The Moduli Space of Points in the Boundary of Quaternionic Hyperbolic Space
Gaoshun Gou Yueping Jiang
math.CV math.AG 32M15, 22E40, 15B33, 14J10











Gaoshun Gou Department of Mathematics, Hunan-University, Changsha 410082, P. R. China 

              gaoshungou@hnu.edu.cn 

             Yueping Jiang Department of Mathematics, Hunan-University, Changsha 410082, P. R. China
              
 ypjiang@hnu.edu.cn
 





The Moduli Space of Points in the Boundary of Quaternionic Hyperbolic Space



     Gaoshun Gou Yueping Jiang 

    
==============================================================================




Let ℱ_1(n,m) be the space of ordered m-tuples of pairwise distinct points in ∂𝐇_ℍ^n up to its isometry group PSp(n,1). It is a real 2m^2-6m+5-∑^m-n-1_i=1m-2  n-1+i dimensional algebraic variety when m>n+1. In this paper, we construct and describe the moduli space of ℱ_1(n,m), in terms of the Cartan's angle and cross-ratio invariants, by applying the Moore's determinant.


32M1522E4015B3314J10



§ INTRODUCTION

Moduli spaces are spaces of solutions of geometric classification problems. They can be viewed as giving a universal space of parameters for the problem.
In this paper, we will study the Moduli space of ℱ_1(n,m) and the Moduli space of the discrete, faithful, totally loxodromic representation family of groups into PSp(2,1).
In projective space, those Moduli spaces are studied by many researchers.

In ℝℙ^n, Hakim-Sanler used Bruhat decomposition to study n+1 points in a certain standard position in <cit.>. They obtained some geometry applications of the structure of double coset space arising in representation theory.

In ℂℙ^n,
the Moduli spaces for the PU(n,1)-congruence classes of ordered distinct points have been well studied. There are three kinds of points: negative, isotropic and positive. The case of negative points was solved by Brehm-Et-Taoui in <cit.>. The moduli space of ordered quadruples of distinct isotropic points in ℂℙ^2 was studied by Falbel, Parker, Platis in <cit.> etc. Parker and Platis <cit.> defined analogous Fenchel-Nielsen coordinates for complex hyperbolic quasi-conformal representations of surface groups. The traces and cross-ratio invariants were used to parameterize the deformation space of complex hyperbolic totally loxodromic quasi-Fuchsian groups. They obtained some important results related to study of totally loxodromic groups. Falbel and Platis <cit.> showed the quotient configuration space ℱ is a real four dimensional variety and proved the existence of natural complex and CR structures on this space. The case of ordered quadruples of distinct isotropic points in ℂℙ^n was investigated by Cunha-Gusevskii in <cit.>, then they <cit.> further studied the moduli space of ordered m-tuple (m≥4) of distinct isotropic points in ℂℙ^n. The condition m=n+1 was discussed by Hakim-Sanler in <cit.>. Moreover, the positive points case was concerned in <cit.>.

In ℍℙ^n, the development of the moduli space for the space ℱ_1(n,m) of ordered distinct points is still in its childhood. The triple of distinct negative points in 𝐇_ℍ^n was considered by Brehm in <cit.>.
Recently, Cao <cit.> studied the moduli space of the triple and the quadruple of isotropic points in ℍℙ^n without associating quaternionic determinant.

It is natural to ask whether the other cases are still true for the moduli space of ordered distinct points in ℍℙ^n.
Therefore, we restrict our attention in this article to the Moduli space of ordered m-tuple (m≥4) of distinct points in ∂𝐇_ℍ^n.

The Gram matrices and invariants associated with a well defined determinant were used to study the moduli space of points
in ℂℙ^n. Thus, if we want to gain additional insight into those of ℍℙ^n, especially to study the relationship between cross-ratio invariants and the Moduli spaces, a suitable quaternionic determinant is essential. In order to bridge this gap, we use the Moore's determinant (see <cit.>).

In this paper, the main purpose is to construct and describe the Moduli space of PSp(n,1)-congruence classes of an ordered m-tuple (m≥4) of distinct points in ∂𝐇_ℍ^n by the special-Gram quaternionic matrices and invariants associated with Moore's determinant. For the purpose, we carry it out in two steps.

Firstly, we overcome the abstractness of the space ℱ_1(n,m) of m-tuple of distinct points in ∂𝐇_ℍ^n. In this place, the special-Gram quaternionic matrices, defined in Definition <ref>, plays an important role.
Let [𝔭] be an element of the space ℱ_1(n,m) of m-tuple 𝔭=(p_1,…,p_m) distinct points in ∂𝐇_ℍ^n. We view [𝔭] as the ordered m-tuple 𝔭 up to the action of PSp(n,1). In Definition <ref>, we give the definition of equivalent relation of special-Gram qutaternionic matrices. Next, we show that a m-tuple 𝔭 corresponding to a normalized special-Gram quaternionic matrix in section 3.1. Then in Proposition <ref> and Corollary <ref>, we will see that [𝔭] is uniquely corresponding to a normalized special-Gram quaternionic matrix. In fact, the Definition <ref> help us to realize this uniqueness. So then, we can directly study the normalized special-Gram quaternionic matrices instead of the space ℱ_1(n,m).

Secondly, we parameterize the normalized special-Gram quaternionic matrices. Here, the quaternionic cross-ratios and a quaternionic Cartan angle as the invariants are necessary.
Let 𝔭=(p_1,p_2,p_3,p_4) be a quadruple of distinct isotropic points in ∂𝐇_ℍ^n. The cross-ratio is defined by 
    𝕏(p_1,p_2,p_3,p_4)=⟨𝐩_3,𝐩_1⟩_1⟨𝐩_3,𝐩_2⟩_1^-1⟨𝐩_4,𝐩_2
    ⟩_1⟨𝐩_4,𝐩_1⟩_1^-1,

where 𝐩_i is the lift of p_i: 𝐩_i∈ℍ^n,1.
The quaternionic Cartan's angular invariant of triple (p_1,p_2,p_3) is given by:

    𝔸_ℍ=𝔸_ℍ(p_1,p_2,p_3)=
    arccos(-⟨𝐩_1,𝐩_2,𝐩_3⟩_1)/|⟨𝐩_1,𝐩_2,𝐩_3⟩_1|,
 
    ⟨𝐩_1,𝐩_2,𝐩_3⟩_1=⟨𝐩_1,𝐩_2⟩_1
    ⟨𝐩_2,𝐩_3⟩_1⟨𝐩_3,𝐩_1⟩_1.
 Where 
    (-⟨𝐩_1,𝐩_2,𝐩_3⟩_1)>0.

