On discreteness of subgroups of quaternionic hyperbolic isometries
Krishnendu Gongopadhyay, Mukund Madhav Mishra, Devendra Tiwari
aa r X i v : . [ m a t h . G T ] J un ON DISCRETENESS OF SUBGROUPS OF QUATERNIONICHYPERBOLIC ISOMETRIES
KRISHNENDU GONGOPADHYAY, MUKUND MADHAV MISHRA, AND DEVENDRA TIWARI
Abstract.
Let H n H denote the n -dimensional quaternionic hyperbolic space. The lineargroup Sp( n,
1) acts by the isometries of H n H . A subgroup G of Sp( n,
1) is called
Zariskidense if it does not fix a point on H n H ∪ ∂ H n H and neither it preserves a totally geodesicsubspace of H n H . We prove that a Zariski dense subgroup G of Sp( n,
1) is discrete if forevery loxodromic element g ∈ G the two generator subgroup h f, gfg − i is discrete, wherethe generator f ∈ Sp( n,
1) is certain fixed element not necessarily from G . Introduction
The classical Jørgensen inequality, see [Jr76], gives a necessary criterion to check dis-creteness of a two generator subgroup of SL(2 , C ) that acts by M¨obius transformations onthe Riemann sphere. This has been generalized to the higher dimensional M¨obius groupthat acts on the n dimensional real hyperbolic space, by several authors. A well-known con-sequence of the generalized Jørgensen inequalities says that a subgroup G of the M¨obiusgroup is discrete if and only if every two generator subgroup is discrete, e.g. [Mar89],[AH90]. There have been several refinements of this result to obtain several discretenesscriteria in M¨obius groups, e.g. [Che04], [WLC05], [GMS18]. Several generalizations of theJørgensen inequality and related discreteness criteria have been obtained in further gener-alized setting like the complex hyperbolic space and normed spaces, e.g. [JKP03], [FH93],[MP03], [MP07].Let H denote the division ring of Hamilton’s quaternions. Let H n H denote the n -dimensionalquaternionic hyperbolic space. Let Sp( n,
1) be the linear group that acts on H n H by isome-tries. In this paper following the above theme of research, we give discreteness criteria fora subgroup of Sp( n, n, g ∈ Sp( n,
1) is elliptic if it has a fixed point on H n H . It is parabolic ,resp. loxodromic (or hyperbolic), if it has a unique fixed point, resp. exactly two fixed pointson the boundary ∂ H n H . A unipotent parabolic element, that is, a parabolic element havingall eigenvalues 1, is called a Heisenberg translation . It is well-known, see [CG74], that anelliptic or loxodromic isometry g is conjugate to a diagonal element in Sp( n, g iselliptic, then up to conjugacy,(1.1) g = diag( λ , . . . , λ n +1 ) , Date : June 24, 2019.2000
Mathematics Subject Classification.
Primary 20H10; Secondary 51M10.
Key words and phrases. hyperbolic space, Jørgensen inequality, discreteness, quaternions.Gongopadhyay acknowledges partial support from SERB MATRICS grant MTR/2017/000355.Tiwari is supported by NBHM-SRF. where for each i , | λ i | = 1, and the eigenvalue λ is such that the corresponding eigenvectorhas negative Hermitian length, while all other eigenvectors have positive Hermitian length.An elliptic element g is called regular if it has mutually disjoint classes of eigenvalues. Aregular elliptic element has a unique fixed point on H n H . If g is loxodromic, then we mayassume up to conjugacy,(1.2) g = diag( λ , ¯ λ − , λ , . . . , λ n +1 ) . with | λ | >
1. One may associate certain conjugacy invariants to isometries as follows.For g is elliptic, we define(1.3) δ ( g ) = max { | λ − | + | λ i − | : i = 2 , . . . , n + 1 } . For g loxodromic, following [CP11], define the following quantities: δ cp ( g ) = max {| λ i − | : i = 3 , . . . , n + 1 } , and M g = 2 δ cp ( g ) + | λ − | + | ¯ λ − − | . Let T s,ζ be a Heisenberg translation in Sp( n, T s,ζ = s ζ ∗ ζ I , where Re ( s ) = | ζ | .A subgroup G of Sp( n,
1) is called
Zariski dense if it does not fix a point on H n H ∪ ∂ H n H ,and neither it preserves a totally geodesic subspace of H n H . With the above notations, weprove the following theorem. Theorem 1.1.
