜⎜⎝

λ1

. . .

λm

⎞⎟⎟⎟⎠ ,

where λi is an eigenvalue of g, 1 ≤ i ≤ m.

Since gn = I, each eigenvalue λi of g satisfies λni = 1, where 1 ≤ i ≤ m. Let n = d0p

t ≥ 2,where d0 and t are non-negative integers, and d0 is prime to p. At least one of the eigenvaluesof g is n-th primitive root of unity, since n is the smallest positive integer. Take λ the n-thprimitive root of unity. If d0 = 1, namely, n = pt, then λ is a primitive pt-th root of unity. ByLemma 2.2, |λ − 1| = p

− 1pt−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 truein the isometric group of complex hyperbolic manifold in 2 or higher dimensions. In the non-Archimedean setting, we prove that this result is true for the finite extension of Qp, but this isnot true for Cp.

The following lemma shows that for some fixed positive integer n, there only exist finitelymany extensions of Qp of degree n.

Lemma 3.7 ([4]) For a given integer n ≥ 1, there are only finitely many extensions Kp ofdegree n of Qp.

Lemma 3.8 Let Kp be a finite extension of degree t of Qp. If H be the union of all finiteextensions Kp with [Kp : Kp] ≤ 2, then the number of primite roots of unity in H is finite.

Proof Since [Kp : Kp] ≤ 2 and [Kp : Qp] ≤ t, we have [Kp : 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 isfinite. By Lemma 3.7, the number of fields of finite extension of Qp of degree less than 2t isfinite. Hence the number of primitive roots of unity in H is finite. �

Lemma 3.9 If λ is the eigenvalue of the elliptic element g of finite order, then p−2

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 rootof unity. By Lemmas 2.1 and 2.2, p−

1p−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|.Hence p−

2p−1 ≤ |tr(g) − 2| ≤ 1. �

If g has an eigenvalue is −1, then the other eigenvalue should also be −1, since the deter-minant 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 Let Kp be a finite extension of Kp with [Kp : Kp] ≤ 2. Then Kp is a finite extension of


Qp. Since Qp is locally compact, Kp is locally compact. By Lemma 3.7, there are only finitelymany such Kp. Let H be union of all such Kp. This implies that H is locally compact.

Take a fixed element g ∈ G\{I}. Suppose that there exists an element h ∈ G which can notcommutate with g. Then g can not be −I or I. So the eigenvalue λ of g satisfies λ2 �= 1. Thusg, h can be conjugated to

g =

⎛⎝ λ 0

0 λ−1

⎞⎠ , h =

⎛⎝ a b

c d

⎞⎠ ∈ SL(2, Kp(λ)).

Then the commutator

[g, h] = ghg−1h−1 =

⎛⎝ ad − bcλ−2 −abλ2 + ab

cd − cdλ−2 −bcλ2 + ad

⎞⎠ .

Therefore, tr[g, h] = 2ad−bc(λ2+ 1λ2 ) = 2−bc[(λ+ 1

λ)2−4] = 2−bc(λ− 1λ)2. By Lemma 3.9,

p−2

p−1 ≤ |bc(λ− 1λ )2| ≤ 1. Since λ2 is also a primitive root of unity and λ2 �= 1, by Lemmas 2.1

and 2.2, we have p−1

p−1 ≤ |λ2 − 1|/|λ| = |λ2 − 1| ≤ 1. Therefore, p−2

p−1 ≤ |bc| ≤ p1

p−1 .

Suppose that there exist infinitely many distinct elements hn which can not commutatewith g, and let hn have the following form

hn =

⎛⎝ an bn

cn dn

⎞⎠ .

Since andn − bncn = 1 and bncn are bounded, andn is also bounded. We also have an, dn arebounded, since an + dn is bounded.

Suppose that bn, cn are bounded, then an, dn, bn, cn are all bounded. Since Qp is locallycompact, an, dn, bn, cn have convergent subsequences. Then hn has the convergent subsequencewhich contradicts the discreteness of G.

Suppose that {bn} or {cn} is unbounded, without loss of generality, we suppose that bn → ∞,as n → ∞. Since p−

2p−1 ≤ |bncn| ≤ p

1p−1 , cn → 0, as n → ∞. Consider the sequence {h1hn}.

