Covolumes of nonuniform lattices in PU(n, 1)

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1 Covoumes of nonuniform attices in PU(n, 1) Vincent Emery Université de Genève 2-4 rue du Lièvre, CP Genève 4, Switzerand Matthew Stover University of Michigan 530 Church Street Ann Arbor, MI February 7, 2012 Abstract This paper studies the covoumes of nonuniform arithmetic attices in PU(n, 1). We determine the smaest covoume nonuniform arithmetic attices for each n, the number of minima covoume attices for each n, and study the growth of the minima covoume as n varies. In particuar, there is a unique attice (up to isomorphism) in PU(9, 1) of smaest Euer Poincaré characteristic amongst a nonuniform arithmetic attices in PU(n, 1). We aso show that for each even n there are arbitrariy arge famiies of nonisomorphic maxima nonuniform attices in PU(n, 1) of equa covoume. 1 Introduction In recent years, voumes of ocay symmetric spaces have received a great dea of attention. One basic probem is to determine the smaest voume quotients of a given symmetric space of noncompact type by the action of a discrete group of isometries. A compete understanding of the smaest voume quotients of the hyperboic pane foows from cassica techniques. However, even for hyperboic 3-space a compete picture of the smaest voume orbifods and manifods was finished ony very recenty [20, 21]. For progress in higher-dimensiona hyperboic spaces see [15]. One observation is that a known smaest voume quotients of rea hyperboic n-space come from attices that are arithmeticay defined. The Supported by SNSF projects PP00P /1 and PBFRP Partiay supported by NSF RTG grant DMS

2 determination of the smaest voume arithmetic hyperboic n-orbifods is now compete [8, 9, 4, 11, 5], and some progress has been made for compex hyperboic space [26, 28, 29, 30]. The purpose of this paper is to study the distribution of voumes of noncompact arithmetic quotients of compex hyperboic space H n C and ocate the smaest amongst them. In other words, we study covoumes of nonuniform arithmetic attices Γ in PU(n, 1). Indeed, PU(n, 1) is the group of hoomorphic isometries of H n C and the covoumes of Hn C /Γ and PU(n, 1)/Γ are reated by an expicit constant (see 5.1). Every nonuniform arithmetic subgroup Γ of PU(n, 1) is defined over Q. More precisey, for each such Γ, there is an associated imaginary quadratic fied = Q ( d ) and a hermitian form on n+1 ; see 2 for the precise construction. We wi say that Γ is associated with /Q to emphasize, even though the agebraic group is defined rationay. We first introduce some notation necessary to state our resuts. For each imaginary quadratic fied, et L (n) be the set of isomorphism casses of nonuniform arithmetic attices in PU(n, 1) associated with /Q. As normaization for the voume we use the Euer Poincaré measure, i.e., the Haar measure µ EP on PU(n, 1) such that µ EP (PU(n, 1)/Γ) = χ(γ) for any attice Γ in PU(n, 1); here χ denotes the Euer Poincaré characteristic in the sense of C. T. C. Wa. Note that χ(γ) = ( 1) n χ(γ) and the compex hyperboic voume of H n C /Γ can be computed from χ(γ) via the Chern Gauss Bonnet Theorem (see for instance [14]). Denote by ν (n) the minimum of χ(γ) among a attices in L (n). Furthermore, et n (n) be the mutipicity of ν (n), that is, the number of isomorphism casses in L (n) of covoume ν (n). Note that since PU(n, 1) has a nontrivia outer automorphism, a singe eement Γ L (n) determines two conjugacy casses of embeddings in PU(n, 1) (corresponding to the two possibe compex structures on the orbifod H n C /Γ), but no more by Mostow Prasad rigidity. To give an exact formua for the covoume, we consider the L-function L = ζ /ζ associated with /Q, where ζ is the Dedekind zeta function of and ζ = ζ Q is the Riemann zeta function. Let D be the absoute vaue of the discriminant of and r the number of finite primes ramified in /Q, i.e., the number of primes dividing D. Finay, et h,n+1 be the order of the subgroup of the cass group of consisting of eements whose order divides n + 1. Theorem 1.1. Suppose that n 2 is even. For each imaginary quadratic fied, a attices in the set L (n) are commensurabe. The minima Euer 2

3 Poincaré characteristic among these attices is given by ν (n) = n n h,n+1 n/2 ζ(1 2j)L ( 2j). j=1 For the number n (n) of attices in L (n) reaizing the minima covoume ν (n) we have 2 r n (n) 2 r h (n), where h (n) is a divisor of h,n+1. In particuar, if h,n+1 = 1 we have n (n) = 2 r. It is a cassica fact that for a j > 0, ζ(1 2j) and L ( 2j) are rationa mutipes of Bernoui and generaized Bernoui numbers. See [17, 16.6]. Since r is the number of primes dividing the discriminant of, it becomes arbitrariy arge as we vary. Indeed, the discriminant of Q( d) (with d square free) equas d or 4d according to whether or not d is congruent to 3 moduo 4. Therefore, we immediatey obtain the foowing coroary. Coroary 1.2. For a even n there exist arbitrariy arge famiies of nonisomorphic maxima attices in PU(n, 1) of the same covoume. Reca that Wang s Theorem [34] states that for each n > 1 there are ony finitey many attices in PU(n, 1) with covoume bounded above by any constant. Therefore, the covoumes in Coroary 1.2 necessariy grow with the size of the famiy. See [12] for an anaogue of this coroary for nonmaxima attices in more genera semisimpe Lie groups. In odd dimensions, our resuts are sighty weaker than Theorem 1.1, due to the more deicate arithmetic of the attices under consideration. In this setting we prove the foowing. Theorem 1.3. Suppose that n 3 is odd and that is imaginary quadratic. Then ν (n) = ( 1) n+1 2 (n + 1)ɛ (n) 2 n h,n+1 ζ( n) (n 1)/2 j=1 ζ(1 2j)L ( 2j), where ɛ (n) 2 is a divisor of 2 r. In particuar ɛ (n) = 2 if /Q is ramified at exacty one rationa prime. Suppose that /Q is ramified at exacty one prime. We have h (n)/2 n (n) h (n), where h (n) is a divisor of h,n+1 if 8 n + 1, and h (n) is a divisor of 2h,n+1 otherwise. One can again express this quantity in terms of Bernoui and generaized Bernoui numbers. Finay, using Theorems 1.1 and 1.3 it is possibe 3