Let G be a Zariski dense subgroup of Sp( n, . (1) Let g ∈ Sp( n, be a regular elliptic element such that δ ( g ) < . If h g, hgh − i isdiscrete for every loxodromic element h ∈ G , then G is discrete. (2) Let g ∈ Sp( n, be loxodromic element such that M g < . If h g, hgh − i is discretefor every loxodromic element h ∈ G , then G is discrete. (3) Let g ∈ Sp( n, be a Heisenberg translation such that | ζ | < . If h g, hgh − i isdiscrete for every loxodromic h in G , then G is discrete. Restricting everything over the complex numbers, the above theorem also holds forSU( n, Corollary 1.2.
Let G be Zariski dense in Sp( n, , resp. SU( n, . (1) Let g ∈ Sp( n, , resp. SU( n, , be a regular elliptic element such that δ ( g ) < . If h g, hgh − i is discrete for every regular elliptic element h ∈ G , then G is discrete. (2) Let g ∈ Sp( n, , resp. SU( n, , be loxodromic element such that M g < . If h g, hgh − i is discrete for every regular elliptic h ∈ G , then G is discrete. (3) Let g ∈ Sp( n, , resp. SU( n, , be a Heisenberg translation such that | ζ | < . If h g, hgh − i is discrete for every regular elliptic h in G , then G is discrete. Note that the above results show that the discreteness of a Zariski dense subgroup G of Sp( n, n, h g, hgh − i , where h ∈ G , but the generator g is fixed and need not be an element from G , and also it is enoughto take h to be loxodromic or regular elliptic. After fixing such a ‘test map’ g , conjugates of N DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES 3 g by generic elements of G determine the discreteness. For isometries of the real hyperbolicspace, similar discreteness criteria using a test map and conjugates of it, have been obtainedin [YZ14], [GMS18, Theorem 1.2] and [GM]. Theorem 1.1 and Corollary 1.2 generalize theseworks in Sp( n,
1) and SU( n,
1) respectively.We have noted down some preliminary notions in section 2. The main result has beenproved in section 4. To prove the results, we shall use some generalized Jørgensen inequali-ties in Sp( n, δ ( g ) as given above. This invariant δ ( g ) isdifferent from the conjugacy invariant δ ct ( g ) used by Cao and Tan. The invariant δ ( g ) maybe considered as a restriction of the Cao-Parker invariant δ cp ( g ) to the subgroups havingat least one generator elliptic. This new invariant also gives quantitatively better bound ina larger domain. We refer to section 3 for more details.2. Preliminaries
The quaternionic hyperbolic space.
