1 Introduction
Quaternionic linear algebra plays a fundamental role across multiple disciplines due to its rich geometric structure and computational advantages over real and complex systems. In mathematics, quaternionic matrices arise in differential geometry and representation theory (see [Reference Díaz, Verjovsky and Vlacci12, Reference Holod and Klimyk19]). In physics, they provide elegant formulations for string theory, twistors, and gauge theories, where their non-commutativity captures essential symmetries (e.g., [Reference Hanson18]). Meanwhile, in computer science and engineering, quaternions enable efficient representations of three-dimensional rotations, making them indispensable in robotic kinematics and real-time graphics (cf. [Reference Dam, Koch and Lillholm11, Reference Rodman27]).
Despite its interdisciplinary utility, quaternionic linear algebra remains underdeveloped compared to its real and complex counterparts. Critical limitations – such as the absence of a determinant-like function and the lack of a comprehensive spectral theory (see [Reference Hanson18]) – hinder progress in both theoretical and applied contexts. These gaps are particularly evident in the study of discrete group dynamics on hyperbolic spaces, where tools from classical linear algebra often fall short. To address these challenges, we turn to the symplectic subgroup
$\operatorname {Sp}(1,n)$
, whose geometric structure naturally circumvents these limitations through its action on quaternionic hyperbolic spaces.
In this article, we focus on
$\operatorname {Sp}(1,n)$
and its dynamical action on the unitary quaternionic ball
$\mathbf {H}_{\mathbb {H}}^n$
. Our primary contribution is an elementary geometric proof of the Cartan decomposition theorem for
$\operatorname {Sp}(1,n)$
(Theorem 1.1); this decomposition serves as the cornerstone for our subsequent analysis of limit sets. Specifically, we show that the action of
$\operatorname {PSp}(1,n)$
on
$\mathbb {H}\mathbb {P}^n$
undergoes a radical structural change as we change from dimension one to the higher-dimensional setting (Theorem 1.2).
To contextualize our results, we first recall the indefinite symplectic group
$\operatorname {Sp}(1,n)$
, defined as the set of invertible quaternionic matrices preserving the Hermitian form H of signature
$(1,n)$
. Explicitly,
$$\begin{align*}\mathrm{Sp}(1,n) = \big\{ g \in \mathrm{GL}(n+1, \mathbb{H}) \mid gHg^* = H \big\}, \end{align*}$$
where
$\mathrm {GL}(n+1, \mathbb {H})$
denotes the general linear group of
$(n+1) \times (n+1)$
invertible matrices over the quaternions
$\mathbb {H}$
;
$g^*$
is the conjugate transpose of g (for a quaternion
$q = a + bi + cj + dk$
, its conjugate is
$\overline {q} = a - bi - cj - dk$
); and H is the Hermitian matrix defining the bilinear form, given by
$$\begin{align*}H = \begin{pmatrix} 0 & 0 & 1 \\ 0 & I_{n-1} & 0 \\ 1 & 0 & 0 \end{pmatrix}. \end{align*}$$
With this in mind, our first result is as follows.
Theorem 1.1 (Cartan Decomposition)
For each
$g\in \mathrm {Sp}(1,n),$
there are elements
$k_+,k_-\in \mathrm {Sp}(n+1)\cap \mathrm {Sp}(1,n)$
, and
$\lambda (g) \geq 0$
such that
$g=k_+\mu (g)k_-$
, and
$\mu (g) \in \mathrm {Sp}(1,n)$
, where
$$\begin{align*}\mu(g)= \begin{pmatrix} e^{\lambda(g)}\\ & I_{n-1}\\ & & e^{-\lambda(g)} \end{pmatrix}. \end{align*}$$
A particularly elegant connection emerges in modeling real hyperbolic 3-space using quaternions. The upper half-space model of
$\mathbb {H}^3_+$
can be described using Hamiltonian quaternions
$\mathbb {H}$
, identified as
$$\begin{align*}\mathbb{H}^3_+ = \{ q \in \mathbb{H} \mid \mathrm{Im}(q)> 0 \}, \end{align*}$$
where the imaginary part is defined in the quaternionic sense. This framework naturally accommodates Möbius transformations through fractional linear maps:
$$\begin{align*}q \mapsto (a q + b)(c q + d)^{-1}, \quad a, b, c, d \in \mathbb{C},\ ad - bc \neq 0, \end{align*}$$
as demonstrated by Ahlfors and Abikoff (see [Reference Abikoff1, Reference Ahlfors, Chavel and Farkas2]). This approach not only establishes a quaternionic analytic foundation for studying discrete isometry groups (cf. [Reference Borel5, Reference Maclachlan and Reid23]) but also interprets Möbius transformation groups as projective transformations acting on
$\mathbb {HP}^n$
. Our next result is a direct application of the Cartan decomposition, and provides a description of limit set for the subgroups in
$\mathrm {PSp}(1,n)$
.
Theorem 1.2 For any non-elementary discrete subgroup
$G \subset \mathrm {PSp}(1,n)$
, the set
$$\begin{align*}\Omega(G) = \mathbb{HP}^n - \bigcup_{p \in \Lambda_{CG}(G)} p^\bot, \end{align*}$$
where
$\Lambda _{CG}(G)$
is the set of accumulation points of an orbit and
$p^\bot = \{[q] \in \mathbb {HP}^n : pQ q^* = 0\}$
, coincides with G’s equicontinuity set and represents the largest open domain where G acts properly discontinuously.
This interplay between quaternionic geometry and group dynamics invites exploration of classical theorems in higher dimensions. Foundational results like Koebe’s Retrosection Theorem and Bers’ Simultaneous Uniformization Theorem – cornerstones of Riemann surface theory and Kleinian groups – raise a natural question: Can these be extended to quaternionic projective spaces? Before addressing this, we briefly recall their classical formulations.