4 to competey determine the minima covoume arithmetic attices in any dimension. This gives the foowing theorem, which generaizes the case n = 2 studied in [30]. Theorem 1.4. For a n > 1, the smaest covoume nonuniform arithmetic attices in PU(n, 1) are associated with Q( 3)/Q. There are exacty two isomorphism casses of minima covoume attices when n is even. There are at most two when n 7 (mod 8). For a other n there is exacty one. Note that for = Q( 3) we have h = r = 1, so that in every dimension we get an expicit formua for the Euer Poincaré characteristic of the attices with smaest covoume. The case n = 1 is exceptiona. Indeed, Q(i) determines the unique commensurabiity cass of nonuniform arithmetic attices in PU(1, 1), which is the conjugacy cass of the moduar group. There is exacty one orbifod of minima voume, namey, the moduar surface. We aso note that ony finitey many commensurabiity casses of nonarithmetic attices in PU(n, 1) are known. Between the seven attices constructed by Deigne and Mostow and the two new exampes of Deraux Parker Paupert [23, 10], there are between six and nine known commensurabiity casses of nonarithmetic attices in dimension 2. There is one known in dimension 3 [23], and there are no exampes for higher n. These do not give attices of smaer covoume than the minima covoume arithmetic attices (see the remark in [30]). We now briefy describe the strategy for proving the above resuts. The first step is to consider Prasad s voume formua [27], which gives the covoume of a so-caed principa arithmetic attice. It is known that any maxima attice is the normaizer of a principa arithmetic attice, and the index is determined by anaysis of Gaois cohomoogy, particuary the resuts of Bore and Prasad [7]. Our resuts on the number of attices reaizing the minima covoume requires further cose anaysis of automorphisms of agebraic groups and Gaois cohomoogy. We cose the paper in 5 by studying the behavior of the smaest covoume as n varies. We prove that the smaest Euer Poincaré characteristic (in absoute vaue) is reaized by the unique attice of smaest covoume in L Q( 3) (9). Let Γ 0 denote this attice. It is commensurabe with the attice acting on H 9 C constructed by Deigne Mostow [23]. This Deigne Mostow attice aso appears in work of Danie Acock [3]. It turns out that Γ 0 aso has smaest overa covoume with respect to the compex hyperboic metric of constant bihoomorphic curvature 1. 4

5 Theorem 1.5. Let Γ 0 < PU(9, 1) be as above, and normaize the hoomorphic sectiona curvatures of H n C to 1. For any n 2, et Γ < PU(n, 1) be a nonuniform arithmetic attice. Then χ(γ) χ(γ 0 ) and vo(h n C /Γ) vo(h 9 C /Γ 0) with equaity if and ony if Γ is isomorphic to Γ 0. The Euer Poincaré characteristic and compex hyperboic covoume of Γ 0 are: χ(γ 0 ) = vo(h 9 C /Γ 0) = 809 5, 746, 705, 367, ; 809π 9 79, 550, 340, 408, We aso determine the asymptotic behavior of the minima covoume and compare it to known voume bounds for compex hyperboic n-orbifods. In particuar, we prove the foowing. Theorem 1.6. As n, the ratio q (n) = ν (n + 1)/ν (n) grows superexponentiay. That is, for each α > 0 there exist n 0 N such that q (n) > e αn for a n n 0. In particuar, ν (n) grows super-exponentiay. Acknowedgments This work was started at the Oberwofach Seminar On arithmeticay defined hyperboic manifods and was competed during the London Mathematica Society EPSRC Durham Symposium Geometry and arithmetic of attices. We thank the organizers of both of these exceent workshops. We aso thank Gopa Prasad for conversations reated to this work and the referees for many hepfu suggestions. 2 Arithmetic subgroups of SU(n, 1) Whie our goa is to anayze covoumes of attices in PU(n, 1), many resuts on the structure of arithmetic subgroups require one to consider attices in a simpy connected agebraic group. Therefore, we must spend much of this paper instead considering attices in SU(n, 1). We ater return to PU(n, 1) via the (n + 1)-fod covering SU(n, 1) PU(n, 1). 2.1 Hermitian forms and attices in SU(n, 1) Let G be a semisimpe Q-agebraic group. Then G(Z) is a attice in the rea Lie group G(R) [6]. Any Γ < G(R) commensurabe with G(Z) is caed an arithmetic attice in G(R). The agebraic groups of interest for us are 5

6 isotropic of type 2 A n [32]. A such groups are constructed as foows (cf., [29, 4.4]). Proposition 2.1. Let Γ < SU(n, 1) be a nonuniform arithmetic attice. Then, there exists an imaginary quadratic fied and a hermitian form h of signature (n, 1) on n+1 such that Γ is commensurabe with G(Z), where G is the Q-agebraic group SU(h) of specia unitary automorphisms of h. We now describe how the cassification of unitary groups over nonarchimedean oca fieds determines the commensurabiity cass of an arithmetic attice. If p is a prime that spits in, i.e., a prime that spits as a product of two distinct prime ideas in, then SU(h) is isomorphic over Q p to SL n+1 (Q p ). If p does not spit, et p be the unique prime idea of over p. Then the competion p of at p is a quadratic extension of Q p and G(Q p ) is. For a nonspit pace, the foowing cassifies the possibe nonarchimedean competions of G. the unitary group of h considered as a hermitian form on n+1 p Proposition 2.2. Let L/Q p be a quadratic extension, h 1, h 2 be hermitian forms on L n+1, and G 1, G 2 the associated Q p -agebraic groups. If n is even, then G 1 (Q p ) = G 2 (Q p ) and they are quasi-spit. If n is odd, then G 1 (Q p ) = G 2 (Q p ) if and ony if det(h 1 ) is congruent to det(h 2 ) moduo N L/Qp (L ) and G j is quasi-spit if and ony if det(h j ) is a norm from L. It foows from the isomorphism between N L/Qp (L )/Q p and Ga(L/Q p ) in eementary oca cass fied theory [31, Ch. 5] that there is exacty one isomorphism cass when n is even and there are exacty two when n is odd. This has the foowing consequence for the cassification of nonuniform arithmetic attices in SU(n, 1). Coroary 2.3. If n is even, associated with each fied there is exacty one commensurabiity cass of nonuniform arithmetic attices in SU(n, 1). When n is odd, commensurabiity casses of nonuniform arithmetic attices in SU(n, 1) are in one-to-one correspondence with eements of Q / N /Q ( ). Let be an imaginary quadratic fied, h a hermitian form on n+1, and T be the finite set of (necessariy non-spit) paces p of Q where det(h) fais to be a oca norm from p, where p is the unique prime of above p. Let G,T be the associated Q-agebraic group. Then G,T fais to be quasi-spit precisey at the paces in T. When T =, which is aways the case for n even, we write G instead. Then G is the unique isotropic Q-agebraic group with the foowing properties: 6