Let {𝕏_1,…,𝕏_d} be the set of cross-ratios associated with an ordered m-tuple 𝔭=(p_1,…,p_m) of distinct points in ∂𝐇_ℍ^n, where d=m(m-3)/2. We observe that 
    {𝕏_1,…,𝕏_d,𝔸_ℍ}

is a parameter system which is invariant under the action of PSp(n,1). This system with a unit pure quaternion 𝔲 and a positive number r forms the minimal parameter system which uniquely determines the PSp(n,1)-congruence classes of m-tuple 𝔭=(p_1,…,p_m), showed in Theorem <ref>.
Then, we have a characterization of the special-Gram quaternionic matrices associated with ordered m-tuple of distinct points in ∂𝐇_ℍ^n. This characterization gives all the relationships between our invariants, showed in Proposition <ref>, such that we can describe the Moduli space of the space ℱ_1(n,m).

Our main results of this paper is concluded as following:
  
Let

    _1(n,m)={w∈ℍ^d×𝔰𝔭(1)×ℝ^2:D^⋆_I^s_m-2(w)≥0, s≤ n-1; 
    D^⋆_I^s_m-2(w)=0, s>n-1},

where d=m(m-3)/2.
Then the space ℱ_1(n,m) is homeomorphic to the set _1(n,m).


Therefore, the Moduli space for ℱ_1(n,m) is described as _1(n,m).

Then, we apply the above theorem to describe the Moduli space of the representation family of groups into PSp(2,1). We have the following theorem.

Let H be a finitely generated group with a fixed ordered set of generators, Rep^o_lox(H,PSp(2,1)) be the discrete, faithful, totally loxodromic representation family of H into PSp(2,1).
Then Rep^o_lox(H, PSp(2,1)) can be identified with an open subset of 
    _1(2,2k)×Υ^k×Θ^2k,

where

    Υ={r∈ℝ| r>0, r≠1},  Θ=[0,π].






Our paper is organised as follows.
In 2 we recall some basic facts in quaternion and quaternionic hyperbolic geometry.
In 3 we give a characterization of special-Gram quaternionic matrices of ordered m-tuple of distinct points in ∂𝐇_ℍ^n.
In 4 we construct and describe the moduli space and prove the main results. Then we apply our results to study of the Moduli space of discrete, faithful, totally loxodromic representation family of groups into PSp(2,1).

When this paper was finished, soon we learnt that Cao <cit.> has also considered similar problem, but he use the different method and have different results from ours. We give more concrete discussion for the problem in terms of cross-ratios invariants.


§ PRELIMINARIES

In this section, we briefly review some facts about quaternions and quaternionic hyperbolic space. We also introduce Moore 's determinant on quaternonic Hermitian matrices.


 §.§ Quaternions and quaternonic matrices

Let ℝ, ℂ, ℍ denote the sets of real, complex and quaternion numbers, respectively.
We write 
    ℍ={q=t+ix+jy+kz | t,x,y,z∈ℝ},
 where 
    i^2=j^2=k^2=-1, ij=k=-ji, jk=i=-kj, ki=j=-ik.

Any q∈ℍ can also be written in the form q=z_1 + jz_2, where z_1,z_2∈ℂ. Note that zj=jz for z∈ℂ, since the center of ℍ is ℝ. Conjugation and modulus is given respectively by 
    q=t+ix+jy+kz=t-ix-jy-kz,  |q|=√(qq)=√(t^2+x^2+y^2+z^2).
 Then

    q_1 q_2=q_2 q_1  for   q_1,q_2∈ℍ.


    (q)=1/2(q+q)=t,


    (q_1q_2)=(q_2q_1)=(q_1q_2)=(q_1q_2)
    .

A pure quaternion is of the form 
    Pu(q)=1/2(q-q)=ix+jy+kz
 and  a unit quaternion is of the form 
    Un(q)=q/|q|∈ S,

where S is the unit sphere in ℍ. Let 𝔰𝔭(1) is the Lie algebra of Lie group Sp(1). Then 
    𝔰𝔭(1)=Pu(ℍ)∩ S
 will be the set of all unit pure quaternion.
Any quaternion is expressible as a power of a pure quaternion and it can be also  written in the form 
    q=|q|e^𝔲θ:=|q|(cosθ + 𝔲sinθ),

where 𝔲∈𝔰𝔭(1) and θ∈[0,π].

Next, we introduce the Moore 's determinant of a Hermitian quaternionic n× n matrix M. Such a matrix has quaternionic entries and  satisfies M^*=M, where M^* is  the conjugate transpose of M. There are several kinds of determinants associated to a quaternionic matrix. Moore 's determinant is the one which is convenient for our purposes. For more information about Moore 's determinant, see <cit.>.

Moore 's determinant was first studied by E. H. Moore and we denote it by det_𝕄. The following definition was given in <cit.>:
  For any Hermitian n× n quaternionic matrix M, the Moore's determinant of M is given by

    det_𝕄(M)=∑_σ∈ s_nε(σ)m_n_11n_12⋯ m_n_1l_1n_11m_n_21n_22⋯ m_n_rl_rn_r1,

where S_n is group of permutations.

Note that any σ∈ S_n can be written as a product of disjoint cycles. We permute each cycle until the leading index is the smallest and  then sort the cycles in increasing order according to the first number of each cycle. That is
 
    σ=(n_11⋯ n_1l_1)(n_21⋯ n_2l_2)⋯ (n_r1⋯ n_rl_r),

where for each i, we let n_i1<n_ij for all j>1, n_11<n_21⋯ <n_r1 and
 ε(σ)=(-1)^n-r.
In fact, we can arrange the cycles in decreasing order according to initial index, used in <cit.>.

We recall the following properties which was proved in <cit.>.

  
Let the quaterninic matrix 
    M=(
            [ M_1   0;   0 M_2;     ])
    ,

where M=M^*, M_1=M_1^*, M_2=M_2^*. Then 
    det_𝕄(M) = det_𝕄(M_1) det_𝕄(M_2).



  
(i) The Moore 's determinant of any  Hermitian complex matrix when this is considered as  Hermitian quaternionic matrix is equal to its usual determinant.
(ii) det_𝕄(U^*MU)=_𝕄(M)_𝕄(U^*U) for any Hermitian quaternionic matrix M and any quaternionic matrix U.

Next is the quaternionic version of Sylvester's principle of inertia. It was studied carefully by Moore in <cit.>. It can also be derived by Lemma <ref> and the Claim 1.1.7 of <cit.> and <cit.>. Note that the complex version was studied by
J.R. Parker in <cit.>.

  [Sylvester's principle of inertia] 
The signature of a Hermitian quaternionic matrix is independent of the means of finding it. In particular, if H_1 and H_2 are two k× k matrices with the same signature then there exists a k× k matrix U so that H_2=U^*H_1U.


The quaternionic generalization of the standard Sylvester criterion was given in Theorem 1.1.13 of <cit.>. We restate it as follows:
  [Sylvester's criterion]
A Hermitian quaternionic matrix M is positive definite if and only if its leading principal minors are all positive.



An analogous theorem holds for characterizing positive semi-definite Hermitian quaternionic matrices, which is showed in following theorem:

  
A Hermitian quaternionic matrix M is positive semi-definite if and only if its leading principal minors are all nonnegative.