Koebe’s Theorem states that every compact Riemann surface of genus
$g \geq 1$
can be represented as
$\Omega /G$
, where G is a Schottky group (a finitely generated, discrete, free group of loxodromic Möbius transformations) and
$\Omega \subset \widehat {\mathbb {C}}$
is a domain with a Cantor-set boundary (see [Reference Koebe22, Reference Maskit24, Reference Mumford, Series and Wright26]). Geometrically, such groups are characterized by disjoint disks
$D_i, D_i'$
in
$\widehat {\mathbb {C}}$
mapped onto each other by their generators (cf. [Reference Maskit24]). However, no analogous characterization exists in higher-dimensional projective spaces (see [Reference Cano6, Reference Conze and Guivarc’h9]), prompting two potential generalizations:
-
• Algebraic Approach: Define the uniformizing group G as a finitely generated, discrete group which is purely loxodromic.
-
• Geometric Approach: Require the group G to admit:
-
– A finite symmetric generating set
$\Sigma $
. -
– A family
$\{A_{a}\}_{a \in \Sigma }$
of compact, pairwise disjoint subsets in
$\mathbb {HP}^n$
satisfying:
$$\begin{align*}\bigcup_{b \in \Sigma \setminus \{a^{-1}\}} a(A_{b}) \subsetneq A_{a}. \end{align*}$$
-
Similarly, Bers’ Theorem (see [Reference Bers3]) asserts that for any closed orientable surface S of genus
$g \geq 2$
with two complex structures X and Y, there exists a discrete group
$\Gamma \subset \mathrm {PSL}(2,\mathbb {C})$
and
$\Gamma $
-invariant Jordan domains
$\Omega _1, \Omega _2 \subset \widehat {\mathbb {C}}$
such that:
-
(1)
$X \cong \Omega _1 / \Gamma $
and
$Y \cong \Omega _2 / \Gamma $
; -
(2)
$\Omega _1 \cup \Omega _2$
is dense in
$\widehat {\mathbb {C}}$
.
Contrasting these classical frameworks with the quaternionic setting, our main results reveal fundamental limitations.
Theorem 1.3 Neither the algebraic nor geometric versions of Koebe’s Retrosection Theorem extend to
$\mathrm {PGL}(n + 1, \mathbb {H})$
-groups acting on
$\mathbb {HP}^n$
for
$n \geq 2$
.
Theorem 1.4 No analog of Bers’ Simultaneous Uniformization Theorem exists for
$\mathrm {PGL}(n + 1,\mathbb {H})$
-groups acting on
$\mathbb {HP}^n$
with
$n \geq 2$
.
The argument of both proofs hinges on two key results:
-
• Structural Rigidity (Klingler and Mok-Yeung, Theorem 6.1): Complete finite-volume quaternionic hyperbolic manifolds admit a unique hyperbolic structure.
-
• Limit Set Characterization (Theorem 1.2).
For instance, assume (for contradiction) that a Bers-type uniformization exists for a compact manifold of the form
$M \sqcup M$
, where
$M = \mathbf {H}^n_{\mathbb {H}}/\Gamma $
and
$\Gamma \subset \mathrm {PSp}(1,n)$
is a discrete group. By rigidity, the uniformizing group
$G \subset \mathrm {PSp}(1,n)$
must satisfy
$G = \Gamma $
(up to conjugation) and have
$\Omega (G) = \mathbf {H}^n_{\mathbb {H}}$
as its only connected component. This leads to the contradiction
$M \sqcup M = M$
. An analogous argument applies to Koebe’s case.
This article is structured as follows: Section 2 begins by reviewing essential results in quaternion linear algebra and quaternionic hyperbolic geometry, while also establishing the notation used in subsequent sections. Building on this foundation, Section 3 presents an elementary proof of the Cartan decomposition for the group
$\mathrm {PSp}(1,n)$
. Section 5 then examines the structure of the limit set for quaternionic hyperbolic groups, offering a detailed characterization. Finally, Section 6 concludes the article by demonstrating that Koebe’s and Bers’ theorems do not hold in the
$\mathrm {PSp}(1,n)$
setting, providing a complete proof of this result.
2 Preliminaries
2.1 Quaternionic linear algebra
The content of this section is based on [Reference Blyth4, Reference Rodman27, Reference Zhang28]. In what follows,
$\mathbb {H}$
will denote the quaternions. It is well known that every element
$q \in \mathbb {H}$
can be expressed as
$$\begin{align*}q = q_0 + q_1\mathbf{i} + q_2\mathbf{j} + q_3\mathbf{k}, \end{align*}$$
where
$q_0, q_1, q_2, q_3 \in \mathbb {R}$
, and the following relations hold:
$$\begin{align*}\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -1, \quad \mathbf{ij} = -\mathbf{ji} = \mathbf{k}, \quad \mathbf{jk} = -\mathbf{kj} = \mathbf{i}, \quad \mathbf{ki} = -\mathbf{ik} = \mathbf{j}. \end{align*}$$
The conjugate of q is the quaternion
$$\begin{align*}q^* = q_0 - q_1 \mathbf{i} - q_2 \mathbf{j} - q_3 \mathbf{k}. \end{align*}$$
Equipped with the norm
$|q| = \sqrt {qq^*}$
,
$\mathbb {H}^n$
becomes a normed
$\mathbb {H}$
-module.
A quaternionic linear subspace of
$\mathbb {H}^n$
is defined as follows.
Definition 2.1 A subset
$W \subseteq \mathbb {H}^n$
is called a quaternionic linear subspace if it satisfies the following conditions:
-
(1) W contains the zero vector, i.e.,
$\mathbf {0} \in W$
. -
(2) For any
$\mathbf {w}_1, \mathbf {w}_2 \in W$
, their sum satisfies
$\mathbf {w}_1 + \mathbf {w}_2 \in W$
. -
(3) For any
$\mathbf {w} \in W$
and any quaternion
$q \in \mathbb {H}$
, the scalar multiple
$\mathbf {w}q \in W$
.
To define the dimension of a quaternionic linear subspace, we introduce the following preliminary concepts:
-
(1) A set of vectors
$\{ \mathbf {w}_1, \mathbf {w}_2, \dots , \mathbf {w}_k \}$
in W is linearly independent if the only solution to (where
$$\begin{align*}\mathbf{w}_1 q_1 + \mathbf{w}_2 q_2 + \dots + \mathbf{w}_k q_k = \mathbf{0} \end{align*}$$
$q_i \in \mathbb {H}$
) is
$q_1 = q_2 = \dots = q_k = 0$
.