7 (i) G (R) = SU(n, 1); (ii) G becomes an inner form over ; (iii) G is quasi-spit over Q p for every rationa prime p. For T, G,T satisfies (i) and (ii), but not (iii). Now, we refine our description of the oca structure of G,T. This wi be crucia for parametrizing and studying the covoumes of the various maxima attices in a commensurabiity cass. This oca structure can be understood by the oca index attached to semisimpe groups over oca fieds, foowing Bruhat Tits theory. We wi use the notation and cassification given in [32, 4]. We begin with G, i.e., the case T =. If p is unramified in /Q the oca index of G /Q p is A n or 2 A n according to whether or not G /Q p spits. Note that the oca index 2 A n differs with the parity of n. Now, suppose that p is ramified in /Q. Then G /Q p has oca index C BC m if n = 2m and B C m if n = 2m 1 (resp. C B 2 for n = 3). The atter index has a nontrivia automorphism, whereas the former is symmetry-free. Now consider the case T. When G,T /Q p is quasi-spit, its oca structure is simiar to G,T /Q p. If G,T /Q p is not quasi-spit it has oca index 2 B C m or 2 A 2m 1 according to whether or not p ramifies in /Q. 2.2 Principa arithmetic attices Let be an imaginary quadratic fied and G,T the Q-agebraic group associated with a hermitian form on n+1 as above. We now describe the principa arithmetic attices in G,T (Q). Let V f be the set of finite primes of Q. For each p V f, et P p be a parahoric subgroup of G,T (Q p ) [33, 3.1], and set P = (P p ) p Vf. We assume that P is coherent, so the restricted direct product p P p defines an open compact subgroup of G,T (A f ), where A f are the finite adees of Q. Embed G,T (Q) in G,T (A f ) diagonay. By construction the subgroup Λ,T (P ) = G,T (Q) p V f P p of G,T (R) = SU(n, 1) is commensurabe with G,T (Z). These are the socaed principa arithmetic attices. 7

8 3 Covoumes of nonuniform attices 3.1 Covoumes of principa arithmetic attices We retain a notation from 2. In particuar, is an imaginary quadratic fied, G,T is the Q-agebraic group determined by a hermitian form on n+1 such that G,T fais to be quasi-spit precisey at the rationa primes in T, and G denotes the unique such group that is quasi-spit at every rationa prime p (i.e., where T = ). Let P,T (resp., P if T = ) be the set of coherent coections P = (P p ) p Vf such that each parahoric subgroup P p is of maxima voume in G,T (Q p ). Then, for P P,T, a subgroups P p are specia [7, A.5]. Moreover, if p T is not ramified in /Q, then P p is necessariy hyperspecia [33, 3.8]. For P P,T we consider the principa arithmetic subgroup Λ,T = Λ,T (P ) (see 2.2). When T =, set Λ = Λ,T. The subgroup Λ,T depends on the choice of P P,T. However, by Prasad s voume formua, its covoume does not depend on P P,T. Let s be n(n + 3)/4 when n is even and (n 1)(n + 2)/4 when n is odd. For µ normaized as in [27] we get that µ(su(n, 1)/Λ ) equas: n D s j=1 j! (2π) j+1 ζ(2)l (3) { ζ(n + 1) n odd L (n + 1) n even (1) where L is the Dirichet L-function attached to the quadratic extension /Q and D is the absoute vaue of the discriminant of. We reate this to the Euer Poincaré measure described in our introduction in 3.6 beow. See aso [7, 4]. The covoume of Λ,T differs from the covoume of Λ by a product of rationa factors λ p > 1 (p T): µ (SU(n, 1)/Λ,T ) = µ (SU(n, 1)/Λ ) p T λ p. (2) More precisey, each factor λ p is given by λ p = p(dim Mp+dim Mp)/2 M p (F p ) M p (F p ), (3) p dim Mp where the reductive F p -groups M p and M p are defined as foows. Let G p (resp., G p ) be the F p -group associated with G,T (resp., the group of which G,T is an inner form) as in [33, 3.5] and R(G p ) (resp., R(G p )) be the 8

9 unipotent radica. Then M p (resp., M p ) is the reductive F p -subgroup so that ( G p = M p R(G p ) resp., Gp = M p R(G p ) ). See [27, 2.2] for more detais. Note that Λ,T (incuding T = ) is of minima covoume among principa arithmetic subgroups of G,T. However, as we sha see, Λ,T is usuay not maxima in SU(n, 1). The normaizer Γ,T = N SU(n,1) (Λ,T ) is a maxima arithmetic subgroup of SU(n, 1) (as usua, we write Γ when T = ). In fact, resuts from [7] can be used to show that the indices [Γ,T : Λ,T ] do not depend on the choice of the coherent coections P P,T and that Γ,T reaizes the minima covoume in its commensurabiity cass (see [11, 12.3] for detais). 3.2 Index computations for normaizers of principa arithmetic attices See [11, Ch. 12] for an expanation of how to estimate the indices [Γ,T : Λ,T ] based on the work of Bore Prasad [7]. The computation of these indices invoves the center of the group G,T, which we denote by C. Let µ n+1 be the -group of units of order n + 1. The center of G,T does not depend on T and is Q-isomorphic to the group Res (1) /Q (µ n+1) of norm 1 eements in the restriction of scaars from to Q. We first restrict our attention to the computation of [Γ : Λ ]. Unti the end of 3.5 we consider the agebraic group G. In particuar, G /Q p is aways quasi-spit. For each finite prime p, the Gaois cohomoogy group H 1 (Q p, C ) acts on the oca Dynkin diagram p of G (Q p ). The corresponding homomorphism is denoted ξ p : H 1 (Q p, C ) Aut( p ). The map H 1 (Q, C ) H 1 (Q p, C ) determines a natura diagona homomorphism ξ : H 1 (Q, C ) p Aut( p ). (4) Let A = ker ( H 1 (Q, C ) H 1 (R, C ) ) and A ξ be the kerne of ξ restricted to A. We then have the foowing. Lemma 3.1. Let Λ and Γ be as above. Then [Γ : Λ ] = n + 1 µ n+1 () A ξ. (5) 9