We first prove necessity. Suppose that A is positive semi-definite and the matrix A_k associated with the principal minors of order k is given by

    A_k=(
            [ a_i_1i_1        ⋯ a_i_1i_k;        ⋮        ⋱        ⋮; a_i_ki_1        ⋯ a_i_ki_k;          ]).

Let Y^* A Y and X^* A_k X be the quadratic forms with matrices A and A_k, respectively. For any X_0=(b_i_1,b_i_2,…,b_i_k)^*≠0, we have Y_0=(c_1,…,c_n)^*, where j=1,2,…,n and if j=i_1,i_2,…,i_k then c_j=b_j, else c_j=0.

Since A is positive semi-definite Y_0^*AY_0≥0. Thus we obtain X_0^*A_kX_0≥0 (i.e. X^*A_kX is positive semi-definite). So there exists a non-degenerate matrix T_k, satisfying

    det_𝕄(T_k^*A_kT_k)
    =det_𝕄(A_k)det_𝕄(T_k^*T_k)≥0.


Next, we prove sufficiency. Suppose that the principal minors of A are all non-negative, choose the k-th leading principal minor and let A_k be its corresponding matrix, given by

    (
      [ a_11 a_12    ⋯ a_1k; a_21 a_22    ⋯ a_2k;    ⋮    ⋮    ⋱    ⋮; a_k1 a_k2    ⋯ a_kk;      ]),

where k=1,2,⋯,n. Then we have

    A_k+λI_k=
    (
    [ a_11+λ   a_12      ⋯   a_1k;   a_21 a_22+λ      ⋯   a_2k;      ⋮      ⋮      ⋱      ⋮;   a_k1   a_k2      ⋯ a_kk+λ;        ]).

According to Proposition 1.1.11 of <cit.> we observe that

    det_𝕄(A_k+λI_k)=λ^k+p_1λ^k-1+⋯+p_k-1λ+p_k,

where p_i is the sum of all principal minors of A_k. By the assumption that all principal minors of A are non-negative, we find that p_i≥0.
Thus if λ>0 then det_𝕄(A_k+λI_k)>0. That is to say, by Theorem <ref>, if λ>0 then A_k+λI_k is a positive-definite matrix.

Now, suppose that A is not positive semi-definite matrix, then there exists a non-zero vector
X_0=(b_1,b_2,…,b_n)^* such that 
    X_0^*AX_0=-c,   c>0.

Let 
    λ=c/X_0^*X_0=c/b_1^2+⋯+b_n^2>0.

Then

    X_0^*(A+λI)X_0=X_0^*AX_0 + X_0^*λI X_0=λX_0^*X_0+X_0^*AX_0=c-c=0
    ,

 a contradiction.


The rank of a quaternionic matrix was studied by Moore and Barnard in <cit.>, see  also <cit.>. We follow the former for our definition:
  Let A be a finite but non-zero quaternionic matrix, then the rank of A, denoted by rank(A), is defined to be the order of a maximal non-singular minor. If A=0 then rank(A)=0.

  For each finite quaternionic matrix A the rank is uniquely determined. It is equal to the number of columns (or rows) of A in a maximal set of left linear independent columns (or rows) of A. Consequently, a finite matrix is non-singular if and only if the matrix is square and its rank is equal to its order.

It could be checked directly that the quaternionic matrices A, A^*, AA^* and A^*A have the same rank.


 §.§ Quaternionic hyperbolic space


 In this section, we shortly review  quaternionic hyperbolic space. For more information, see <cit.>.

Let the Hermitian quaternionic matrices

    H_1   =(
         [     1     0     0;     0 I_n-1     0;     0     0    -1;       ])
     , 
     H_2=(
       [     0     0     1;     0 I_n-1     0;     1     0     0;       ])
     ,
    
       H_3   =(
         [    -1     0     0;     0 I_n-1     0;     0     0     1;       ])
       , 
       H_4=(
        [     0    -1     0;    -1     0     0;     0     0 I_n-1;       ]).

Denote by ℍ^n,1 the (n+1)-dimensional ℍ-vector space equipped with the Hermitian form of signature (n,1) given by Φ_i(𝐳,𝐰)=⟨𝐳,𝐰⟩_i=𝐰^*H_i𝐳. Here, 𝐳,𝐰 are column vectors in ℍ^n,1 and i=1,2,3,4. Let ℙ denote a natural right projection from ℍ^n,1∖{0} onto the quaternionic projective space ℍℙ^n.
Define subsets V_-^i, V_0^i, V_+^i of V=ℍ^n,1∖{0} as follows:

    V_-^i={V:⟨𝐳,𝐳⟩_i < 0},
          
    V_0^i={V:⟨𝐳,𝐳⟩_i = 0},
          
    V_+^i={V:⟨𝐳,𝐳⟩_i > 0}.

We say that 𝐳∈ℍ^n,1 is negative, isotropic or positive if 𝐳 is in V_-, V_0 or V_+, respectively. Motivated by relativity, these are sometimes called time-like, light-like and space-like. Their projections to ℍℙ^n are called negative, isotropic and positive points, respectively.

The quaternionic hyperbolic n-space ℍℙ^n is then defined by 𝐇_ℍ^n=ℙ(V_-^i) and its boundary is ∂𝐇_ℍ^n=ℙ(V_0^i).
When i=1 or 3, we have a ball model for quaternionic hyperbolic space and when i=2 or 4, we have a Siegel domain model. We may pass between them using a corresponding Cayley transform.

The metric on 𝐇_ℍ^n is given by

    ds^2=-4/⟨𝐳,𝐳⟩_i^2
    =(
       [   ⟨𝐳,𝐳⟩_i  ⟨d𝐳,𝐳⟩_i;  ⟨𝐳,d𝐳⟩_i ⟨d𝐳,d𝐳⟩_i;           ])
    .

Equivalently, it is given by the distance function ρ where

    cosh^2(ρ(ℙ𝐳,ℙ𝐰)/2)=
    ⟨𝐳,𝐰⟩_i⟨𝐰,𝐳⟩_i/⟨𝐳,𝐳⟩_i⟨𝐰,𝐰⟩_i.

We consider the Lie group Sp(n,1)={M∈ GL_n+1(ℍ):M^*H_iM=H_i}. The isometry group of 𝐇_ℍ^n is the group PSp(n,1)= Sp(n,1)/±I_n+1. This group is a non-compact real semi-simple Lie group.

We recall the following Proposition which was showed in <cit.>:
  
Sp(n,1) acts transitively on V_- and doubly transitively on V_0.

According to the above Proposition, we conclude:
  
If  𝐳_1,𝐳_2 are two distinct isotropic vectors in ℍ^n,1 then we have ⟨𝐳_1,𝐳_2⟩_i≠ 0.



