Jørgensen ’ s Inequality and Algebraic Convergence Theorem in Quaternionic Hyperbolic Isometry Groups

and Applied Analysis 3


Introduction
Jørgensen's inequality [1] gives a necessary condition for a nonelementary two-generator subgroup of SL(2, C) to be discrete, which involves the traces of one of the generators and the commutator of both generators, as follows.
Theorem A. Let where [, ] =  −1  −1 is the commutator of  and  and tr(⋅) is the trace function.
Jørgensen's inequality has been generalized in many ways in real hyperbolic space [2,3], complex hyperbolic space [4][5][6], and quaternionic hyperbolic space [7][8][9] and plays an important role in studying discreteness and algebraic convergence for real, complex, or quaternionic hyperbolic isometry group [10][11][12][13][14].However, due to the noncommutative multiplication of the quaternions, Jørgensen's inequality in quaternionic hyperbolic isometry groups is relatively more complicated.To carry the results holding in real or complex hyperbolic geometry over to the quaternionic hyperbolic geometry, one sometimes has to reconsider these results involving the use of commutativity or the fact that purely imaginary complex numbers are isomorphic to R.
In quaternionic hyperbolic space, the first step to generalize Jørgensen's inequality was taken by Kim and Parker [7] who gave a quaternionic hyperbolic version of Basmajian and Miner's stable basin theorem.Subsequently, Markham [9] and Kim [8] independently gave versions of Jørgensen's inequality for Sp (2,1).Recently, Cao, Tan, and Parker, [15,16] obtained analogues of Jørgensen's inequality for nonelementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic or loxodromic.
Shimizu's lemma deals with two-generator subgroup ⟨, ⟩ with  being parabolic element and there are some generalizations to quaternionic hyperbolic space [7][8][9] for some special kinds of parabolic elements.But for  being screw parabolic, we only have analogues in the setting of 2dimensional complex hyperbolic space [4,6] and so forth.This gap is the main obstacle to investigate the discreteness and algebraic convergence theorem of groups in quaternionic hyperbolic space.
Our first aim is to erect generalizations of Jørgensen's inequality for two-generator nonelementary subgroup with some special kinds of elements in higher dimensional quaternionic hyperbolic isometry group Sp(, 1).
On the other hand, convergence of nonelementary subgroups of real or complex hyperbolic isometry groups is also another important problem.Let G be the -dimensional sense-preserving Möbius group (R then one says that { , } algebraically converges to   = ⟨ 1 ,  2 , . . .,   ⟩.
The problem that under which condition the limit group   is also a Kleinian group if, for each ,  , is a Kleinian group was intensely studied.Using the well-known Jørgensen's inequality, Jørgensen and Klein [17] proved the following.
where G is (R 2 ), then the limit group   is also a Kleinian group.
However, the examples in [18] show that Theorem B could not be extended to -dimensional cases ( ≥ 3) without any modifications.The reason for this phenomenon is that there is a distinction between the fixed point sets of elliptic elements in (R 2 ) and (R  ) ( ≥ 3).The reasoning mechanism in (R 2 ) mainly relies on the fact that each elliptic element has only two fixed points in R 2 .Because the fixed point set of an elliptic element of (R  ) ( ≥ 3) may be empty set or subset of R  , we cannot use the same reasoning mechanism as in  = 2.By adding some condition(s) to control the fixed point set of elliptic element and using generalized Jørgensen's inequality, several authors have obtained their analogues in (R  ) when  ≥ 3.
Theorem C. Let   be the algebraic limit group of a sequence of -generator Kleinian groups of (R  ) of uniformly bounded torsion.Then   is a Kleinian group.
Martin also asked how one might weaken the hypothesis of uniformly bounded torsion.Fang and Nai [19] first gave condition A to consider such a question.Recently, Wang [20] and Yang [21] used EP-condition and condition A, respectively, to weaken Martin's uniformly bounded torsion and proved the following.
Theorem D. Let  < ∞ and   be the algebraic limit group of a sequence of -generator Kleinian groups { , } of (R  ).
If { , } satisfies EP-condition (or condition A), then   is a Kleinian group.
See details for the definitions of uniformly bounded torsion, EP-condition, and condition A in [18,19,21].
In [10], Cao gave a convergence theorem about algebraic limit group of complex Kleinian groups under IP-condition, as follows.
Theorem E. Let   be the algebraic limit group of complex Kleinian groups { , } of (1, ; C).If { , } satisfies IPcondition, then   is a complex Kleinian group.
Our second aim is to investigate analogous condition mentioned above that an algebraic convergence theorem holds in the quaternionic hyperbolic space.We define the concept of uniformly bounded torsion as follows: a subset  of Sp(, 1) is said to have uniformly bounded torsion if there exists an integer  such that ∀ ∈ , ord () ≤  or ord () = ∞. ( where ‖ ⋅ ‖ is the Hilbert-Schmidt norm of an element. Theorem 4. Let   be the algebraic limit group of quaternionic Kleinian groups { , } of Sp (, 1).If { , } satisfies I-condition, then   is a quaternionic Kleinian group.