Since

h1hn =

⎛⎝ a1an + b1cn a1bn + b1dn

anc1 + d1cn bnc1 + d1dn

⎞⎠ ,

then tr[h1hn] = a1an + b1cn + bnc1 + d1dn. Since b1, c1 are nonzero and an, dn are bounded,then a1an + b1cn + bnc1 + d1dn → ∞, as n → ∞. Therefore, when n is sufficiently large, h1hn

is a loxodromic element which contradicts that G has elliptic elements only. Hence there donot exist infinitely many elements which can not commutate with g. Suppose that h ∈ G cancommutate with g. Then h and g can be conjugated to diagonal matrices simultaneously. Sinceeigenvalues of h ∈ G are primitive roots of unity in Kp, then by Lemma 3.8, there exist finitelymany 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 infiniteextensions of Qp. We can find infinitely many primitive roots, but the distance between all theprimitive roots and unity is larger than a fixed positive number.


Example 3.11 Let ζi be the primitive pi-th root of unity, where i = 1, 2, 3, . . .. If G isgenerated by

gi =

⎛⎝ ζi 0

0 ζ−1i

⎞⎠ ,

then G is discrete.

Proof Since G is generated by gi, then for each g ∈ G, g = gε1i1

gε2i2

· · · gεnin

, where εi ∈ {−1, 1}.Then eigenvalues of g are ζε1

i1ζε2i2

· · · ζεnin

and the reciprocal which are roots of unity. Hence eachelement 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 isdifferent 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-Archimedeanmetric 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 diagonalmatrix. Let B be any element of SL(2, Qp) so that, when acting on Qp ∪ {∞} via Mobiustransformations, 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 ofJørgensen’s inequality for SL(m, Cp) by algebraic method.

Lemma 4.2 Let λ be the n-th primitive root of unity with [Qp(λ) : Qp] ≤ 2, where n ≥ 2.Then

(1) If p > 3, then |λ − 1| = 1;

(2) If p = 3, then |λ − 1| ≥ 1√3;

(3) If p = 2, then |λ − 1| ≥ 12 .

Proof Let n = d0pd, 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 theirreducible polynomial Xpd−1

Xpd−1−1= 0. Hence [Qp(λ) : Qp] = pd−1(p − 1). It is divided into

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) ≤ 2implies that d = 1. Therefore, λ is the 3-th primitive root of unity, and then |λ − 1| = 1√

3.

If p = 2, then [Qp(λ) : Qp] = 2d−1(2 − 1) ≤ 2 deduces that d = 1, or d = 2. If λ is the2-th primitive root of unity, then |λ − 1| = 1

2 . If λ is the 22-th primitive root of unity, then|λ − 1| = 2−

12 . We draw our conclusion. �

According to our results, the discrete subgroup does not contain any parabolic elementwhich yields that a generator A ∈ SL(2, Cp) can be conjugated to a diagonal matrix. If thesubgroup G is generated by −I and B ∈ SL(2, Cp), then the group G = {(−1)iBj} is verytrivial. Hence we do not consider −I as the generator.


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−1B−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 ellipticelements of finite order. Let λ and 1

λ be eigenvalues of A. If A is a loxodromic element, thenthe absolute value of one of the eigenvalues is larger than 1. Without loss of generality, we canassume that |λ| > 1. Since |λ− 1

λ | = |λ| > 1, we have |tr2(A)− 4| = |(λ+ 1λ)2 − 4| = |λ− 1

λ |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 impliesthat |tr2(A)− 4| = |(λ+ 1

λ)2 − 4| = |λ2 − 1|2/|λ|2 ≥ 1; when p = 3, by Lemma 4.2, |λ− 1| ≥ 1√3

implies that |tr2(A) − 4| = |(λ + 1λ )2 − 4| = |λ2 − 1|2/|λ|2 ≥ 1

3 ; when p = 2, by Lemma 4.2,|λ2 − 1| ≥ 1

2 implies that |tr2(A) − 4| = |(λ + 1λ )2 − 4| = |λ2 − 1|2/|λ|2 ≥ 1

4 .

Let μ and μ−1 be eigenvalues of [A, B]. Similar to the analysis above, when [A, B] is aloxodromic 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| ≥ 1

3 ; 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, thenmin{|tr2(A) − 4|, |tr[A, B] − 2|} ≥ p−

2p−1 .

Proof If [A, B] = I, then ABA−1B−1 = I. This implies that AB = BA which means B canfix or interchange the fixed point of A which contradicts the hypothesis. Hence [A, B] �= I.

Let λ and 1λ 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, wecan assume that |λ| > 1. Hence |λ − 1

λ | = |λ| > 1, and then |tr2(A) − 4| = |(λ + 1λ )2 − 4| =

|λ − 1λ |2 = |λ| > 1. If A is an elliptic element of finite order, then λ is the n-th primitive

root of unity. Let n = d0pd, 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 ofunity, where d is a positive integer, then by Lemma 2.2, |λ2 − 1| = p

− 1pd−1(p−1) ≥ p−

1p−1 . Hence

|tr2(A)−4| = |(λ+ 1λ)2−4| = |λ2−1|2/|λ|2 ≥ p−

2p−1 . When d0 > 1, since λ2 is the d0p

d-th rootof unity, then by Lemma 2.1, |λ2 − 1| = 1. Hence |tr2(A) − 4| = |λ2 − 1|2/|λ|2 = 1 ≥ p−

2p−1 .