We begin with some background material onquaternionic hyperbolic geometry. Much of this can be found in [CG74, KP03].Let H n, be the right vector space over H of quaternionic dimension ( n + 1) (so realdimension 4 n + 4) equipped with the quaternionic Hermitian form for z = ( z , ..., z n ) , w =( w , ..., w n ), h z, w i = − (¯ z w + ¯ z w ) + Σ ni =2 ¯ z i w i . Thus the Hermitian form is defind by the matrix J = − − I n − . Equivalently, one may also use the Hermitian form given by the following matrix whereverconvenient. J = (cid:18) − I n (cid:19) . Following Section 2 of [CG74], let V = n z ∈ H n, − { } : h z , z i = 0 o , V − = n z ∈ H n, : h z , z i < o . It is obvious that V and V − are invariant under Sp( n, ∼ on H n, by z ∼ w if and only if there exists a non-zero quaternion λ so that w = z λ . Let[ z ] denote the equivalence class of z . Let P : H n, − { } −→ HP n be the right projection map given by P : z z , where z = [ z ]. The n dimensional quaternionic hyperbolic spaceis defined to be H n H = P ( V − ) with boundary ∂ H n H = P ( V ).In the model using J , there are two distinct points 0 and ∞ on ∂ H n H . For z = 0, theprojection map P is given by P ( z , z , . . . , z n +1 ) = ( z z − , . . . , z n +1 z − ) , K. GONGOPADHYAY, M. M. MISHRA, AND D. TIWARI and accordingly we choose boundary points(2.1) P (0 , , . . . , , t = 0 . (2.2) P (1 , , . . . , , t = ∞ . In the model using J , we mark P (1 , , . . . , , t as the origin 0 = (0 , , . . . , t of thequaternionic hyperbolic ball. The Bergmann metric on H n H is given by the distance formulacosh ρ ( z, w )2 = h z , w ih w , z ih z , z ih w , w i , where z, w ∈ H n H , z ∈ P − ( z ) , w ∈ P − ( w ) . The above forumula is independent of the choice of z and w .Now consider the non-compact linear Lie groupSp( n,
1) = { A ∈ GL( n + 1 , H ) : A ∗ J i A = J i } . An element g ∈ Sp( n,
1) acts on H H n = H n H ∪ ∂ H n H as g ( z ) = P g P − ( z ). Thus the isometrygroup of H n H is given by PSp( n,
1) = Sp( n, / { I, − I } . Cao-Parker Inequality.
Recall that the quaternionic cross ratio of four distinctpoints z , z , z , z on ∂ H n H is defined as:[ z , z , z , z ] = h z , z ih z , z i − h z , z ih z , z i − , where z i denote the lift to H n +1 of a point z i on ∂ H n H . We note the following lemmaconcerning cross ratios. Lemma 2.1. [CP11]
Let , ∞ ∈ ∂ H n H stand for the (0 , , . . . , t and (1 , , . . . , t ∈ H n, under the projection map P , respectively and let h ∈ PSp( n, be given by (2.5). Then | [ h ( ∞ ) , , ∞ , h (0)] | = | bc | , | [ h ( ∞ ) , ∞ , , h (0)] | = | ad | , | [ ∞ , , h ( ∞ ) , h (0)] | = | bc || ad | . Now, Cao and Parker’s theorem may be stated as follows.
Theorem 2.2. (Cao and Parker) [CP11]
Let g and h be elements of Sp( n, such that g isloxodromic element with fixed points u, v ∈ ∂ H n H , and M g < . If h g, h i is non-elementaryand discrete, then (2.3) | [ h ( u ) , u, v, h ( v )] | | [ h ( u ) , v, u, h ( v )] | ≥ − M g M g . A Shimizu’s Lemma in
Sp( n, . We use the Hermitian form J in this section. Upto conjugacy, we assume that an Heisenberg translation fixes the boundary point 0, i.e. itis of the form(2.4) T s,ζ = s ζ ∗ ζ I , where Re ( s ) = | ζ | . N DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES 5
Let A be an element in Sp( n, A to be of the following form.(2.5) A = a b γ ∗ c d δ ∗ α β U , where a, b, c, d are scalars, γ, δ, α, β are column matrices and U is an element in M ( n − , H ).Then, it is easy to compute that A − = ¯ d ¯ b − β ∗ ¯ c ¯ a − α ∗ − δ − γ U ∗ . The following theorem follows by mimicking the arguments of Hersonsky and Paulin in[HP96, Appendix]. However, Hersonsky and Paulin proved it over the complex numbers. Towrite it down over the quaternions, only slight variation is needed, and is straight-forward.We skip the details.
Theorem 2.3.