Several Lemmas
Let F denote the field R, C, or H.We adopt the same notations and definitions as in [7,16,22,23] such as   H , (1,  + 1; F), discrete groups, limit sets, and elementary and nonelementary.
We first discuss some properties of elliptic elements.As in [22], for an elliptic element , let Λ 0 and Λ  ,  = 1, 2, . . .,  be its negative and positive class of eigenvalues, respectively.Let fix() denote the set of fixed point(s) of  in   F .Then the fixed point set of  in   F contains only one fixed point if Λ 0 ̸ = Λ  ,  = 1, 2, . . ., , and is a totally geodesic submanifold which is equivalent to coincides with exact  of the class Λ  ,  = 1, 2, . . ., .In the latter case, the fixed point set of  in   F is   F or   C , and we define dim(fix()) = .The elliptic elements with only one fixed point in   F are called regular elliptic elements, while the other elements are called boundary elliptic elements.We call an elliptic element  an irrational rotation if  i ∈ Λ  with irrational  for some .
Since (1,  + 1; F) does not act effectively in   F , one always consider its projective group PU(1, ; F) = (1, ; F)/(1, ; F).It is well known that the -dimensional Möbius group () is isomorphic to the identity component of PU(1,  + 1; R), the projective orthogonal group.Each elliptic element is conjugate to an element with the form diag ( 0 , ) ∈  (1; F) ×  (; F) , where  0 ∈ Λ 0 .(8) When F = R and  is even, there are elliptic elements with  0 = 1 and the eigenvalues of positive class form /2 conjugated pairs of complex numbers of norm 1.Those elements correspond to the so-called fixed-point-free elements in (R −1 ).However, when  is odd, by our above isomorphism, ( − 1)-dimensional Möbius group ( − 1) cannot contain any fixed-point-free elements.In contrast to real hyperbolic space, we have regular elliptic elements in any dimensional complex and quaternionic hyperbolic space.
Using the quaternionic version in [24] of Schur's unitary triangularization theorem, we can prove the following lemma.
Proof.Let the right complex eigenvalues of  be   =    ( = 1, . . .,  + 1).By Schur's unitary triangularization theorem of quaternionic version in [24], there is a matrix  ∈ ( + 1; H) such that By the above lemma, we know that if the sequence {  } of nontrivial unitary quaternionic transformations converges to the identity, then the orders of   converge to infinity.So a family of groups { , } satisfies I-condition if there is no sequence {   }(   ∈  ,  ) converging to the identity, such that card(fix(  )) = ∞ for each .
When working in the matrix algebra, one has two choices, whether to use the spectral norm or the Hilbert-Schmidt norm.Following the ideas of Martin [2], we choose the Hilbert-Schmidt norm to construct our version of Jørgensen's inequality (Theorem 3) in Sp(, 1).
Lemma 11.Suppose that  and  ∈ Sp (, 1) generate a discrete and nonelementary group.Then where ‖ ⋅ ‖ is the Hilbert-Schmidt norm of an element.
Proof of Lemma 11.We can choose C to be the subspace of H spanned by {1, i}.With respect to this choice of C we can write H = C ⊕ Cj; that is, every element  ∈ H can be uniquely expressed as Similarly,  ∈ Sp(, 1) can be expressed as  =  1 +  2 j, where  1 ,  2 ∈  +1 (C), the set of ( + 1) × ( + 1) complex matrices.This gives an embedding We call () the complex representation of .Obviously,  is an isomorphism between Sp(,
Proof of Theorem 4. We divide our proof into two parts.
(1) We first prove that   is discrete.
Suppose that   is not discrete.Then there is a sequence {  } of   such that   ,  } are sequence of boundary elliptic elements which converges to the identity.This is a contradiction to our assumption of Icondition, while, for the first case, each  ,  shares a fixed point in   H .This is also a contradiction.The above proves the discreteness of   .
Since  , is discrete and nonelementary, there exist two loxodromic elements   and ℎ i having no common fixed points.Since   and ℎ  are words of the generators { , }, we can get the limit  and  by the word convergence of   and ℎ  , respectively.It remains to prove that ⟨, ℎ⟩ ⊂   is nonelementary.
And we call a nonelementary and discrete subgroup  of Sp(, 1) a quaternionic Kleinian group.For a sequence of subgroups { , } of Sp(, 1), we introduce the following condition.

)
If  contains a parabolic element but no loxodromic element, then  is elementary if and only if it fixes a point in   H ; (2) if  contains a loxodromic element, then  is elementary if and only if it fixes a point in H or a point-pair {, } ⊂   H ; (3)  is purely elliptic; that is, each nontrivial element of  is elliptic; then  is elementary and fixes a point in   H .By Lemma 6, we have the following lemmas.Lemma 7. If  ⊂ Sp (, 1) is discrete nilpotent group without elliptic element, then  is elementary.Lemma 8 (cf.[2, Lemma 2.8]).Let  and  be two distinct points in   H .If  ∈ Sp (, 1) interchanges  and , then ‖ −  +1 ‖ ≥ √ 2.