10 Proof. By [7, Prop. 2.9] and [11, Lemme 12.1] we have [Γ : Λ ] = C (R)/ C (Q) A ξ. (6) The group C (Q) = Res (1) /Q (µ n+1)(q) corresponds to the set of eements x µ n+1 () for which N /Q (x) = 1. Since /Q is imaginary quadratic, we have N /Q (µ n+1 ()) = 1 and hence C (Q) = µ n+1 (). The same argument shows that C (R) is isomorphic to µ n+1 (C) and hence has order n+1. Then This proves the emma. C (R)/ C (Q) = n + 1 µ n+1 (). (7) 3.3 Actions on p Define the foowing subgroup of : L = { x / N /Q (x) (Q ) n+1}. (8) The first Gaois cohomoogy groups H 1 (Q, C ) and H 1 (R, C ) are described by foowing diagram, with exact rows: 1 µ n+1 (Q) H 1 (Q, C ) L/( ) n+1 1 = 1 µ n+1 (R) H 1 (R, C ) 1 1 From this we see that the first row spits and the subgroup A H 1 (Q, C ) appearing in 3.2 can be identified with L/( ) n+1. Then we get an action of L on every diagram p. The kerne A ξ fits into the foowing exact sequence: 1 A ξ L/( ) n+1 p Aut( p ) 1. (9) Now suppose that p is a finite prime. The group H 1 (Q p, C ) is described by the exact sequence: where 1 N p H 1 (Q p, C ) W p 1, (10) N p = µ n+1 (Q p )/ N /Q (µ n+1 ( Q p )), (11) ( ) W p = ker ( Q p ) /(( Q p ) ) n+1 N /Q Q p /(Q p ) n+1. (12) 10

11 Reca that Q p is isomorphic to p p p, which is a fied if p is either inert or ramified in /Q, or equa to two copies of Q p if p = pp spits. In the atter case the norm N /Q on Q p = p p is obtained by mutipying the two components together, and the group W p is isomorphic to Q p /(Q p ) n+1. Lemma 3.2. The subgroup N p H 1 (Q p, C ) acts triviay on p for each prime p, i.e., N p H 1 (Q p, C ) ξp. Proof. A proof for p not ramified in /Q is given in [7, 5.3]. Suppose that p is ramified in /Q. If n = 2m is even the oca index of G /Q p is C BC m, for which we have Aut( p ) = 1. So in this case the emma triviay hods. Assume that n is odd. Let T be the Q p -group defined as the kerne of the map G m Res /Q (G m ) G m (x, y) x n+1 N /Q (y) 1, where G m denotes the mutipicative group. We get a commutative diagram with exact rows: 1 C Res /Q (µ n+1 ) N /Q µ n C Res /Q (G m ) θ T 1, the homomorphism θ being θ(z) = (N /Q (z), z n+1 ), and the embedding µ n+1 T by x (x, 1). Passing to cohomoogy, the first row gives the sequence (10), whereas the second gives an isomorphism between H 1 (Q p, C ) and a quotient of T(Q p ) = { (x, y) Q p p x n+1 = N /Q (y) }. We refer to [18, Prop. (30.13)] for detais. Thus, an eement of H 1 (Q p, C ) can be represented by an equivaence cass [x, y] of eements (x, y) T(Q p ). Let V be a p -vector space of dimension n + 1 and h the hermitian form on V given by the matrix [ 0 Id Id 0 where Id is the identity matrix of dimension n+1 2. If G denotes the adjoint group of G, then G (Q p ) is PGU(V, h) = GU(V, h)/ p, where GU(V, h) is the group of unitary simiitudes of (V, h). By [18, 31.A] the connecting homomorphism G (Q p ) H 1 (Q p, C ) has the form ], g mod p [µ(g), N /Q (det(g))], (13) 11

12 where µ(g) is the mutipier of g GU(V, h). Let x µ n+1 (Q p ). The eement of g G (Q p ) given by the diagona matrix diag(1,..., 1, x,..., x) mod p is ceary contained in a compact torus of G (Q p ). Hence it acts triviay on p (see [33, 2.5]), as does its image in H 1 (Q p, C ). From (13) we see that this image is [x, 1] H 1 (Q p, C ). This eement is aso obtained as the image of x under the composition µ n+1 (Q p ) T(Q p ) H 1 (Q p, C ). We concude that µ n+1 (Q p ) and its quotient N p act triviay on p. 3.4 Determining A ξ By Lemma 3.2, the homomorphism ξ p defines an action of W p on p. We now describe the kerne of this action. Let Ram be the set of finite primes of Q which ramify in /Q. This is a finite non-empty set. Reca from the introduction that # Ram = r. A nonarchimedean vauation v of is normaized if v( ) = Z. In the foowing, for a prime p of above p, the vauation v p wi denote the normaized vauation associated with p. We denote the p-adic integers of p by O p. Since our group G is quasi-spit over every p-adic fied Q p, we have the foowing consequence of [7, Lemma 2.3 and Prop. 2.7]. Lemma 3.3. Let p be a finite prime with p Ram. Then the kerne of the homomorphism W p Aut( p ) is given by Wp O = ker O p ( p ) n+1 /( p ) n+1 N /Q Q p /(Q p ) n+1. (14) p p In particuar, x L acts triviay on p if and ony if v p (x) (n + 1)Z for each prime p of above p. Remark 3.4. For p Ram we aso define W O p as in (14). As noticed in [7, Lemma 5.4], for any p Ram and x L we have v p (x) (n + 1)Z. The same argument shows that W p = W O p for p Ram. We denote by L n+1 the subgroup of L consisting of eements x with v(x) (n + 1)Z for every normaized nonarchimedean vauation v of. From Lemma 3.3 and Remark 3.4, the sequence (9) yieds the exact sequence 1 A ξ L n+1 /( ) n+1 Aut( p ). (15) p Ram 12