-
(2) A basis for W is a linearly independent set that spans W under right scalar multiplication. The dimension of W, denoted
$\dim _{\mathbb {H}} W$
, is the number of vectors in such a basis.
Given two quaternionic vector spaces V and W, a function
$T: V \to W$
is called a quaternionic linear transformation if, for all vectors
$\mathbf {u}, \mathbf {v} \in V$
and all scalars
$p, q \in \mathbb {H}$
, it satisfies
$$\begin{align*}T(\mathbf{u} p + \mathbf{v} q) = T(\mathbf{u})p + T(\mathbf{v})q. \end{align*}$$
As in the real or complex case, if
$V = \mathbb {H}^n$
and
$W = \mathbb {H}^m$
, a right
$\mathbb {H}$
-linear transformation T can be represented by an
$m \times n$
quaternionic matrix A acting on column vectors via right multiplication:
$$\begin{align*}T(\mathbf{v}) = A \mathbf{v}. \end{align*}$$
Unlike complex linear maps, quaternionic linear maps do not naturally form a quaternionic vector space due to non-commutativity. Instead, they constitute a real vector space. We introduce the following notation:
-
•
$\mathrm {M}(n, \mathbb {H})$
denotes the space of
$\mathbb {H}$
-linear transformations from
$\mathbb {H}^n$
to
$\mathbb {H}^n$
. -
•
$\mathrm {GL}(n, \mathbb {H})$
denotes the Lie group of
$\mathbb {H}$
-linear isomorphisms of
$\mathbb {H}^n$
.
A quaternionic Hermitian matrix is a square matrix A with quaternion entries that satisfies
$A = A^*$
, where
$A^*$
is the quaternionic conjugate transpose of A. A Hermitian matrix
$A \in \mathrm {M}(n, \mathbb {H})$
is positive semi-definite if, for every
$v \in \mathbb {H}^n \setminus \{0\}$
,
$$\begin{align*}v (Av)^* \in \mathbb{R}^+. \end{align*}$$
Theorem 2.1 (See [Reference Farenick and Pidkowich15, Proposition 5.2 and Corollary 5.3])
Let
$A \in \mathrm {M}(n, \mathbb {H})$
. Then:
-
(1) The matrix
$A^*A$
is positive semi-definite. -
(2) If
$A \in \mathrm {M}(n, \mathbb {H})$
is a positive semi-definite invertible matrix, then there exists a unitary matrix U (i.e.,
$UU^* = U^*U = \mathrm {Id}_n$
) such that is a diagonal matrix with positive real entries.
$$\begin{align*}U^* A U = D \end{align*}$$
2.2 Quaternionic projective geometry
2.2.1 Quaternionic projective space and subspaces
The content of this section is based on [Reference Coxeter10]. The quaternionic projective space
$\mathbb {HP}^n$
is defined as
$$\begin{align*}\mathbb{HP}^n = (\mathbb{H}^{n+1} \setminus \{0\}) / \mathbb{H}^*, \end{align*}$$
where
$\mathbb {H}^*$
acts by right scalar multiplication. If
$[\cdot ] \colon \mathbb {H}^{n+1} \setminus \{0\} \to \mathbb {HP}^n$
is the quotient map, a non-empty subset
$\mathcal {L} \subset \mathbb {HP}^n$
is called a quaternionic projective subspace of dimension k if there exists an
$\mathbb {H}$
-linear subspace
$\widetilde {\mathcal {L}} \subset \mathbb {H}^{n+1}$
of dimension
$k+1$
such that
$$\begin{align*}[\widetilde{\mathcal{L}} \setminus \{0\}] = \mathcal{L}. \end{align*}$$
Quaternionic projective subspaces of dimension
$n-1$
are called hyperplanes, and those of dimension
$1$
are called quaternionic lines.
As in the classical case,
$e_1, \ldots , e_{n+1}$
will denote the standard basis of
$\mathbb {H}^{n+1}$
. Now, for a set of points
$P \subset \mathbb {HP}^n$
, we define
$$\begin{align*}\langle \langle P \rangle \rangle = \bigcap \left\{ \mathcal{L} \subset \mathbb{HP}^n \mid \mathcal{L} \text{ is a quaternionic projective subspace containing } P \right\}. \end{align*}$$
By definition,
$\langle \langle P \rangle \rangle $
is itself a quaternionic projective subspace.
The Grassmannian of quaternionic lines,
$\mathrm {Gr}(1,n)$
, is the space of all one-dimensional quaternionic projective subspaces of
$\mathbb {HP}^n$
, endowed with the Hausdorff topology. It can be embedded into the quaternionic projective space
$\mathbb {P}(\bigwedge ^2 \mathbb {H}^{n+1})$
via the Plücker embedding:
$$\begin{align*}\iota \colon \mathrm{Gr}(1,n) \to \mathbb{P}\left(\bigwedge^2 \mathbb{H}^{n+1}\right), \quad \iota(V) = [v_1 \wedge v_2], \end{align*}$$
where
$\langle \langle v_1, v_2 \rangle \rangle = V$
.
2.2.2 Projective and pseudo-projective transformations
A quaternionic projective transformation
$\mathbf {g}$
is an invertible map on
$\mathbb {HP}^n$
of the form
$$\begin{align*}\mathbf{g}[w] = [g(w)], \end{align*}$$
where g is an
$\mathbb {H}$
-linear isomorphism of
$\mathbb {H}^{n+1}$
. Here, g is called a lift of
$\mathbf {g}$
. This definition is well-defined because the matrix group acts on the left while the projection acts on the right.