§ A CHARACTERIZATION OF SPECIAL-GRAM QUATERNIONIC MATRICES

In this section, we consider the quaternionic hyperbolic space realized by the ball model related to the quadratic form H_1. An analogous method was used in the complex case in <cit.>. Here, we also have to take into account the non-commutativity of quaternions. 


 §.§ Special-Gram quaternionic matrix

Let 𝔭=(p_1,…,p_m) be an ordered m-tuple of pairwise distinct isotropic points in ∂𝐇_ℍ^n, n≥ 2 and let  𝔭̃=(𝐩_1,…,𝐩_m) be its lift:  𝐩_i∈ℍ^n,1.

  
The special-Gram quaternionic matrix associated to 𝔭 is 
    G=G(𝔭̃)=(g_ij)=(⟨𝐩_j,𝐩_i⟩_1),

where we assume that 𝐩_1 is the standard lift (see <cit.>) of p_1.


Note that g_ij is zero for i=j and non-zero for i≠ j, by Proposition <ref>.
It is straightforward to check that special-Gram quaternionic matrices are invariant under the action of the isometry group, i.e. 
    G(𝔭̃)=G(T(𝔭̃))=(T(𝐩_1),… ,T(𝐩_m)), T∈ PSp(n,1).


Still, G depends on the choice of lifts 𝐩_i for i=2,…,m.

If  D=  diag(1,λ_2,…,λ_m) is the diagonal quaternionic m× m matrix with quaternionic entries λ_i≠ 0 and 𝔭̃^'=(𝐩_1,𝐩_2λ_2,…,𝐩_mλ_m), then we have
G=G(𝔭̃^')=(⟨ p_jλ_j,p_iλ_i⟩_1)=(λ̅_i⟨ p_j,p_i⟩_1λ_j), in other words G=D^*GD.

We give the following definition of equivalence for special-Gram quaternionic matrices.
  
Two special-Gram quaternionic matrices M and M are equivalent, if there exists a non-singular diagonal matrix D=  diag(1,λ_2,…,λ_m) such that M=D^*MD.


Let 𝐩_1 be the standard lift of p_1, we will find that each ordered m-tuple 𝔭 of pairwise distinct points in ∂𝐇_ℍ^n is assigned an equivalence class of special-Gram quaternionic matrices. Let G and G are two equivalence special-Gram quaternionic matrices associated with an m-tuple 𝔭. By Lemma <ref>, we have 
    det_𝕄(G)=λ det_𝕄(G),

where λ>0. Note that the sign of det_𝕄(G) is independent of the chosen lifts 𝐩_i.

We have the following proposition, compared with Proposition 2.1 of <cit.>:
  
Let 𝔭=(p_1,…,p_m) be an ordered m-tuple of pairwise distinct points in ∂𝐇_ℍ^n. Then the equivalence class of a special-Gram quaternionic matrix associated to 𝔭 contains a unique matrix G(𝔭)=(g_ij)=(⟨𝐩_j,𝐩_i⟩_1) with g_ii=0, g_1j=1 for j=2,…,m.


Let 𝔭̃=(𝐩_1,…,𝐩_m) be a lift of 𝔭=(p_1,…,p_m). We know that g_ij≠0 whence i≠ j due to the fact that p_i are distinct and isotropic.
We re-scale 𝐩_j appropriately, replacing 𝐩_j by 𝐩_jλ_j, to obtain that g_1j=1 where
λ_1=1 and λ_j=⟨𝐩_j,𝐩_1⟩_1^-1 with j=2,…,m.

Let 𝔭̃^'=(𝐩_1,𝐩_2μ_2,…,𝐩_mμ_m) is the another lift of 𝔭=(p_1,…,p_m). We start from 𝔭̃^' to find the matrix (g^'_ij), where (g^'_ij) is in the equivalence class of a special-Gram quaternionic matrix associated to 𝔭 with g^'_ii=0 and g^'_1j=1 for j=2,…,m. Then we will find  (g_ij)=(g^'_ij). The uniqueness is clear.



    If we choose arbitrary lift for p_1, then two different lifts of 𝔭=(p_1,…,p_m), 𝔭̃=(𝐩_1,…,𝐩_m) and 𝔭̃^'=(𝐩_1μ_1,𝐩_2μ_2,…,𝐩_mμ_m), can't deduce to the same one matrix we need, i. e. (g_ij)≠(g^'_ij). In fact, we may deduce the matrix (g_ij) by different method from the one used in Proposition <ref> and observe that g^'_lj= g_lj/|μ_1|^2 for some fixed l. Hence (g_ij)=(g^'_ij) only if |μ_1|=1.


The unique matrix defined in Proposition <ref> is called a normalized special-Gram quaternionic matrix and is denoted by
G(𝔭) or simply  by G if there is no danger of confusion. It follows that there is a correspondence between the space of ordered m-tuples of pairwise distinct isotropic points and the space of normalized special-Gram quaternionic matrices.


 §.§ The Characterization of special-Gram quaternionic matrices

In this section, we discuss some properties of special-Gram quaternionic matrices associated with m-tuple of isotropic pairwise distinct points, m>1.

The Hermitian form Φ_1(𝐳,𝐰) on a subspace W⊂ℍ^n,1 is degenerate, if there exists a non-zero vector w∈ W such that ⟨ w,v⟩_1=0 for all v∈ W. We call the subspace W  degenerate, otherwise non-degenerate. Clearly, w must lie in V_0.

  
Let W be a degenerate subspace of ℍ^n,1 and 
    R(W)={𝐰∈ W|⟨𝐰,𝐮⟩_1,∀𝐮∈ W}.

Then R(W) is a line in V_0.


Let k= dim(W). If k=1 then the proof is completed. So we suppose 2≤ k≤ n+1. We will prove that R(W) can not have two linearly independent isotropic vectors. Supposing the contrary, i.e., there exist two such vectors 𝐰_1,𝐰_2 ∈ R(W). Then on one hand we have ⟨𝐰_1,𝐰_2⟩_1=0 by definition.  On the other hand, since 𝐰_1,𝐰_2∈ V_0, we have that  ⟨𝐰_1,𝐰_2⟩_1≠0 by Proposition <ref>. This completes the proof.


Lemma <ref> implies the following theorem.
  
Let W⊂ℍ^n,1.
For the Hermitian form Φ_1 (but as well as for all other Φ_i, i=2,3,4 as well) acting  on ℍ^n,1, the restriction Φ_1| W loses at most one dimension and the Witt index of a totally isotropic subspaces is 1.


We recall the following definition which was given in <cit.>:
  A subspace W⊂ℍ^n,1 is called hyperbolic if the restriction Φ_1| W is non-degenerate and indefinite; it is elliptic if Φ_1| W is positive definite; and it is parabolic if Φ_1| W is degenerate.


Let W be a (k+1)-dimensional subspace of ℍ^n,1, 1≤ k≤ n and denote its signature by (n_+,n_-,n_0), where n_+ (resp. n_-, n_0) is the number of positive (reps. negative, zero) eigenvalues of the Hermitian matrix of W. Then Theorem <ref> implies the following corollary:

The restriction of the Hermitian product Φ_1 on ℍ^n,1 to W only has signature (k,1,0), (k+1,0,0), (k-1,1,0) or (k,0,0).