Suppose T s,ζ be an Heisenberg translation in Sp( n, and A be an elementin Sp( n, of the form (2.5). Suppose A does not fix . Set (2.6) t = Sup {| b | , | β | , | γ | , | U − I |} , M = | s | + 2 | ζ | . If M t + 2 | ζ | < , then the group generated by A and T s,ζ is either non-discrete or fixes . This is the simplest quaternionic version of the Shimizu’s lemma for two generator sub-groups of Sp( n,
1) with a unipotent parabolic generator. More generalized versions of theShimizu’s lemma in Sp( n,
1) has been obtained by Kim and Parker in [KP03, Theorem4.8], and Cao and Parker [CP18]. Though the above version of the quaternionic Shimizu’slemma is simpler, it is weaker than the versions of Kim and Parker, and Cao and Parker.However, we find it easier to apply for our purpose.2.4.
Useful Results.
A subgroup G of Sp( n,
1) is called elementary if it has a finite orbitin H n H ∪ ∂ H n H . If all of its orbits are infinite then G is non-elementary. In particular, G isnon-elementary if it contains two non-elliptic elements of infinite order with distinct fixedpoints. Theorem 2.4. [CG74]
Let G be a Zariski dense subgroup of Sp( n, . Then G is eitherdiscrete or dense in Sp( n, . Cao-Tan Inequality Revisited
Theorem 3.1.
Let g and h be elements of Sp( n, . Suppose that g is a regular ellipticelement with fixed point q , and δ ( g ) as in (4.2). If (3.1) cosh ρ ( q, h ( q ))2 δ ( g ) < , then the group h g, h i generated by g and h is either elementary or not discrete. The proof of the above theorem is a variation of the proof of [CT10, Theorem 1.1]. Theinitial computations are very similar, except that at a crucial stage we replace the Cao-Taninvariant by δ ( g ) and observe that it still works. We sketch the proof for completeness. Wefollow similar notations as in [CT10]. We shall use the ball model, i.e. Hermitian from J is being used is what follows: K. GONGOPADHYAY, M. M. MISHRA, AND D. TIWARI
Proof.
Using conjugation, we may assume that g is of the following form (1.2) having fixedpoint q = (0 , . . . , t ∈ H n H and h = ( a i,j ) i,j =1 ,...,n +1 = (cid:18) a , βα A (cid:19) . For L = diag ( λ , . . . , λ n +1 ), write g as: g = (cid:18) λ L (cid:19) . Then cosh ρ ( q, h ( q ))2 = | a , | , δ ( g ) = max { | λ − | + | λ i − | : i = 2 , . . . , n + 1 } . The inequality (3.1) becomes,(3.2) | a , | δ ( g ) < . Let h = h and h k +1 = h k gh − k . We write h k = ( a ( k ) i,j ) i,j =1 ,...,n +1 = a ( k )1 , β ( k ) α ( k ) A ( k ) ! . If for some k , β ( k ) = 0, as in the proof of [CT10, Theorem 1.1], it follows that h g, h i iselementary. So, assume β ( k ) = 0 and the group h g, h i is non-elementary. Then followingexactly similar computations as in the proof of [CT10, Theorem 1.1], one can see that:(3.3) | a ( k +1)1 , | ≤ | a ( k )1 , | + | β ( k ) | − n +1 X i =2 | a ( k )1 , | | a ( k )1 ,i | (2 − | u − u i | ) , where u i = a ( k )1 ,i − λ i a ( k )1 ,i , i = 2 , . . . n + 1 . Noting that | a ( k )1 , | − | β ( k ) | = 1, by (3.3) we have | a ( k +1)1 , | − ≤ | a ( k )1 , | n +1 X i =2 | a ( k )1 ,i | | u − u i | (3.4) ≤ | a ( k )1 , | n +1 X i =2 | a ( k )1 ,i | | (cid:18) | u − | + | u i − | (cid:19) (3.5) ≤ | a ( k )1 , | n +1 X i =2 | a ( k )1 ,i | (cid:18) | u − | + | u i − | (cid:19) . (3.6)Therefore(3.7) | a ( k +1)1 , | − ≤ ( | a ( k )1 , | − | a ( k )1 , | δ ( g ) . Now, it follows by induction that | a ( k +1)1 , | < | a ( k )1 , | , and(3.8) | a ( k +1)1 , | − < ( | a , | − | a , | δ ( g )) k +1 . N DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES 7
Since | a , | δ ( g ) < | a ( k )1 , | →
1. Now, as in the last part of the proof of [CT10, Theorem1.1], β ( k ) → , α ( k ) , A ( k ) ( A ( k ) ) ∗ → I n . By passing to its subsequence, we may assume that A ( k t ) → A ∞ , a ( k t )1 , → a ∞ . Thus h k +1 converges to h ∞ = (cid:18) a ∞ A ∞ (cid:19) ∈ Sp( n, , which implies that h g, h i is not discrete. This completes the proof. (cid:3) Using embedding of SL(2 , C ) in Sp(1 ,
1) and then applying similar arguments as in theproof of [CT10, Theorem 1.2], we have the following corollary that may be thought ofa generalized version of the classical Jørgensen inequality in SL(2 , C ) for two generatorsubgroups with an elliptic generator. Corollary 3.2.