The induced homomorphism
$[[\,\cdot \,]] \colon \mathrm {GL}(n+1, \mathbb {H}) \to \mathrm {Aut}(\mathbb {HP}^n)$
is surjective, with kernel consisting of nonzero real scalar multiples of the identity. Thus, the group of quaternionic projective transformations of
$\mathbb {HP}^n$
is isomorphic to:
$$\begin{align*}\mathrm{PSL}(n+1, \mathbb{H}) := \mathrm{GL}(n+1, \mathbb{H}) / \{\alpha \mathrm{Id} \mid \alpha \in \mathbb{R}^*\}. \end{align*}$$
For a nonzero linear transformation
$M \colon \mathbb {H}^{n+1} \to \mathbb {H}^{n+1}$
, let
$\mathrm {Ker}(M)$
denote its kernel and
$\mathrm {Ker}([[M]])$
its projectivization. Then, M induces a well-defined map:
$$\begin{align*}[[M]] \colon \mathbb{HP}^n \setminus \mathrm{Ker}([[M]]) \to \mathbb{HP}^n, \quad [[M]]([v]) = [M(v)], \end{align*}$$
called a quaternionic pseudo-projective transformation. The space of such transformations forms a compactification of
$\mathrm {PGL}(n+1, \mathbb {H})$
, as shown below.
Lemma 2.2 (Compactness Properties)
-
(1) Given a sequence
$(g_m) \subset \mathrm {PGL}(n+1, \mathbb {H})$
, there exists a subsequence
$(h_m)$
and a pseudo-projective transformation h such that
$h_m \to h$
uniformly on compact subsets of
$\mathbb {HP}^n \setminus \mathrm {Ker}(h)$
. -
(2) Let
$(h_m) \subset \mathrm {GL}(n+1, \mathbb {H})$
be a sequence converging pointwise to
$h \neq 0$
. Then,
$([[h_m]])$
converges to
$[[h]]$
.
Proof For each
$m \in \mathbb {N}$
, let
$g_m \in \mathrm {GL}(n+1, \mathbb {H})$
be a lift of
$\mathbf {g}_m$
. Define
$$\begin{align*}|g_m| = \max \{ |g_{ij}^{(m)}| \mid i, j = 1, \ldots, n+1 \}. \end{align*}$$
Since
$|g_m|^{-1} g_m$
is also a lift of
$\mathbf {g}_m$
and bounded sequences in
$\mathrm {M}(n+1, \mathbb {H})$
admit convergent subsequences, we may assume that
$(|g_m|^{-1} g_m)$
converges to a nonzero matrix
$g \in \mathrm {M}(n+1, \mathbb {H})$
. Thus,
$$\begin{align*}\lim_{m \to \infty} |g_m|^{-1} g_m = g \end{align*}$$
in the compact-open topology.
Now, let
$K \subset \mathbb {HP}^n \setminus \mathrm {Ker}([[g]])$
be compact, and define
$$\begin{align*}\mathbf{K} = \{ k \in \mathbb{H}^{n+1} \mid [k] \in K \} \cap \mathbb{S}^n_{\mathbb{H}}, \end{align*}$$
where
$\mathbb {S}^n_{\mathbb {H}}$
is the unit sphere in
$\mathbb {H}^{n+1}$
. Then,
$$\begin{align*}\lim_{m \to \infty} g_m|_{\mathbf{K}} = g|_{\mathbf{K}} \end{align*}$$
in the compact-open topology, proving the claim.
In the complex case, the space of pseudo-projective transformations has proven useful for precisely describing the equicontinuity region of families of projective transformations. For a detailed treatment, see [Reference Cano, Loeza and Ucan-Puc7].
2.3 Quaternionic hyperbolic groups
All the material in this section has been adapted from [Reference Kim and Parker20]. Consider
$\mathbb {H}^{n,1}$
, the quaternionic vector space of quaternionic dimension
$n + 1$
endowed with the quaternionic Hermitian form:
$$\begin{align*}\langle z, w \rangle = w^* H z, \end{align*}$$
where
$$\begin{align*}H = \begin{pmatrix} & & 1 \\ & I_{n-1} & \\ 1 & & \end{pmatrix}. \end{align*}$$
We define the following subspaces of
$\mathbb {H}^{n,1}$
:
$$\begin{align*}\begin{cases} V_- = \{z \in \mathbb{H}^{n,1} : \langle z, z \rangle < 0\}, \\ V_0 = \{z \in \mathbb{H}^{n,1} \setminus \{0\} : \langle z, z \rangle = 0\}, \\ V_+ = \{z \in \mathbb{H}^{n,1} : \langle z, z \rangle> 0\}. \end{cases} \end{align*}$$
The quaternionic hyperbolic n-space is defined as
$\mathbf {H}^n_{\mathbb {H}} = [V_-] \subset \mathbb {HP}^n$
. This space is projectively equivalent to the quaternionic unitary ball, and its boundary is
$[V_0]$
. In analogy with the complex case, we call this model the Siegel domain.
As in the real and complex cases, given a projective subspace
$P \subset \mathbb {HP}^n$
, we define its orthogonal complement as:
$$\begin{align*}P^\bot = \left\{ [w] \in \mathbb{HP}^n \mid \langle w, v \rangle = 0 \text{ for all } [v] \in P \right\}. \end{align*}$$
For a point
$p \in \mathbb {HP}^n$
, the orthogonal complement
$p^\bot $
is a quaternionic hyperplane that is:
-
• osculating to
$\partial \mathbf {H}^n_{\mathbb {H}}$
at p if
$p \in [V_0]$
; -
• secant to
$\mathbf {H}^n_{\mathbb {H}}$
if
$p \in [V_-]$
; -
• disjoint from
$\mathbf {H}^n_{\mathbb {H}}$
if
$p \in [V_+]$
.
The metric on
$\mathbf {H}^n_{\mathbb {H}}$
is given by the distance function
$\rho $
, where
$$\begin{align*}\cosh^2 \left( \frac{\rho([z], [w])}{2} \right) = \frac{\langle z, w \rangle \langle w, z \rangle}{\langle z, z \rangle \langle w, w \rangle}. \end{align*}$$
The quaternionic indefinite group of signature
$(1,n)$
, in symbols
$\text {Sp}( 1,n)$
, is the subgroup of
$\text {GL}(n + 1, \mathbb {H})$
that preserves the Hermitian form
$\langle z, w \rangle $
under left action. Thus,
$g \in \text {Sp}( 1,n)$
satisfies
$$\begin{align*}g^* H g = H. \end{align*}$$
We define
$\text {PSp}( 1,n)$
as the quotient of
$\text {Sp}( 1,n)$
by its center, which consists of real scalar matrices. Since
$\det (g)$
has unit norm, the only such scalars are
$\pm 1$
, so:
$$\begin{align*}\text{PSp}(1,n) = \text{Sp}( 1,n)/\{\pm I\}. \end{align*}$$
The group
$\text {PSp}( 1,n)$
is the group of isometries of
$\mathbf {H}^n_{\mathbb {H}}$
. Following the complex case, we refer to its subgroups as quaternionic hyperbolic groups.