Let 𝔭=(p_1,…,p_m) be an ordered m-tuple of pairwise distinct points in ℍℙ^n and let L(𝐩_i,…,𝐩_m)⊂ℍ^n,1 be the subspace spanned by the distinct vectors 𝐩_i,…,𝐩_m, where as usual 𝐩_i∈ℍ^n,1 is the lift of the isotropic point p_i, i=1,…,m. Moreover assume that dim(L(𝐩_i,…,𝐩_m))=k+1, then the following three cases exhaust all possibilities, compared with Cunha <cit.>:

  
  * L(𝐩_i,…,𝐩_m) is hyperbolic if it has signature (k,1,0), where 1≤ k≤ n;
  
  * L(𝐩_i,…,𝐩_m) is elliptic if it has signature (k+1,0,0), where 1≤ k≤ n;
  
  * L(𝐩_i,…,𝐩_m) is parabolic if it has signature (k,0,1) or (k-1,1,1), where 1≤ k≤ n-1.


Let G(𝔭̃)=(g_ij)=(⟨𝐩_j,𝐩_i⟩_1) be the special-Gram matrix associated to an m-tuple 𝔭=(p_1,…,p_m) of pairwise distinct points in the quaternionic projective space ℍℙ^n, where 𝐩_i is the lift of p_i∈ℍℙ^n, i=1,….m. By Corollary 6.2 of <cit.> and Theorem <ref>, the following holds:

    G(𝔭̃)=(
                                [ ⟨𝐩_1,𝐩_1⟩_1 ⟨𝐩_2,𝐩_1⟩_1           ⋯ ⟨𝐩_m,𝐩_1⟩_1; ⟨𝐩_1,𝐩_2⟩_1 ⟨𝐩_2,𝐩_2⟩_1           ⋯ ⟨𝐩_m,𝐩_2⟩_1;           ⋮           ⋮           ⋱           ⋮; ⟨𝐩_1,𝐩_m⟩_1 ⟨𝐩_2,𝐩_m⟩_1           ⋯ ⟨𝐩_m,𝐩_m⟩_1;             ])
    =C^*JC,

where C∈ GL_m(ℍ) and the Hermitian matrix J= diag(α_1,…,α_m), with α_i=± 1 or 0 and G(𝔭̃) has the same signature as J, which is obtained by restricting Φ_1 on ℍ^n,1 to the subspace L(𝐩_i,…,𝐩_m).

Therefore the signature of the special-Gram qutaternionic matrix G(𝔭̃) may be given according to the type of L(𝐩_i,…,𝐩_m):

  
  * If L(𝐩_i,…,𝐩_m) is hyperbolic, then G(𝔭̃) has signature (n_+,n_-,n_0) with 1≤ n_+≤ n, n_-=1, and n_++1+n_0=m.
  
  * If L(𝐩_i,…,𝐩_m) is elliptic, then G(𝔭̃) has signature (n_+,n_-,n_0) with 1≤ n_+≤ n, n_-=0, and n_++n_0=m.
  
  * If L(𝐩_i,…,𝐩_m) is parabolic, then G(𝔭̃) has  signature (n_+,n_-,n_0) with 1≤ n_+≤ n, 1≤ n_0≤ n, n_-=0 or 1, and n_++n_-+n_0=m.

The above conditions shall be called the signature conditions.

We now focus on the case that 𝔭=(p_1,…,p_m) is an ordered m-tuple of pairwise distinct points in ∂𝐇_ℍ^n, with the lift 𝔭̃=(𝐩_i,…,𝐩_m). Since the Moore's determinant of the upper left hand 2×2-block of G(𝔭̃) is -1, we see that L(𝐩_i,…,𝐩_m) always has signature with n_-=1. Thus any special-Gram quaternionic matrix associated to an m-tuple of pairwise distinct isotropic points has exactly signature (n_+,n_-,n_0) with n_-=1, 1≤ n_+≤ n, and 1+n_++n_0 =m.
Then the following holds:
  
Let G=(g_ij) be a Hermitian quaternionic matrix with g_ii=0, g_ij≠0 for i≠ j, m>1. Then G is a special-Gram quaternionic matrix associated with some ordered m-tuple 𝔭=(p_1,…,p_m) of distinct isotropic points in ∂𝐇_ℍ^n if and only if rank(G)≤ n+1 and G has the signature (n_+,n_-,n_0) with n_-=1, 1≤ n_+≤ n, and 1+n_++n_0 =m.


The proof is similar to the proof of the Proposition 2.2 in <cit.>. For completeness, we give it as follows:

Let G=(g_ij) be a Hermitian quaternionic matrix with g_ii=0, g_ij≠0 for i≠ j, m>1. According to Theorem <ref>, there exists a matrix U∈ GL_m(ℍ) such that U^*GU=B, where B=(b_ij) is the diagonal m× m-matrix such that b_ii=1 for 1≤ i≤ n_+, b_ii=-1 for i=n_++1 and b_ij=0 for all others. Now let A=(a_ij) be the (n+1)× m-matrix such that a_ii=1 for 1≤ i≤ n_+, a_ii=-1 for i=n_++1 and a_ij=0 for all others. Then we easily obtain that B=A^*H_1A. Thus the i-th column vector of the matrix AU^-1 can be defined to be 𝐩_i. We see that ⟨𝐩_j,𝐩_i⟩_1=g_ij. Finally, we find the isotropic points p_i=π(𝐩_i), this is our desired.


The following proposition is in <cit.>:
  Let W be a subspace of ℍ^n,1. Each linear isometry of W into ℍ^n,1 can be extended to an element of Sp(n,1).

We thus have:
  
Let 𝔭=(p_1,…,p_m) and 𝔭^'=(p^'_1,…,p^'_m) be two m-tuples of pairwise distinct points in ∂𝐇_ℍ^n. Then 𝔭 and 𝔭^' are congruent in PSp(n,1) if and only if their associated special-Gram quaternionic matrices are equivalent.



Let 𝔭=(p_1,…,p_m) and 𝔭^'=(p^'_1,…,p^'_m) be two m-tuples of pairwise distinct points in ∂𝐇_ℍ^n, and let G(𝔭) and G(𝔭^') be their normalized special-Gram quaternionic matrices. Then 𝔭 and 𝔭^' are congruent in PSp(n,1) if and only if G(𝔭)=G(𝔭^').


Let G(𝔭)=(g_ij) be the matrix

    G(𝔭)=(
          [    0    1    1    1    ⋯    1;    1    0 g_23 g_24    ⋯ g_2m;    1 g_23    0 g_34    ⋯ g_3m;    1 g_24 g_34    0    ⋯ g_4m;    ⋮    ⋮    ⋮    ⋮    ⋱    ⋮;    1 g_2m g_3m g_4m    ⋯    0;      ]),

where g_ij≠0 for i≠ j.
Then the normalized special-Gram quaternionic matrix associated with an ordered m-tuple of pairwise distinct isotropic points is necessarily of the above form.
  