13 Proposition If n is even, then A ξ is isomorphic to L n+1 /( ) n If n is odd, denote by ɛ the index of A ξ in L n+1 /( ) n+1. Then ɛ is a divisor of 2 r with ɛ 2. In particuar, if r = 1 we have ɛ = 2. Proof. If n = 2m is even, for p Ram we have that G /Q p has C BC m as oca index, and so Aut( p ) = 1. The first statement foows immediatey from (15). For n = 2m 1 and p Ram, the oca index of G /Q p is B C m, for which Aut( p ) has order 2. Then ɛ must be a divisor of 2 r. For p Ram et π p be any uniformizer with respect to the pace p above p. We know from [22, 4.2] that π p π 1 p L is a generator of p (where π p is the Gaois conjugate of π p in ). When p is odd, the eement d is a uniformizer, where = Q( d), and we obtain that 1 L n+1 is a generator of Aut( p ). This proves that ɛ 2 when Ram contains an odd prime p. This is aways the case, except when d is 1 or 2. For d = 2, using the same uniformizer π p = d we aso see that ɛ = 2. For d = 1 and π (2) = 1+i we have i L n+1 as generator of 2, and thus ɛ = 2 as we. 3.5 The covoume of SU(n, 1)/Γ We denote by I (resp., P ) the group of fractiona (resp., principa) ideas of and by C = I /P the cass group of. Since the group of integra units in L is µ(), we have the exact sequence (see [11, 12.6] for detais) 1 µ()/µ() n+1 L n+1 /( ) n+1 (P L I n+1 )/P n+1 1, (16) where P L is the subgroup of principa ideas which can be written as (x) with x L. For an idea (x) = a n+1 P L I n+1, its idea norm N /Q (x) = [O : (x)] is obviousy in (Q ) n+1. Since is imaginary quadratic this norm is just N /Q (x). This proves that P L I n+1 equas P I n+1. The quotient of this atter group by P n+1 is isomorphic to the subgroup of C consisting of those eements with order dividing n + 1 (see [7, proof of Prop. 0.12]). Let h,n+1 be the order of this subgroup. Returning to the sequence (16), we see that the order of L n+1 /( ) n+1 is h,n+1 µ()/µ() n+1. The exact sequence 1 µ n+1 () µ() xn+1 µ() n

14 impies that µ()/µ() n+1 equas the order of µ n+1 (). We concude that L n+1 /( ) n+1 = h,n+1 µ n+1 (). (17) This equaity is the ast step in the computation of [Γ : Λ ]. It foows from the equaities (5), (7), (17) and Proposition 3.5 that [Γ : Λ ] is equa to (n + 1)h,n+1 if n is even, and ɛ 1 (n + 1)h,n+1 if n is odd. Together with (1), we get the foowing. Proposition If n is even, then µ(su(n, 1)/Γ ) equas 1 D n(n+3) 4 (n + 1)h,n+1 n j=1 2. If n is odd, then µ(su(n, 1)/Γ ) is ɛ D (n 1)(n+2) 4 (n + 1)h,n+1 j! (2π) j+1 ζ(2)l (3) ζ(n)l (n + 1). n j=1 where ɛ is as in Proposition 3.5. j! (2π) j+1 ζ(2)l (3) L (n)ζ(n + 1), 3.6 Bounding [Γ,T : Λ,T ] and computing ν (n) Suppose that n is odd, and consider the attice Γ,T (see 3.1). The index [Γ,T : Λ,T ] can be expressed by the same formua as in (5), where A H 1 (Q, C ) now acts on the oca Dynkin diagrams p associated with G,T. Lemma 3.3 is not vaid for primes p where G,T /Q p fais to be quasispit. In particuar A ξ does not need to be a subgroup of L n+1 /( ) n+1. However, G,T /Q p is not quasi-spit ony for a finite number of primes p, so A ξ wi be commensurabe with L n+1 /( ) n+1, and it is sti possibe to bound [Γ,T : Λ,T ]. A carefu anaysis (see [11, 12.5] for detais) shows that here we have the foowing bound. Lemma 3.7. For any Λ,T, [Γ,T : Λ,T ] (n + 1) #ˆT+1 h,n+1, (18) where ˆT denotes the set of unramified primes p where G,T /Q p is not quasispit. 14

15 Ceary ˆT T, where T is the set of primes defined in 3.1. Note that the voume of SU(n, 1)/Γ,T then has a factor of λ p /(n + 1) for every p ˆT, where the factors λ p are given in Equation (3). One can compute the factors λ p from [25, Tabe 1] and see that λ p /(n+1) > 1 for every p and every n 3. It foows then from Lemma 3.7 and the expicit expression for [Γ : Λ ] that the covoume of Γ is aways smaer than the covoume of Γ,T. We are now ready for the foowing. Computation of ν (n) for Theorems 1.1 and 1.3. Let Γ < PU(n, 1) be a attice of minima covoume for its commensurabiity cass. It foows from the above that Γ is the image in PU(n, 1) of Γ. As in [7, 4], we have that the Euer Poincaré measure on SU(n, 1) equas χ(cp n )µ = (n + 1)µ, since CP n is the compact dua to H n C. Since SU(n, 1) covers PU(n, 1) with degree n + 1, we see that µ EP (PU(n, 1)/Γ) = (n + 1) 2 µ(su(n, 1)/Γ ). The equations for ν (n) in Theorems 1.1 and 1.3 now foow from Proposition 3.6 foowed by the standard functiona equations for ζ and L (see [31, 15]). 3.7 The proof of Theorem 1.4 In this section, we prove the first part of Theorem 1.4. The case n = 2 appears in [30]. Proposition 3.8. For a n 2 the minima covoume nonuniform arithmetic attice in SU(n, 1) is Γ for = Q( 3) and T =. Proof. Suppose that the attice of smaest covoume comes from some other imaginary quadratic fied. By 3.6, this attice is some Γ. Let 0 = Q( 3). We now give an upper bound for the discriminant of. First, set s = n(n + 3)/4 for n even and s = (n 1)(n + 2)/4 for n odd. Then, assume that 1 [Λ : Γ ] Ds n j=1 { j! ζ(n + 1) n odd (2π) j+1 ζ(2)l (3) L (n + 1) n even 15