As in real and complex hyperbolic geometry, nontrivial elements of
$\text {PSp}( 1,n)$
are classified as:
-
(1) Loxodromic if it fixes exactly two points in
$\partial \mathbf {H}^n_{\mathbb {H}}$
. -
(2) Parabolic is it fixes exactly one point in
$\partial \mathbf {H}^n_{\mathbb {H}}$
. -
(3) Elliptic if it fixes at least one point in
$\mathbf {H}^n_{\mathbb {H}}$
.
The following notion of a limit set for quaternionic hyperbolic groups was introduced by Chen and Greenberg [Reference Chen and Greenberg8].
Definition 2.2 For a subgroup
$G \subset \text {PSp}(1,n)$
, the Chen–Greenberg limit set
$\Lambda _{CG}(G)$
is the set of accumulation points in
$\partial \mathbf {H}^n_{\mathbb {H}}$
of the orbit
$Gx$
for any
$x \in \mathbf {H}^n_{\mathbb {H}}$
.
As with Fuchsian groups,
$\Lambda _{CG}(G)$
is independent of the choice of x and contains either 1, 2, or infinitely many points. A group G is called non-elementary if
$\Lambda _{CG}(G)$
is infinite.
Finally, we define the compact symplectic group and its projectivization, which will be used throughout the text:
$$\begin{align*}\begin{array}{l} \text{Sp}(n) = \{ g \in \text{GL}(n, \mathbb{H}) \mid gg^* = I_{n} \},\\ \text{PSp}(n) = \{ [g] \in \text{PGL}(n, \mathbb{H}) \mid gg^* = I_{n} \}. \end{array} \end{align*}$$
3 Cartan’s decomposition
In this section, we present a geometric proof of the Cartan decomposition theorem for
$\mathrm {Sp}(1,n)$
. Succinctly, this decomposition is achieved by first changing from the Siegel domain model to the unitary quaternionic ball
$\mathbb {B}_{\mathbb {H}}^{n}$
. Then, we construct an operator
$S(z)$
that maps the origin
$0$
to a point
$z \in \mathbb {B}_{\mathbb {H}}^{n}$
. Now, by setting
$ h = g\,S(z)^{-1},$
we ensure that h fixes
$0$
and thus belongs to
$\mathrm {Sp}(n) \times \mathrm {Sp}(1)$
. Next, by taking the logarithm of
$S(z)$
and diagonalizing it, we rewrite the hyperbolic component in exponential form
$S(z) = \exp (X) k,$
where X is in the Lie algebra of hyperbolic translations and
${k \in \mathrm {Sp}(n) \times \mathrm {Sp}(1)}$
, thereby clearly separating the hyperbolic translation from the compact rotations.
Consider the quadratic form on
$\mathbb {H}^{n,1}$
of signature
$(1,n)$
induced by the matrix
$$\begin{align*}Q = \begin{pmatrix}-1 & 0 \\ 0 & I_n\end{pmatrix}.\end{align*}$$
We define
$$\begin{align*}\operatorname{Sp}_Q(1,n)= \left\{ g \in \text{GL}(n, \mathbb{H}) \mid g Qg^* = Q \right\}. \end{align*}$$
The groups
$\operatorname {Sp}_Q(1,n)$
and
$\operatorname {Sp}(1,n)$
are conjugate via the matrix
$$\begin{align*}\begin{pmatrix} 2^{-1/2} & 2^{-1/2} & 0 \\ 0 & 0 & I_{n-1} \\ -2^{-1/2} & 2^{-1/2} & 0 \end{pmatrix}. \end{align*}$$
For simplicity in calculations, we present Cartan’s Decomposition Theorem for
$\operatorname {Sp}_Q(1,n)$
. In what follows, we consider the following model of quaternionic hyperbolic space:
$$\begin{align*}\mathbb{B}_{\mathbb{H}}^n = \left\{ z \in \mathbb{H}^{n} \mid z^* z < 1 \right\}. \end{align*}$$
Note that for
$g \in \operatorname {Sp}_Q(1,n)$
, the relation
$gQg^* = Q$
implies that g can be written in block form as
$\begin {pmatrix}a & b \\ c & d\end {pmatrix}$
, where
$a \in \mathbb {H}$
,
$b \in \mathbb {H}^n$
, c is an n-dimensional row vector, and
$d \in M(n,\mathbb {H})$
satisfy, cf. [Reference Kim and Parker20]:
$$ \begin{align*} aa^{*} - bb^{*} &= 1 = a^{*}a - c^{*}c, \\ -ac^{*} + bd^{*} &= 0 = -a^{*}b + c^{*}d, \\ -cc^{*} + dd^{*} &= I_n = -b^{*}b + d^{*}d. \end{align*} $$
Theorem 3.1 Given
$g = \begin {pmatrix}a & b \\ c & d\end {pmatrix} \in \operatorname {Sp}_Q(1,n)$
, the map
$z \mapsto (a + zc)^{-1}(b + zd)$
defines a right group action on
$\mathbb {B}_{\mathbb {H}}^{n}$
.
Proof For
$g, h \in \operatorname {Sp}(1,n)$
, direct substitution shows
$((z)g)h = (z)(gh)$
, confirming the action property.
Lemma 3.2 For
$z \in \mathbb {B}_{\mathbb {H}}^{n} $
written as a row vector:
-
(1)
$I_n - z^{*}z \in \mathrm {GL}(n,\mathbb {H})$
is a positive-definite quaternionic Hermitian matrix. -
(2)
$(1 - zz^{*})^{1/2}z = z(I_n - z^{*}z)^{1/2}$
. -
(3)
$(1 - zz^{*})^{1/4}z = z(I_n - z^{*}z)^{1/4}$
.
Proof Positive definiteness follows from
$v^{*}(I_n - z^{*}z)v = \|v\|^2 - \|zv\|^2> 0$
. The equalities hold due to the spectral properties of analytic functions.