Let G=(g_ij) be a Hermitian quaternionic m× m-matrix, m>2, with that g_ii=0, g_1j=1 for j=2, …, m and g_ij≠0 for i≠ j. Then there exists a matrix in GL_m(ℍ) which transforms G into the matrix

    (
      [   0   1   0;   1   0   0;   0   0 G^⋆;     ]),

where G^⋆ is Hermitian quaternionic (m-2)× (m-2)-matrix given by

    G^⋆=(
           [    -(g_23+g_23) -g_23-g_24+g_34               ⋯ -g_23-g_2m+g_3m; -g_23-g_24+g_34    -(g_24+g_24)               ⋯ -g_24-g_2m+g_4m;               ⋮               ⋮               ⋱               ⋮; -g_23-g_2m+g_3m -g_24-g_2m+g_4m               ⋯    -(g_2m+g_2m);                 ]).



Let 
    T=(
             [     1     0 -g_23 -g_24     ⋯ -g_2m;     0     1    -1    -1     ⋯    -1;     0     0     1     0     ⋯     0;     0     0     0     1     ⋯     0;     ⋮     ⋮     ⋮     ⋮     ⋱     ⋮;     0     0     0     0     ⋯     1;       ])

then

    T^*GT=(
      [   0   1   0;   1   0   0;   0   0 G^⋆;     ]).



The matrix G^⋆ is called the associated matrix to G.

Let G=(g_ij) be a Hermitian quaternionic m× m-matrix satisfying the conditions of Proposition <ref> and let G^⋆ be the associated matrix to G. Then

    _𝕄(G)=-_𝕄(G^⋆)/_𝕄(T^*T), and rank(G)=rank(G^⋆)+2.



The following Theorem follows by Proposition <ref>.
  
Let G=(g_ij) be a Hermitian quaternionic m× m-matrix, m>2, such that g_ii=0, g_1j=1 for j=2,…,m, and g_ij≠0 for i≠ j. Let G^⋆ be the associated matrix to G. Then G is a special-Gram quaternionic matrix associated to some ordered m-tuple 𝔭=(p_1,…,p_m) of pairwise distinct points in ∂𝐇_ℍ^n if and only if rank(G^⋆)≤ n-1 and G^⋆ is positive semi-definite.


Using Theorem <ref>, we may express Theorem <ref> equivalently as:
  
Let G=(g_ij) be a Hermitian quaternionic m× m-matrix, m>2, such that g_ii=0, g_1j=1 for j=2,…,m, and g_ij≠0 for i≠ j. Let G^⋆ be the associated matrix to G. Then G is a special-Gram quaternionic matrix associated to some ordered m-tuple 𝔭=(p_1,…,p_m) of pairwise distinct  points in ∂𝐇_ℍ^n if and only if rank(G^⋆)≤ n-1 and all principal minors of G^⋆ are non-negative.


We call the conditions in Theorem <ref> the Moore's determinant conditions.


§ THE MODULI SPACE



 §.§ Invariants

We first  recall Cartan's angular invariant and quaternionic cross-ratio. One may find more information in <cit.>.

Let 𝔭=(p_1,p_2,p_3) be an ordered triple of pairwise distinct  points in ∂𝐇_ℍ^n. The quaternionic Cartan's angular 𝔸_ℍ(𝔭) of 𝔭 is defined by

    𝔸_ℍ(𝔭)=
    arccos(-⟨𝐩_1,𝐩_2,𝐩_3⟩_1)/|⟨𝐩_1,𝐩_2,𝐩_3⟩_1|,

where 𝐩_i∈ℍ^n,1 are the lifts of p_i, ⟨𝐩_1,𝐩_2,𝐩_3⟩_1=⟨𝐩_1,𝐩_2⟩_1
⟨𝐩_2,𝐩_3⟩_1⟨𝐩_3,𝐩_1⟩_1 and (-⟨𝐩_1,𝐩_2,𝐩_3⟩_1)>0.

Rewrite -⟨𝐩_1,𝐩_2,𝐩_3⟩_1 as |⟨𝐩_1,𝐩_2,𝐩_3⟩_1|e^𝔲θ, where 𝔲∈𝔰𝔭(1). Then θ=𝔸_ℍ(𝔭). Therefore, the quaternionic Cartan's angular  can also be given by:

    𝔸_ℍ(𝔭)=(-⟨𝐩_1,𝐩_2,𝐩_3⟩_1).


One can prove that 0≤𝔸_ℍ(𝔭)≤π/2
and 𝔸_ℍ(𝔭) independent of the chosen lifts and the order of three points.

We give the definition of quaternionic cross-ratio, following <cit.>. See <cit.> for more cross-ratio.

Let 𝔭=(p_1,p_2,p_3,p_4) be a quadruple of pairwise distinct  points in ∂𝐇_ℍ^n. Their cross-ratio is defined by 
    𝕏(p_1,p_2,p_3,p_4)=⟨𝐩_3,𝐩_1⟩_1⟨𝐩_3,𝐩_2⟩_1^-1⟨𝐩_4,𝐩_2
    ⟩_1⟨𝐩_4,𝐩_1⟩_1^-1,

where 𝐩_i are the lifts of p_i.
Observe that 
    𝕏(p_1λ_1,p_2λ_2,p_3λ_3,p_4λ_4)=λ_1⟨𝐩_3,𝐩_1⟩_1⟨𝐩_3,𝐩_2⟩_1^-1⟨𝐩_4,𝐩_2
    ⟩_1⟨𝐩_4,𝐩_1⟩_1^-1λ_1^-1.


Let now 𝔭=(p_1,…,p_m) be an m-tuple of pairwise distinct points in ∂𝐇_ℍ^n.
For short, let 
    𝕏_2j=𝕏(p_1,p_2,p_3,p_j), 𝕏_3j=𝕏(p_1,p_3,p_2,p_j), 𝕏_kj=𝕏(p_1,p_k,p_3,p_j),

where m≥4, 4≤ k ≤ m-1, k<j. It is clear that the number of the above cross-ratios is equal to m(m-3)/2.