Let g and h are elements in SL(2 , C ) . Let g = (cid:18) e iθ e − iθ (cid:19) , θ ∈ [0 , π ] , h = (cid:18) a bc d (cid:19) . Let || h || = | a | + | b | + | c | + | d | . If h g, h i is non-elementary and discrete, then (3.9) 4 sin θ (cid:18) || h || + 2 (cid:19) ≥ . Proof.
Let ˆ g be the image of g in Sp(1 , δ (ˆ g ) = 4 sin θ , and cosh ( ρ (0 , ˆ h (0)2 ) = || h || . This gives the proof. (cid:3) Comparison of the conjugacy invariants.
Let g be ellliptic, up to conjugacy, inSp( n, g = diag ( λ , . . . , λ n +1 ) , where | λ i | = 1 for all i . Cao and Tan used the following conjugacy invariant instead of δ ( g ): δ ct ( g ) = max {| λ i − λ | : i = 2 , . . . , n + 1 } . We have δ ( g ) = max { | λ − | + | λ i − | : i = 2 , . . . , n + 1 } . For all j , let λ j = e iθ j , θ j ∈ [0 , π ]. Note that | e iθ − | + | e iφ − | = 2 (cid:0) | sin θ | + | sin φ | (cid:1) , K. GONGOPADHYAY, M. M. MISHRA, AND D. TIWARI this implies δ ( g ) = 2 max {| sin θ | + | sin θ j +1 | : j = 1 , . . . , n } = max { θ θ j +1 j = 1 , . . . , n } = max { θ + θ j +1 θ − θ j +1 j = 1 , . . . , n } . On the other hand, the expression for Cao-Tan invariant in [CT10] is δ ct ( g ) = max (cid:26) θ ± θ j +1 j = 1 , . . . , n (cid:27) . Recall that by [CT10, Corollary 1.2], under the hypothesis of the above corollary,(3.10) 4 sin θ (cid:18) || h || + 2 (cid:19) ≥ . Comparing the two sine terms in the LHS of the inequalities (3.9) and (3.10), we see thatsin ( θ/ ≤ sin θ, for θ ∈ [0 , π/ , and our inequality (3.9) is stronger than the inequality (3.10) of Cao and Tan. But when θ ∈ (2 π/ , π ], then sin ( θ/ > sin θ , and consequently the inequality of Cao and Tan isbetter in this subinterval. So, except the last one-third of the interval [0 , π ], our version ofthe Jørgensen inequality in SL(2 , C ) is better.4. Proof of Theorem 1.1
Proof.