Definition 3.1 For
$z \in \mathbb {B}_{\mathbb {H}}^{n}$
, define:
$$\begin{align*}S(z) = \begin{pmatrix} (1 - zz^{*})^{-1/2} & (1 - zz^{*})^{-1/2}z \\ z^{*}(1 - zz^{*})^{-1/2} & (I_n - z^{*}z)^{-1/2} \end{pmatrix}. \end{align*}$$
Theorem 3.3 (Properties of
$S(z)$
)
The matrix
$S(z)$
satisfies:
-
(1)
$S(z) \in \operatorname {Sp}_Q(1,n)$
; -
(2)
$(0)S(z) = z$
; -
(3)
$S(z)$
is positive semi-definite.
Proof To prove (1), observe:
$$\begin{align*}\begin{aligned} -1 &= (1 - zz^{\ast})^{-1/2}(-1 + zz^{\ast})(1 - zz^{\ast})^{-1/2} \\ &= -(1 - zz^{\ast})^{-1} + (1 - zz^{\ast})^{-1/2} z z^{\ast}(1 - zz^{\ast})^{-1/2}, \\ 0 &= -(1 - zz^{\ast})^{-1}z + (1 - zz^{\ast})^{-1/2} z(I_n - z^{\ast}z)^{-1/2}, \\ I_n &= -z^{\ast}(1 - zz^{\ast})^{-1}z + (I_n - z^{\ast}z)^{-1}. \end{aligned} \end{align*}$$
Thus,
$S(z)$
satisfies the defining relations of
$\operatorname {Sp}_Q(1,n)$
.
For (2), compute
$(0)S(z) = (1 - zz^{\ast })^{1/2}(1 - zz^{\ast })^{-1/2}z = z$
.
For (3), let
$A = \begin {pmatrix}(1 - zz^{\ast })^{-1/4} & 0 \\ 0 & (I_n - z^{\ast }z)^{-1/4}\end {pmatrix}$
. For
$v \in \mathbb {B}^n_{\mathbb {H}} \setminus \{\mathbf {0}\}$
, let
$w = A^{-1}v$
. Then
$$\begin{align*}v^* S(z) v = w^* \begin{pmatrix}(1 - zz^{\ast})^{1/2} & z \\ 0 & I_n\end{pmatrix} \begin{pmatrix}(1 - zz^{\ast})^{1/2} & 0 \\ z^* & I_n\end{pmatrix} w \geq 0.\\[-41pt] \end{align*}$$
Lemma 3.4 If
$g \in \operatorname {Sp}(1,n)$
fixes
$0$
, then
$g \in \operatorname {Sp}(n+1)$
.
Proof If
$(0)g = 0$
, then
$c = 0$
in the block matrix, reducing g to
$\operatorname {Sp}(n+1)$
.
Theorem 3.5 Every
$g {\kern-1pt}\in{\kern-1pt} \operatorname {Sp}(1,n)$
decomposes uniquely as
$g {\kern-1pt}={\kern-1pt} hS(z)$
with
${h {\kern-1pt}\in \operatorname {Sp}(n{\kern-1pt}+{\kern-1pt}1)}$
.
Proof Let
$z = (0)g$
. Then,
$h = gS(z)^{-1}$
fixes
$0$
, hence
$h \in \operatorname {Sp}(n+1)$
.
Theorem 3.6 For each
$z \in \mathbb {B}_{\mathbb {H}}^n$
, there exists
$\beta \in \mathbb {H}^n$
such that
$$\begin{align*}\ln(S(z)) = \begin{pmatrix}0 & \beta^* \\ \beta & 0\end{pmatrix}. \end{align*}$$
Moreover, if
$\beta \neq 0$
, there exists
$A \in \mathrm {Sp}(n)$
such that:
$$\begin{align*}S(z) = \begin{pmatrix}1 & 0 \\ 0 & A^{-1}\end{pmatrix} \begin{pmatrix}\cosh(|\beta|) & \sinh(|\beta|) & 0 \\ \sinh(|\beta|) & \cosh(|\beta|) & 0 \\ 0 & 0 & I_{n-1}\end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & A\end{pmatrix}. \end{align*}$$
Proof Since
$S(z)$
is positive semi-definite, diagonalize it as
$S(z) = U^*DU$
with U unitary and D diagonal. Define
$\ln (S(z)) = U^* \ln (D) U$
. The identity
$$\begin{align*}Q (\ln(S(z)))^* Q = -\ln(S(z)) \end{align*}$$
implies that
$\ln (S(z))$
has the stated form. If
$\beta \neq 0$
, apply Gram–Schmidt (Theorem 4.3 in [Reference Farenick and Pidkowich15]) to obtain an orthonormal basis
$\{v_0, \ldots , v_{n-1}\}$
with
$v_0 = \beta |\beta |^{-1}$
. Let A have columns
$v_i$
; then
$A \in \mathrm {Sp}(n)$
and
$$\begin{align*}\begin{pmatrix}1 & 0 \\ 0 & A^{-1}\end{pmatrix} \ln(S(z)) \begin{pmatrix}1 & 0 \\ 0 & A\end{pmatrix} = \begin{pmatrix}0 & e_1^* |\beta| \\ e_1 |\beta| & 0\end{pmatrix}. \end{align*}$$
Exponentiating yields the decomposition.
4 Proof of Theorem 1.1
From Theorem 3.6, every
$g \in \operatorname {Sp}_Q(1,n)$
decomposes as:
$$\begin{align*}g = k_1 \begin{pmatrix}\cosh(t) & \sinh(t) & 0 \\ \sinh(t) & \cosh(t) & 0 \\ 0 & 0 & I_{n-1}\end{pmatrix} k_2, \end{align*}$$
where
$k_1, k_2 \in \operatorname {Sp}(n+1) \cap \operatorname {Sp}_Q(1,n)$
and
$t \geq 0$
. Diagonalizing the hyperbolic component yields the exponential form
$e^{A(g)}$
.