16 q n + 1 3s n j=1 { j! ζ(n + 1) n odd (2π) j+1 ζ(2)l 0 (3) L 0 (n + 1) n even When n is odd, we can cance the factor of ζ(n + 1) from each side. For a n, we cance the factors of j!/(2π) j+1 to get D s [Λ : Γ ] ζ(2)l (3) L (n + t) 3s q n + 1 ζ(2)l 0 (3) L 0 (n + t), (19) where t = 0 for n odd and t = 1 for n even. Comparing terms of each series expansion, we see that ζ(k)l 0 (k + 1) ζ(2)l 0 (3) for any k 2. Therefore, we have 3 s q n + 1 ζ(2)l 0 (3) L 0 (n + t) 3s q n + 1 (ζ(2)l 0 (3)) n t 2 = 3s q ( ) n t 2π 5 2 n /2 ( 2 3s 2π 5 n + 1 One the other hand, from Proposition 3.6, we have 3 11/2 D s [Λ : Γ ] ζ(2)l (3) L (n + t) ) n t 2 D s (n + 1)h,n+1 ζ(2)l (3) L (n + t). Then we have h,n+1 h, and a key ingredient of the proof of the Brauer Siege Theorem impies that ( ) m/2 D h µ m(m 1)(m 1)! 4π 2 ζ (m) for any integer m > 1 and imaginary quadratic fied, where µ is the order of the group of roots of unity in and ζ is the zeta function of. See the proof of Lemma 1 in 1 of Chapter XVI of [19]. For not Q(i) or Q( 3), µ = 2. From here forward assume D > 4, so ( ) m/2 D h 2m(m 1)(m 1)! 4π 2 ζ (m) for a m > 1. Taking m = 2, we get D s [Λ : Γ ] ζ(2)l (3) L (n + t) 16.

17 Ds 1 π 2 ζ (2) ζ(2)l (3) L (n + t). Then ζ(2k)l (2k + 1) > 1 for a k > 1 and ζ (2) ζ(2) = π 2 /6, so we have D s [Λ : Γ ] ζ(2)l (3) L (n + t) 6D s 1. We concude that if gives smaer covoume than Q( 3), then ( ) n t 2π 5 2(s 1) D /2 An immediate computation shows that this is bounded above by 4 for a n 4, so we need ony consider Q(i). For Q(i), we repace µ with 4 in the above computations and see that Q(i) must give arger covoume than Q( 3) as we, provided that n 5. One then checks by hand that the smaest covoume comes from Q( 3) for n = 2, 3, 4. This competes the proof of the proposition. 4 Counting mutipicities In this section we address the question of determining the number of isomorphism casses of arithmetic subgroups of G of minima covoume. In we wi first count conjugacy casses of arithmetic subgroups of G (R), and then in 4.5 we show how this impies our resuts on counting isomorphism casses stated in Counting conjugacy casses Let us fix an identification G (R) = SU(n, 1). The arithmetic subgroup Λ = Λ (P ) and its normaizer Γ = Γ (P ) in SU(n, 1) depend on the choice of a coherent coection P P (see 3.1). Though the covoume does not depend on this choice, two different coherent coections of parahorics with the same oca types may determine attices that are not conjugate under the action of PU(n, 1) by inner automorphisms. In this section it wi be usefu to consider Λ and Γ as functions Λ : P {attices in G (Q)}, Γ : P {attices in G (R)}. 17

18 Then PU(n, 1) acts by conjugation on both Λ (P ) and Γ (P ). Let G be the adjoint group of G. For any fied extension K/Q the group G (K) corresponds to the group of inner automorphisms of G defined over K. For K = R we have G (R) = PU(n, 1). The group G (Q) acts on the set P diagonay by conjugating each component P p G (Q p ) of a coherent coection P = (P p ) p Vf P. Lemma 4.1. The number n of PU(n, 1)-conjugacy casses in Γ (P ) is equa to the number of G (Q)-conjugacy casses in P. Proof. Since arithmetic subgroups are Zariski dense in the Q-group G they can ony be conjugated by inner automorphisms that are in G (Q). Thus we must count the number of G (Q)-conjugacy casses in Γ (P ). Let P and P be two coherent coection in P. Since Γ is by definition the normaizer of Λ, if Λ (P ) and Λ (P ) are conjugate, then Γ (P ) and Γ (P ) are aso conjugate by the same eement. The converse is aso true and foows from maximaity of Λ (P ) and Λ (P ) in G (Q). This proves that n is equa to the number of G (Q)-conjugacy casses in Λ (P ). By Strong Approximation, every coherent coection P can be recovered by the principa arithmetic subgroup Λ (P ). In particuar, if Λ (P ) and Λ (P ) are conjugate, then P and P are conjugate. This finishes the proof. Choose P = (P p ) P. The adeic group G (A f ) acts on P, each component g p of g = (g p ) p Vf G (A f ) acting by conjugation on P p. The group G (Q) is diagonay embedded in G (A f ) as usua. Then the action of the adeic group restricted to G (Q) coincides with the action of G (Q) on P as above. Fix a coherent coection P P. For each p V f we denote by P p the stabiizer of P p in G (Q p ). We can identify the G (A f )-conjugacy cass of P with the set G (A f )/ p P p. The G (A f )-conjugacy cass of P in P is divided into G (Q)-conjugacy casses, which are in bijection with the doube cosets G (Q)\G (A f )/ p V f P p. (20) The cardinaity of this set is caed the cass number of G reative to P, and is known to be finite (see for exampe [7, Prop. 3.9]). 18

19 4.2 Computing the cass number For each prime p, consider the connecting homomorphism δ p : G (Q p ) H 1 (Q p, C ), (21) where, as in 3, the group C denotes the center of G. Let p V f H 1 (Q p, C ) be the restricted direct product with respect to the groups δ p (P p ). The group H 1 (Q, C ) is diagonay embedded in p V f H 1 (Q p, C ). Foowing [28, 5.2] or [5, 6.1] we know that there is a bijection between the doube coset (20) and the group p V C (P ) = f H 1 (Q p, C ) δ ( G (Q) ) p δ p(p p ), (22) where δ is the connecting homomorphism from G (Q) to H 1 (Q, C ). Note that the quotient on the right is an abeian group. Since P is a coherent coection in P we have that each P p has specia type. An automorphism of the oca Dynkin diagram p of G (Q p ) that fixes a specia type is necessariy trivia [11, Prop. 12.2]. It foows that for each prime p the group δ p (P p ) is precisey the kerne H 1 (Q p, C ) ξp of the map ξ p introduced in 3.2. In particuar, this kerne is independent of P p. Thus the order of C = C (P ) does not depend on P P. We wi denote this order by h and ca it the cass number of G. Proposition 4.2. Let r be the number of ramified primes in /Q. Then { n 2 r h = if n is even; if n is odd. h Proof. By Lemma 4.1, n is given by the number of G (Q)-conjugacy casses in the set P. By the above discussion, the set of these casses is divided into G (A f )-conjugacy casses, where each G (A f )-conjugacy casses identifies exacty h of the G (Q)-conjugacy casses. Therefore, we ony need to determine the number of G (A f )-conjugacy casses in P. If p Ram, the type of P p for P P is hyperspecia and G (Q p ) acts transitivey on hyperspecia types [33, 2.5]. If n is odd and p Ram then G (Q p ) permutes the two specia points of the oca Dynkin diagram p. It foows that a coherent coections of P are conjugate in this case. When n is even we have Aut( p ) = 1 for p Ram and, in particuar, the two specia points of p are not conjugate. This shows that for even n there are as many G (A f )-conjugacy casses in P as there are possibe choices of a specia point in each p for p Ram. That is, there are 2 r G (A f )-conjugacy casses. 19