Let ζ and ζ−1 be the eigenvalues of the [A, B]. Since [A, B] �= I, ζ �= 1. If [A, B] is aloxodromic element, then the absolute value of one of the eigenvalues is larger than 1. Withoutloss of generality, we can assume that |ζ| > 1. Hence |ζ − 1

ζ | = |ζ| > 1 which implies that|tr[A, B] − 2| = |(ζ + 1

ζ ) − 2| = |ζ − 1|2/|ζ| = |ζ| > 1. If [A, B] is an elliptic element of finiteorder, then ζ is an n-th primitive root of unity. Let n = d0p

d, where d0 is a positive integerwhich is prime to p, and d is a non-negative integer. There are two different cases. If d0 = 1,


namely, ζ is the pd-th root of unity, then by Lemma 2.2, |ζ − 1| = p− 1

pd−1(p−1) ≥ p−1

p−1 . Hence|tr[A, B] − 2| = |ζ + 1

ζ − 2| = |ζ − 1|2/|ζ| ≥ p−2

p−1 . If d0 > 1, namely, ζ is a d0pd-th primitive

root of unity, then by Lemma 2.1, |ζ − 1| = 1. Therefore,

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

p−1 . �

Furthermore, we give a Jørgensen’s inequality of discrete subgroup of SL(m, Cp) as an isola-tion of identity. Although we can not control the matrix by the trace in higher dimensions, wecan 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 Mobius transform in higherdimensions.

Theorem 4.5 ([6, Theorem 4.5]) Let f and g be Mobius transformations of Sn. If f and g

together generate a discrete non-elementary group, then max{‖gifg−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 elementis too close to the identity, then the eigenvalue is very close to 1. In [6], Martin discussed thegroup generated by finitely many elements, and estimated the maximum distance between thegenerator and the identity.

Lemma 4.6 If g ∈ SL(m, Cp) and ‖g − I‖ < p−m

p−1 , then all eigenvalues of g are in D(1, r),where r < p−

1p−1 .

Proof Let g = (bij) ∈ SL(m, Cp). Since ‖g − I‖ < p−m

p−1 , then |bij − δij | < p−m

p−1 , whereδij = 1, if i = j; otherwise δij = 0, if i �= j.

Then eigenpolynomial

|λI − g| =

∣∣∣∣∣∣∣∣∣∣∣∣

λ − b11 −b12 . . . −b1m

−b21 λ − b22 · · · −b2m

.... . . . . .

...

. . . . . . −bm(m−1) λ − bmm

∣∣∣∣∣∣∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣∣∣∣∣

(λ − 1) + 1 − b11 −b12 . . . −b1m

−b21 (λ − 1) + 1 − b22 · · · −b2m

.... . . . . .

...

. . . . . . −bm(m−1) (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

by product or addition. By the ultrametric property, we have |ai| ≤ max{|cij |} < p−m

p−1 . Since|ai| 1

m−i ≤ p−m

(m−i)(p−1) ≤ p−1

p−1 , there exists a positive number r satisfying 0 < r < p−1

p−1 suchthat |ai| < rm−i. By Lemma 2.7, each eigenvalue of g is in D(1, r). �

Theorem 4.7 If a subgroup G of SL(m, Cp) is discrete, then for each g ∈ G\{I}, ‖g − I‖ ≥p−

mp−1 .


Proof By Theorem 3.3, we know that each element in G is either a loxodromic element or anelliptic element of finite order. If g in G is a loxodromic element, then there exists at least oneeigenvalue λ whose absolute value is larger than 1. Hence |λ − 1| = |λ| > 1. If g is an ellipticelement of finite order, then there exists at least one eigenvalue λ which is an n-th primitiveroot of unity. Let n = d0p

d, where d0 is a positive integer which is prime to p, and d is anon-negative integer. If d0 = 1, then by Lemma 2.2, |λ− 1| = p

− 1pd−1(p−1) ≥ p−

1(p−1) . If d0 > 1,

then by Lemma 2.1, |λ − 1| = 1. Therefore, |λ − 1| ≥ p−1

p−1 . This shows that λ, which isone of the eigenvalues of g, satisfies |λ − 1| ≥ p−

1(p−1) . Hence by Lemma 4.6, we know that

‖g − I‖ ≥ p−m

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)

the discrete subgroups and jÃ¸rgensenâs inequality for sl(m,â p )

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