5 Quaternionic hyperbolic groups
In this section, we prove Theorem 1.2. The proof relies on the Cartan decomposition theorem and the
$\lambda $
-lemma (see Lemma 5.2 below), which together provide rigorous control over the dynamics of quaternionic lines. To begin, we introduce the following definition, establishing a conceptual framework for controlling the orbits of compact sets.
Definition 5.1 (cf. [Reference Frances16])
Let
$(g_m) \subset \mathrm {PSp}(1,n)$
be a sequence of distinct elements, and let
$x \in \mathbb {HP}^n$
. Then:
-
(1) We say that
$(g_m)$
tends simply to infinity if: -
• The sequences of compact factors in the Cartan decomposition converge.
-
•
$(\lambda (g_m))$
converges to infinity.
-
-
(2) For a sequence
$(g_m) \subset \mathrm {PSp}(1,n)$
tending simply to infinity, we define where the union is taken over all sequences
$$\begin{align*}\mathcal{D}_{(g_m)}(x) = \bigcup \{\text{accumulation points of } (g_m(x_m))\}, \end{align*}$$
$(x_m)$
converging to x.
The following lemma is straightforward.
Lemma 5.1 For
$p \in \partial \mathbf {H}^n_{\mathbb {H}}$
, define:
-
•
$\pi _\beta (p) = \{\ell \in Gr(1,n) : p \in \ell \subset p^\bot \}$
. -
• The metric
$d_p: \pi _\beta (p) \times \pi _\beta (p) \to \mathbb {R}_+$
by where
$$\begin{align*}d_p(\ell_1, \ell_2) = \arccos \left( \sqrt{ \frac{\langle q_1, q_2 \rangle \langle q_2, q_1 \rangle}{\langle q_1, q_1 \rangle \langle q_2, q_2 \rangle} } \right), \end{align*}$$
$q_1, q_2 \in \mathbb {H}^{n+1} - \{\mathbf {0}\}$
satisfy
$\ell _i = \langle \langle p, [q_i] \rangle \rangle $
.
Then,
$(\pi _\beta (p), d_p)$
is a metric space isometric to
$(\mathbb {HP}^{n-1}, d_{n-1})$
, where
$d_{n-1}$
is the standard Fubini–Study metric.
This lemma naturally induces a metric on concurrent lines lying in an osculating plane to
$\partial \mathbf {H}_{\mathbb {H}}^n$
. Observe that this metric reproduces the construction of both the Fubini metric and the intrinsic metric of quaternionic hyperbolic space.
In what follows,
$\mathrm {Isom}(p,q)$
denotes the set of isometries from
$(\pi _\beta (p), d_p)$
to
$(\pi _\beta (q), d_q)$
, where
$p, q \in \mathbf {H}^n_{\mathbb {H}}$
. The next lemma is crucial for this article.
Lemma 5.2 (
$\lambda $
-Lemma, cf. [Reference Frances16])
Let
$G \subset \mathrm {PSp}(1,n)$
be a discrete group, and let
${(g_m) \subset G}$
be a sequence tending simply to infinity. Then, there exist quaternionic pseudo-projective transformations
$g_+$
and
$g_-$
such that:
-
(1)
$g_m^{\pm 1}$
converges to
$g_\pm $
. -
(2) The sets
$\mathrm{Im}(g_+)$
and
$\mathrm{Im}(g_-)$
are points in the Chen–Greenberg limit set of G. Moreover: -
•
$\mathrm{Im}(g_+)^\bot = \mathrm{Ker}(g_-)$
; -
•
$\mathrm{Im}(g_-)^\bot = \mathrm{Ker}(g_+)$
.
-
Additionally, there exists a projective equivalence
$\phi \in \mathrm {Isom}(\mathrm{Im}(g_-), \mathrm{Im}(g_+))$
such that for
$\ell \in \pi _\beta (\mathrm{Im}(g_\mp ))$
and
$w \in \ell - \mathrm{Im}(g_\mp )$
, we have
$$\begin{align*}\mathcal{D}_{(g_m^{\pm 1})}(w) = \phi^{\pm 1}(\ell). \end{align*}$$
Proof We first prove part (1). By Cartan’s decomposition theorem, there exist sequences
$(\alpha _m) \in \mathbb {R}_+$
,
$(\kappa _m^+), (\kappa _m^-) \subset K = \mathrm {Sp}(n+1) \cap \mathrm {PSp}(1,n)$
such that:
$$\begin{align*}g_m = \left[ \kappa_m^+ \begin{pmatrix} e^{\alpha_m} & & \\ & I_{n-2} & \\ & & e^{-\alpha_m} \end{pmatrix} \kappa_m^- \right]. \end{align*}$$
Assume
$\kappa _m^\pm $
converges to
$\kappa _\pm \in K$
. Then,
$$\begin{align*}\lim_{m \to \infty} g_m = g_+ = \left[ \kappa_+ \begin{pmatrix} 1 & \\ & \mathbf{0} \end{pmatrix} \kappa_- \right], \quad \lim_{m \to \infty} g^{-1}_m = g_- = \left[ \kappa^{-1}_- \begin{pmatrix} \mathbf{0} & \\ & 1 \end{pmatrix} \kappa^{-1}_+ \right]. \end{align*}$$
Consequently,
-
•
$\mathrm{Im}(g_+) = \kappa _+([e_1])$
,
$\mathrm{Im}(g_-) = \kappa ^{-1}_-([e_{n+1}]) \in \Lambda _{CG}(G)$
; -
•
$\mathrm{Ker}(g_+) = \mathrm{Im}(g_-)^\bot $
; -
•
$\mathrm{Ker}(g_-) = \mathrm{Im}(g_+)^\bot $
.
Now, define the isometry
$\phi : \pi _\beta (\mathrm{Im}(g_-)) \to \pi _\beta (\mathrm{Im}(g_+))$
by
$\phi (\ell ) = \kappa _+ (H(\kappa _-(\ell )))$
.
For
$\ell \in \pi _\beta (\mathrm{Im}(g_-))$
,
$w \in \ell - \mathrm{Im}(g_-)$
, and
$(w_m) \subset \mathbb {HP}^n$
converging to w, we have
$[\kappa _- (w_m)] \to [\kappa _- (w)] \in \langle \langle [e_2], \ldots , [e_{n+1}] \rangle \rangle - \{[e_{n+1}]\}$
.