By direct computations, we have the following proposition:
  
Let 𝔭=(p_1,…,p_m) be an m-tuple of pairwise distinct points in ∂𝐇_ℍ^n and G(𝔭)=(g_ij) be the normalized special-Gram quaternionic matrix of 𝔭. Then the following relations hold:

    𝔸_ℍ   =𝔸_ℍ(p_1,p_2,p_3)=arg(-g_23),
    𝕏_2j   =𝕏(p_1,p_2,p_3,p_j)=g_23^ -1g_2j,
    𝕏_3j   =𝕏(p_1,p_3,p_2,p_j)=g_23^ -1g_3j,
    𝕏_kj   =𝕏(p_1,p_k,p_2,p_j)=g_2k^ -1g_kj,
      and  
    g_23   =-re^-𝔲𝔸_ℍ,
    
    g_2j   =
    -re^-𝔲𝔸_ℍ𝕏_2j,
    
    g_3j   =
    -re^𝔲𝔸_ℍ𝕏_3j,
    
    g_kj   =-r𝕏_2ke^𝔲𝔸_ℍ𝕏_kj,

where r=|g_23|  and 𝔲∈𝔰𝔭(1) and all indices are in accordance with the indices of the above-defined  cross-ratios.


According to Corollary <ref>, we have the following result which is useful for us to study the Moduli space:
  
The congruence class of 𝔭 in PSp(n,1) is uniquely determined by a positive number r and a unit pure quaternion 𝔲∈𝔰𝔭(1) and the invariants given the above 𝕏_2j,𝕏_3j,𝕏_kj,𝔸_ℍ.



 §.§ Moduli space and proof of the theorem <ref>

Let I^s_n={(i_1,…,i_s) | 1≤ i_1< i_2…<i_s≤ n} and let A_I^s_n denote the sub-matrix of an n× n-matrix A, formed by choosing the elements of the original matrix from the rows whose indices are in i_1,…,i_s and columns whose indexes are in i_1,…,i_s.

Let 𝔭=(p_1,…,p_m) be an m-tuple of pairwise distinct points in ∂𝐇_ℍ^n and G(𝔭)=(g_ij) be the normalized special-Gram quaternionic matrix of 𝔭. Let G^⋆ be the associated (m-2)× (m-2)-matrix to G(𝔭). The principal minors of G^⋆ are _𝕄(G^⋆_I^s_m-2).

By Proposition <ref>, we treat _𝕄(G^⋆_I^s_m-2) as functions of 
    (𝕏_2j,𝕏_3j,…,𝕏_kj,𝔲,𝔸_ℍ,r).

Identify w=(q_1,…,q_d,𝔲,t_1,t_2) with (𝕏_2j,𝕏_3j,…,𝕏_kj,𝔲,𝔸_ℍ,r), where q_i∈ℍ is nonzero, i=1,…,d and d=m(m-3)/2.

We define the map 
    D^⋆_I^s_m-2:ℍ^m(m-3)/2×𝔰𝔭(1)×ℝ^2→ℝ,

given by

    w↦ D^⋆_I^s_m-2(w)=_𝕄(G^⋆_I^s_m-2).


Let ℱ_1(n,m) be the quotient configuration space of ordered m-tuples of pairwise distinct points in ∂𝐇_ℍ^n, that is, the space of m-tuples cut by the action of PSp(n,1). Let [𝔭]∈ℱ_1(n,m). Then, according to Theorem <ref> and Proposition <ref>, we define the map

    τ_1:ℱ_1(n,m)→ℍ^m(m-3)/2×𝔰𝔭(1)×ℝ^2,

given by

    [𝔭]↦ w.

  The lower index 1 means that we use the Hermitian form H_1 for the entries of the special-Gram matrix. Likewise, the index 2, 3, 4.

Let 
    _1(n,m)={w∈ℍ^d×𝔰𝔭(1)×ℝ^2:D^⋆_I^s_m-2(w)≥0, s≤ n-1; 
    D^⋆_I^s_m-2(w)=0, s>n-1},

where w=(q_1,…,q_d,𝔲,t_1,t_2), 0≠ q_i∈ℍ, 𝔲∈𝔰𝔭(1), t_1∈[0,π/2], t_2>0 for all i=1,…,d=m(m-3)/2.

Now let's prove our main result:

 

Our purpose is to find a homeomorphic map ℱ_1(n,m)→_1(n,m). It will be the map τ_1:ℱ_1(n,m)→_1(n,m) as above.
Let _1(n,m) be equipped with the topology inherited from ℍ^m(m-3)/2×𝔰𝔭(1)×ℝ^2. Hence we only need to prove that  τ_1 is bijective.

Injectivity follows straightforwardly by Theorem <ref>.
It is only necessary to show that the map τ_1 is surjective.

If w∈_1(n,m), we can construct a Hermitian quaternionic m× m-matrix G=(g_ij) with g_ii=0 and g_1j=1 for j=2,…,m.
We see that  w=(q_1,…,q_d,𝔲,t_1,t_2) identify with (𝕏_2j,𝕏_3j,…,𝕏_kj,𝔲,𝔸_ℍ,r). Then we can fix the other entries of G using Proposition <ref>.
If this G satisfies Moore's conditions, see Theorem <ref>, then G is the normalized special-Gram quaternionic matrix for some ordered m-tuple of pairwise distinct  points 𝔭 in ∂𝐇_ℍ^n. In other words, w uniquely corresponds to a point [𝔭]∈ℱ_1(n,m). This proves that τ_1 is surjective.


We call _1(n,m) the moduli space for ℱ_1(n,m).  The following corollary is clear.

ℱ_1(n,m) is a real 2m^2-6m+5-∑^m-n-1_i=1m-2  n-1+i dimensional algebraic variety when m>n+1.


  ℱ_1(n,m) is homeomorphic to ℱ_i(n,m) since we may pass between two different Hermitian forms of the same signature by using Cayley transformations, where i=2,3,4. Therefore _i(n,m) shares the same topology with _1(n,m).



 §.§ Moduli space for representation family of totally loxodromic groups and proof of the theorem <ref>

This section is motivated by the last section of <cit.>. However, our results are different, even when we restrict ourselves to the complex hyperbolic space. Our target space is the deformation space of discrete, faithful, totally loxodromic and finitely generated groups in PSp(2,1).
  A subgroup of PSp(n,1) is called totally loxodromic if it comprises only loxodromic elements and the identity.


The following important proposition comes from <cit.>.
  
Let G be a totally loxodromic subgroup of Sp(n,1) such that the quaternionic dimension dim_ℍ(M(G)) of the smallest G-invariant totally geodesic submanifold M(G), is even. Then G is discrete.


Let G_0 be a totally loxodromic subgroup of PSp(n,1) and dim_ℍ M(G_0) be even. By the preceding proposition, G_0 is discrete. For example, the totally loxodromic subgroups of PSp(2,1) that are generated by finite distinct generators without common fixed points. These subgroup are discrete subgroups of PSp(2,1).

Let H=⟨ h_1,h_2,…,h_k⟩ be a finitely generated group with a fixed ordered set of generators ⟨ h_1,h_2,…,h_k⟩ and G a topological group. The set of homomorphisms Hom(H,G) naturally sits inside G^k via the evaluation map f:Hom(H,G)→ G^k given by ρ↦ (ρ(h_1),…,ρ(h_k)). Hence, Hom(H,G) has an induced topology by f.

  The representation family of discrete, faithful, totally loxodromic representations of H into G is

    Rep^o_lox(H,G)={ρ∈Hom(H,G):ρ injective; ρ(H) discrete, loxodromic}.