20 4.3 More on the cass number Foowing 4.2 we know that h is given by the size of the group C = p V f H 1 (Q p, C ) δ ( G (Q) ) p H1 (Q p, C ) ξp. Using Lemma 3.2, we can factor out the product p N p from the numerator and denominator. With the description of δ ( G (Q) ) in 4.2 we obtain an isomorphism C = p V f W p L/( ) n+1 p W ξ p, (23) where W ξp is the kerne of the homomorphism W p Aut( p ) induced by ξ p. For p Ram we have W ξp = Wp O (cf., Lemma 3.3). If the dimension n is even we have Aut( p ) = 1 for p Ram. Using Remark 3.4 we see that W ξp = Wp O in this case as we, so C = p V f W p L/( ) n+1 p W O p. (24) Now suppose that n is odd. For p Ram the subgroup W ξp is of index two in W p = Wp O. However, if # Ram = 1 the proof of Proposition 3.5 shows that L/( ) n+1 W ξp = L/( ) n+1 Wp O. Therefore, assuming r = 1, Equation (24) sti hods. 4.4 Ideic formuation p Let I be the group of finite idees of. We denote by I the subgroup of integra finite idees [31, 9]. The norm map N /Q extends to idees as a map N /Q : I I Q, defined component-wise, where I Q is the group of idees of Q. Consider the kerne ( ) N /Q I L = ker I I Q /I n+1 Q. (25) p 20

21 Equation (24), which hods when n is even or r = 1, can be rewritten using idees as C I L /I n+1 = L/( ) n+1 I In+1 ( = I L / LI L In+1 /I n+1 ), (26) with L embedded diagonay in I L, and IL = I I L. Reca h,n+1 is the order of the subgroup of the cass group C consisting of those eements with order dividing n + 1. We then have the foowing. ( Lemma 4.3. The order of the group I L / 8 n + 1, this order divides h,n+1. LI L In+1 Proof. By definition of L and I L, we have the exact sequence 1 L I L N /Q Q /(Q ) n+1 φ ) divides 2h,n+1. If p V f Q p /(Q p ) n+1, where φ is the natura diagona map. This shows that the index of L in I L has the same size as the kerne of φ. According to [24, Thm ] this kerne has order at most 2, and is trivia if 8 n + 1. Each idee of determines a fractiona idea of by a standard procedure described, for exampe, in [31, 9]. This induces a homomorphism I L I /P I n+1, (27) whose kerne is ( I L )IL In+1. Now I /P I n+1 is the quotient C /C n+1 of the cass group C, whose order is easiy seen to be h,n+1. Together with the index of L in I L this concudes the proof. We sum up the resuts about h in the foowing proposition. Proposition If n is even, then h is a divisor of h,n+1 2. Suppose that n is odd and r = 1. Then h is a divisor of 2h,n+1. Moreover, if 8 n + 1, then h is a divisor of h,n+1. 21

22 4.5 Counting up to isomorphism Any nonuniform attice in PU(n, 1) associated with /Q and of minima covoume among them corresponds to the projection of a subgroup Γ G (R). It is cear that the number n = n (n) of isomorphism casses of such attices is at most equa to the number n of PU(n, 1)-conjugacy casses. Since PU(n, 1) is of index 2 in its automorphism group, Mostow-Prasad rigidity shows that n is at east equa to n /2. However, when n is even and p Ram, the two specia vertices on p correspond to nonisomorphic parahoric subgroups in G (Q p ), and thus different choices at this prime wi give nonisomorphic attices. This shows that for n even n is at east equa to 2 r. From Propositions 4.2 and 4.4 we thus obtain the foowing resut. Proposition If n is even, then n 2 r and 2 r 1 h n 2 r h, where h is a divisor of h,n Suppose that n is odd and r = 1. Then h /2 n h, where h is a divisor of 2h,n+1. Moreover, if 8 n+1, then h is a divisor of h,n+1. 5 Theorems and comparison with geometric voume estimates 5.1 Overa minima covoume We now prove Theorem 1.5 assuming Theorem 1.6, which we prove in the seque. By Theorem 1.4, a nonuniform arithmetic attice Λ < PU(m, 1) such that χ(γ) > χ(λ) for any Γ < PU(n, 1) is necessariy associated with the extension Q( 3)/Q. Using the formuas for ν Q( 3) (n) given in Theorem 1.1 and 1.3, and the fact that for n high enough the covoume grows (Theorem 1.6), we find that the minimum appears for m = 9, where the attice Γ 0 of minima covoume is unique. Equip compex hyperboic space with the metric under which hoomorphic sectiona curvatures are 1. We appy Chern Gauss Bonnet (see [14, 26]) to obtain vo(h n ( 4π)n C /Γ) = χ(γ). (28) (n + 1)! Repeating the computations for the Euer characteristic with the compex hyperboic voume, one sees that repacing µ EP by vo does not change the 22