Given g , let F g denote the subgroup of Sp( n,
1) that stabilizes the set of fixed pointsof g . The subgroup F g is closed in Sp( n, G is not discrete. Then G is dense in Sp( n, L forms an open subset of Sp( n, L \ F g is also an opensubset in Sp( n, g be a regular elliptic. We shall use the ball model. Up to conjugacy, we mayassume that q = 0 is a fixed point of g , and it is of the form (1.1). Since, G is dense inSp( n, { h m } in L ∩ G such that h m → I .For each m , the element h m gh − m is also a regular elliptic with fixed point h m ( q ). Let(4.1) h m gh − m = ( a ( m ) i,j ) = a ( m )1 , β ( m ) α ( m ) A ( m ) ! . Then h m gh − m → g . In particular, a m , → λ , where | λ | = 1. Since q = 0 is a fixed point of g , the left hand side of (3.1) becomes | a ( m )1 , | δ ( g ). The group h g, h m gh − m i is clearly discrete.If possible, suppose h g, h m gh − m i is elementary. Since loxodromic elements have no fixedpoint on H n H , h m (0) = 0. Thus, g and h m gh − m do not have a common fixed point. Then itmust keep two boundary points p , p invariant, and hence will keep invariant the quater-nionic line l passing through p and p . Then g | l acts as a regular elliptic element ofIsom( l ) ≈ Sp(1 , q must belong to l , otherwise g would have at least two fixedpoints contradicting regularity of g . Now, note that g | l is also an elliptic element thatfixes p , p and q . Then it can be seen that with respect to a chosen basis p and p , g | l N DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES 9 must be of the form g | l = diag ( λ, λ ), where | λ | = 1. This implies g has an eigenvalue classrepresented by λ / of multiplicity at least 2. This again contradicts the regularity of g .So, the group h g, h m gh − m i must be non-elementary. By Theorem 3.1, | a ( m )1 , | δ ( g ) ≥ . But | a ( m )1 , | → δ ( g ) <
1. So, the above is a contradiction. This proves part (1).We shall use the Siegel domain model for proving the other assertions.(2) Let g be loxodromic. Up to conjugacy, let 0 and ∞ are the fixed points of g , and so g is of the form (1.2). Since G is dense in Sp( n, { h n } of loxodromicelements in ( L \ F g ) ∩ G such that h n → g . Let(4.2) h n gh − n = a n b n γ ∗ n c n d n δ ∗ n α n β n U n . Since, h n ∈ L \ F g , g and h n can not have a common fixed point, and neither canhave a two point invariant subset. So, and h g, h n gh − n i is non-elementary for each n . ByTheorem 2.2, | a n d n | | b n c n | ≥ − M f M f . But b n c n → n → ∞ , hence 1 − M f M f ≤ , which is a contradiction.(3) Let g be a Heisenberg translation. Up to conjugacy, let 0 be the fixed point of g and it is of the form (2.4). As g ∈ ¯ G , there exist a sequence of loxodromic elements { h n } ∈ ( L \ F g ) ∩ G such that h n → g. Let h n gh − n be of the form (4.2). Since h n gh − n → g , it follows that t n → g and h n gh − n have no fixed points in common, h g, h n gh − n i is discrete and non-elementary, hence by Theorem 2.3, M t n + 2 | ζ | > . But t n → n → ∞ . Thus for large n , | ζ | ≥ . This is a contradiction as | ζ | < is given.This proves the theorem. (cid:3) Proof of Corollary 1.2.
Note that the set of regular elliptic elements in Sp( n, E .(1) Let g be a regular elliptic. We shall use the ball model. Up to conjugacy, we mayassume g is of the form (1.1), and thus g (0) = 0. Since, G is dense in Sp( n, { h m } in ( E \ F g ) ∩ G such that h m → I . For each m , the element h m gh − m is also a regular elliptic with fixed point h m (0). Let h m gh − m isof the form (4.1). The group h g, h m gh − m i is clearly discrete. We claim that it is also notelementary. For otherwise, g and h m gh − m must have a common fixed point that will bedifferent from 0 and h m (0), which will contradict the regularity of the isometries. Now, byTheorem 3.1, | a ( m )1 , | δ ( g ) ≥
1. Since | a ( m )1 , | → δ ( g ) <
1, this is a contradiction. Thisproves part (1).
Using similar arguments as in the proof of Theorem 1.1, (2) and (3) follow.
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Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sec-tor 81, SAS Nagar, Punjab 140306, India
E-mail address : [email protected], [email protected] Department of Mathematics, Hansraj College, University of Delhi, Delhi 110007, India
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