We endow Rep^o_lox(H,G) with the topology of point-wise convergence. In this topology a sequence formed by the homomorphisms ρ_j:H→ G,j=1,2,…, converges to a homomorphism ρ:H→ G if and only if for each h∈ H the sequence ρ_1(h),ρ_2(h),… converges to ρ(h) in the topological group G.

In remainder of this section we assume G=PSp(2,1).

The points of Rep^o_lox(H,G) are identified with the G-conjugation equivalence classes of G_0, where G_0=ρ(H). Let G_0=⟨ g_1,…,g_k⟩ with g_i=ρ(h_i), then  g_i≠ g_j for all i≠ j.

We now describe the moduli space of Rep^o_lox(H,G). First, we need to verify that the G-conjugation equivalence class of each loxodromic element g_i can be uniquely determined by some parameters.

Cao and Gongopadhyay proved the following theorem on the conjugation classification for the complex and the quaternionic hyperbolic plane in <cit.>:

  
Let 𝔽=ℂ or ℍ and denote by Û(2,1:𝔽) the isometry group preserving the Hermitian form ⟨𝐳,𝐰⟩_4=-(z_0w_1+z_1w_0)+
z_2w_2 which gives the Siegel domain model. Suppose that A∈Û(2,1:𝔽)
is a loxodromic element. Then A is conjugate to an element of the form

    L=L(β,θ)=([    re^iβ        0        0;        0 r^-1e^iβ        0;        0        0     e^iθ ])
    ,   r>0,  r≠1,

such that 0≤β, θ≤π when 𝔽=ℍ, and π≤β, θ≤π when 𝔽=ℂ.


Note that the ball model and the Siegel domain model for hyperbolic space are interchanged by Cayley transformations. By Theorem <ref>, we obtain that every loxodromic equivalence class with respect to the conjugation action of Sp(2,1) or U(2,1) can be uniquely determined by r, β and θ parameters.

 

According to Proposition <ref>, we associate the quintuple (p^+_i,p^-_i,r,β,θ) to each g_i, where p^+_i and p^-_i are respectively the attracting and the repelling fixed points of g_i. Since G_0 is discrete, all fixed points p^+_1,p^-_1,…,p^+_k,p^-_k are distinct.

Then

    (p^+_1,p^-_1,…,p^+_k,p^-_k;r_1…,r_k;β_1,θ_1…,β_k,θ_k),

may be associated to G_0=⟨ g_1,…,g_k⟩.

Now, according to Theorem <ref>, the proof is completed.

If we restrict the Theorem <ref> to complex space, then Θ=[-π,π].


The author G. Gou wishes to express his thanks to Professor I. D. Platis for his warmly help, and Professor J. R. Parker, Professor E. Falbel and professor W. Cao for viewing the preprint of the paper and their suggestions. Besides, authors would like to thank NSFC (No. 11371126) and NSFC (No. 11701165) for financial support. We also thank the referees and the useful references.


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Alesker29S. Alesker: Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables. Bull. Sci. Math, 127: 1-35, (2003)
Apanasov2007B. N. Apanasov , I. Kim: Cartan angular invariant and deformations of rank 1 symmetric spaces. Sbornik Math, 198(2): 147-169, (2007)
Aslaksen15H. Aslaksen: Quaternionic determinants, Math. Intelligencer, 18(3): 57-65, (1996)
Barn.Moore28R. W. Barnard and E. H. Moore: General analysis. Part 1: Memoirs of the American Philosophical Society, (1935)
Brehm2U. Brehm: The shape invariant of triangles and trigonometry in two-point homogeneous spaces. Geom. Dedicata, 33: 59-76, (1990)
Brehm3U. Brehm , B. Et-Taoui: Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces. Manuscr. Math, 96(1): 81-95, (1998)
CaoW. S. Cao: Congruence classes of points in quaternionic hyperbolic space. Goem. Dedicata, 180: 203-228, (2016)
Cao2012W. S. Cao, K. Gongopadhyay: Algebraic characterization of isometries of the complex and the quaternionic hyperbolic planes. Geom. Dedicata, 157: 23-39, (2012)
Cao2W. S. Cao: The moduli space of points in quaternionic projective space. preprint.


Chen1974S. S. Chen, L. Greenberg: Hyperbolic Spaces, Contributions to Analysis. New York: Academic Press (1974)
Cunha4H. Cunha, N. Gusevskii: The moduli space of quadruples of points in the boundary of complex hyperbolic space. Transform. Groups, 15(2): 261-283, (2010)
Cunha5H. Cunha, N. Gusevskii: The moduli space of points in the boundary of complex hyperbolic space. J. Geom. Anal., 22: 1-11, (2012)
Cunha6H. Cunha, F. Dutenhefner, N. Gusevskii, R. Santos Thebaldi: The moduli space of complex geodesics in the complex hyperbolic plane. J. Geom. Anal., 22: 259-319, (2012)
HakimJ.2000J. Hakim, H. Sandler: Standard position for objects in hyperbolic space. J. Geom., 68: 100-113, (2000)
HakimJ.16J. Hakim, H. Sandler: The moduli space of n + 1 points in complex hyperbolic n-space. Geom. Dedic., 97: 3-15, (2003)
Falbel8E. Falbel, I. D. Platis: The PU(2, 1) confguration space of four points in S^3 and the cross-ratio variety. Math. Ann., 340(4): 935-962, (2008)

KGongopadhyay19 K. Gongopadhyay, S. Parsad: Classification of quaternionic hyperbolic isometries. Conform. Geom. Dyn. 17, 68-76, (2013)
N.Jacobson22N. Jacobson: An application of E. H. Moore's determinant of a Hermitian matrix. Bull. Amer. Math. Soc., 45: 745-748, (1939)
I.Kim17I. Kim , P. Pansu: Local rigidity in quaternionic hyperbolic space. Journal of the European Mathematical Society (JEMS), 11(6): 1141-1164, (2009)
KimI.18I. Kim, J. R. Parker: Geometry of quaternionic hyperbolic manifolds. Math. Proc. Camb. Philos. Soc., 135: 291-320, (2003)
E.H.Moore7 E. H. Moore: On the determinant of an hermitian matrix of quaternionic elements. Bull. Amer. Math. Soc., 28: 161-162, (1922)
Parker2008 J. R. Parker, I. D. Platis: Complex hyperbolic Fenchel-Nielsen coordinates. Topology, 47(2): 101-135, (2008)

Parker2007 J. R. Parker: Hyperbolic Spaces. The Jyväskylä Notes, (2007)
Platis2014I. D. Platis: Cross-ratios and the Ptolemaean inequality in boundaries of symmetric spaces of rank 1. Geometr. Dedicata., 169: 187-208, (2014)
Zhang1997F. Z. Zhang: Quaternions and matrices of quaternions, Linear Algebra Appl., 251: 21-57, (1997)