23 fact that the smaest voume appears in dimension m = 9. This competes the proof of Theorem Asymptotic behavior of the minima covoume We proceed with the proof of Theorem 1.6. Proof of Theorem 1.6. Let /Q be an imaginary quadratic extension. The ratio q (n) = ν (n + 1)/ν (n) is given by: q (n) = D ± n+1 2 ɛ (n) 1 n + 2 n + 1 (n + 1)! F (n + 2), (2π) n+2 where F is L when n is odd and is ζ when n is even, and the signs ± depend on the parity of n. Since is fixed, ɛ (n) is bounded. We aso have that im n n + 2 F (n + 2) = 1. n + 1 Therefore, for n arge and up to a constant, q (n) grows ike ( ) ±1/2 n+2 D (2π) n+2 (n + 1)!, which is super-exponentia. Note that the anaogous statement for compex hyperboic voumes is immediate, since the ratio of the additiona factors given in (28) decreases ineary in n. It aso foows that the minima voume of a noncompact arithmetic compex hyperboic n-manifod grows super-exponentiay in n. We now anayze the difference between our resuts and other estimates for voumes of noncompact compex hyperboic orbifods. A ower bound for the voume of a compex hyperboic n-manifod with k cusps, due to Hwang [16], is (4π) n ( ) n + 1 k 1, n!(p (4) P (2)) P (4) P (2) where P () = (n + n + )!. n!(n + )! This bound comes from agebraic geometry, and requires smoothness. This bound improves upon earier estimates due to Parker [26] and Hersonsky Pauin [14]. Note that if k is bounded, this estimate decays to 0 as n. 23

24 Comparison with Theorem 1.6 raises the question of how the number of cusps grows as n. More recenty, generaizing work of Adeboye in rea hyperboic space [1], Fu, Li, and Wang [13] gave a ower bound for the voume of a compex hyperboic n-orbifod depending ony on the argest order of a torsion eement of its fundamenta group. Adeboye s work has aso been extended to compex hyperboic space by work in progress by Adeboye Wei [2]. These bounds aso go to zero as n goes to infinity. It woud be interesting to better understand the gap between these geometric voume bounds and our arithmetic resuts. References [1] I. Adeboye. Lower bounds for the voume of hyperboic n-orbifods. Pacific J. Math., 237(1):1 19, [2] I. Adeboye and G. Wei. On voumes of compex hyperboic orbifods. In preparation, [3] D. Acock. The Leech attice and compex hyperboic refections. Invent. Math., 140(2): , [4] M. Beoipetsky. On voumes of arithmetic quotients of SO(1, n). Ann. Sc. Norm. Super. Pisa C. Sci. (5), 3(4): , [5] M. Beoipetsky and V. Emery. On voumes of arithmetic quotients of PO(n, 1), n odd. To appear in Proc. Lond. Math. Soc., [6] A. Bore and Harish-Chandra. Arithmetic subgroups of agebraic groups. Ann. of Math. (2), 75: , [7] A. Bore and G. Prasad. Finiteness theorems for discrete subgroups of bounded covoume in semi-simpe groups. Inst. Hautes Études Sci. Pub. Math., (69): , [8] T. Chinburg and E. Friedman. The smaest arithmetic hyperboic three-orbifod. Invent. Math., 86(3): , [9] T. Chinburg, E. Friedman, K. N. Jones, and A. W. Reid. The arithmetic hyperboic 3-manifod of smaest voume. Ann. Scuoa Norm. Sup. Pisa C. Sci. (4), 30(1):1 40,

25 [10] M. Deraux, J. Parker, and J. Paupert. Census of the compex hyperboic sporadic triange groups. Experiment. Math., 20(4): , [11] V. Emery. Du voume des quotient arithmétiques de espace hyperboique. Ph.D. thesis, Univ. Fribourg, [12] V. Emery. Arbitrariy arge famiies of spaces of the same voume. To appear in Geom. Dedicata, [13] X. Fu, L. Li, and X. Wang. A ower bound for the voumes of compex hyperboic orbifods. Geom. Dedicata, 155(1):21 30, [14] S. Hersonsky and F. Pauin. On the voumes of compex hyperboic manifods. Duke Math. J., 84(3): , [15] T. Hid. The cusped hyperboic orbifods of minima voume in dimensions ess than ten. J. Agebra, 313(1): , [16] J.-M. Hwang. On the voumes of compex hyperboic manifods with cusps. Internat. J. Math., 15(6): , [17] K. Ireand and M. Rosen. A cassica introduction to modern number theory, voume 84 of Graduate Texts in Mathematics. Springer-Verag, second edition, [18] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tigno. The book of invoutions, voume 44 of American Mathematica Society Cooquium Pubications. American Mathematica Society, [19] S. Lang. Agebraic number theory, voume 110 of Graduate Texts in Mathematics. Springer-Verag, second edition, [20] R. Meyerhoff. The cusped hyperboic 3-orbifod of minimum voume. Bu. Amer. Math. Soc. (N.S.), 13(2): , [21] P. Miey. Minimum voume hyperboic 3-manifods. J. Topo., 2(1): , [22] A. Mohammadi and A. Saehi-Gosefidy. Discrete vertex transitive actions on Bruhat Tits buidings. To appear in Duke Math. J., [23] G. D. Mostow. On discontinuous action of monodromy groups on the compex n-ba. J. Amer. Math. Soc., 1(3): ,

26 [24] J. Neukirch, A. Schmidt, and K. Wingberg. Cohomoogy of number fieds, voume 323 of Grundehren der Mathematischen Wissenschaften. Springer-Verag, second edition, [25] T. Ono. On agebraic groups and discontinuous groups. Nagoya Math. J., 27: , [26] J. Parker. On the voumes of cusped, compex hyperboic manifods and orbifods. Duke Math. J., 94(3): , [27] G. Prasad. Voumes of S-arithmetic quotients of semi-simpe groups. Inst. Hautes Études Sci. Pub. Math., (69):91 117, [28] G. Prasad and S.-K. Yeung. Fake projective panes. Invent. Math., 168(2): , [29] G. Prasad and S.-K. Yeung. Arithmetic fake projective spaces and arithmetic fake Grassmannians. Amer. J. Math., 131(2): , [30] M. Stover. Voumes of Picard moduar surfaces. Proc. Amer. Math. Soc., 139(9): , [31] H. P. F. Swinnerton-Dyer. A brief guide to agebraic number theory, voume 50 of London Mathematica Society Student Texts. Cambridge University Press, [32] J. Tits. Cassification of agebraic semisimpe groups. In Agebraic Groups and Discontinuous Subgroups, pages Amer. Math. Soc., [33] J. Tits. Reductive groups over oca fieds. In Automorphic forms, representations and L-functions, Part 1, pages Amer. Math. Soc., [34] H. C. Wang. Topics on totay discontinuous groups. In Symmetric spaces, pages Pure and App. Math., Vo. 8. Dekker,