Assuming
$[\kappa _- (w_m)] = [w_{1m}, \ldots , w_{n+1,m}]$
and
$[\kappa _- (w)] = [0, w_2, \ldots , w_{n+1}]$
with
$\sum _{j=2}^n |w_j| \neq 0$
, we compute
$$\begin{align*}\begin{pmatrix} e^{\alpha_m} & & \\ & I_{n-2} & \\ & & e^{-\alpha_m} \end{pmatrix} \begin{pmatrix} w_{1m} \\ v_m \\ w_{n+1,m} \end{pmatrix} = \begin{pmatrix} e^{\alpha_m} w_{1m} \\ v_m \\ e^{-\alpha_m} w_{n+1,m} \end{pmatrix}, \end{align*}$$
where
$v_m = (v_{2m}, \ldots , v_{nm})$
.
Thus, the accumulation points of
$(g_m(w_m))$
lie on the line
$\phi (\ell )$
. The result follows from this computation.
In the following,
$\mathrm{Im}(g_+)$
(resp.,
$\mathrm{Im}(g_-)$
) will be called the attracting point (resp., repelling point associated with
$(g_m)$
. The map
$\phi $
will be called the transition map associated with
$\mathrm{Im}(g_+)$
,
$\mathrm{Im}(g_-),$
and
$(g_m)$
.
The
$\lambda $
-lemma serves as a fundamental tool for analyzing the dynamics of compact sets, with particular emphasis on line dynamics. Its key dynamical implication can be stated as follows.
Proposition 5.3 Given a discontinuity region
$\Omega $
for a discrete group
$G \subset \mathrm {PSp}(1,n)$
and a sequence of distinct elements
$\{g_n\}_{n\in \mathbb {N}} \subset G$
, then (after possibly passing to a subsequence) there exist supporting hyperplanes
$T_1$
and
$T_2$
tangent to
$\partial \mathbf {H}^n_{\mathbb {H}}$
such that for any osculating line
$\ell $
of
$\partial \mathbf {H}^n_{\mathbb {H}}$
contained in
$T_1$
with
$\ell \not \subset \mathbb {HP}^n \setminus \Omega $
, there exists a corresponding osculating line
$\phi (\ell )$
at
$\partial \mathbf {H}^n_{\mathbb {H}}$
satisfying
$$\begin{align*}\phi(\ell) \subset T_2 \cap (\mathbb{HP}^n \setminus \Omega), \end{align*}$$
moreover, the map
$\phi $
is induced by an element of
$\mathrm {PSp}(n)$
.
With this in mind, it is not difficult to convince oneself that if we could guarantee the existence of a sequence of distinct elements for which the hyperplanes
$ T_1 $
and
$ T_2 $
coincide and the transition map
$ \phi $
is the identity, then Theorem 1.2 would be proven. However, in order to achieve this, it is necessary to consider limiting processes in which there is no control over the transition maps. For this reason, it is necessary to extend the maps to global transformations that allow us to keep precise control over the dynamics of the lines. The following definitions are intended to capture these ideas.
Definition 5.2 Let
$G\subset \mathrm {PSp}(1,n)$
be a discrete and non-elementary group and
$x,y\in \Lambda _{CG}(G)$
, then:
$$\begin{align*}G(y,x)=\{\gamma:y^{\perp}(y)\to x^{\perp}(x)\mid\gamma\text{ is a transition map associated with } x,y, (\gamma_m)\}.\end{align*}$$
Now, it is immediate that:
Corollary 5.4 Given a discrete group
$G\subset \mathrm {Psp}(1,n)$
and
$x,y\in \Lambda _{CG}(G)$
, we have that
$G(y,x)$
is non-empty.
Definition 5.3 Let
$G\subset \mathrm {PSp}(1,n)$
be a discrete group,
$x,y\in \Lambda _{CG}(\Gamma )$
,
$\mu :y^{\perp }(y)\to x^{\perp }(x)$
be a transition map associated with
$x,y,$
and
$(\gamma _m)$
; and T a projective transformation of
$P(\bigwedge ^{2}\mathbb {H}^{n+1})$
. We say that T is a Cartan extension for
$\mu $
with respect to
$(\gamma _{m})$
. If:
-
(1) The Cartan decomposition of
$\gamma _{m}$
is
$k_{m}^{+}A_{m}k_{m}^{-}$
. -
(2) We have
$k_{m}^{\pm }\xrightarrow [m\to \infty ]{}k^{\pm }$
, where
$k^{+},k^{-}\in K=\mathrm {PSp}(n+1)\cap \mathrm {PSp}(1,n)$
. -
(3) Also
$T=(k^{+}Hk^{-})\wedge (k^{+}Hk^{-})$
.
The proof of the following result is straightforward, so we omit it here.
Lemma 5.5 Let
$G \subset \mathrm {PSp}(1,n)$
be a discrete group, and let:
-
•
$x_+, x_- \in \Lambda _{CG}(G)$
be the attracting and repelling points, respectively; -
•
$\phi $
be the transition map associated with
$x,y$
and
$(g_m) \subset G$
; -
• T be the Cartan extension of
$\phi $
with respect to
$(g_m) \subset G$
.
Then,
$T(\ell ) = \phi (\ell )$
for each
$\ell \in \pi _\beta (x_-)$
.
Definition 5.4 Let
$G \subset \mathrm {PSp}(1,n)$
be a non-elementary discrete subgroup,
$x^+ \in \Lambda _{CG}(G)$
, and
$\phi \in G(x^+, x^+)$
. We say
$\phi $
is a limit transition map if:
-
(1) There exists a sequence
$(x_m^-) \subset \Lambda _{CG}(G)$
of distinct elements converging to
$x^+$
. -
(2) For each
$m \in \mathbb {N}$
, there is a sequence
$(g_{mj})_j \subset G$
tending simply to infinity with: -
• Attracting point
$x^+$
; -
• Repelling point
$x_m^-$
.
-
-
(3)
$\lambda (g_{mj}) \to \infty $
as
$|(m,j)| \to \infty $
. -
(4) There exist sequences
$(\kappa _{mj}^+)_j, (\kappa _{mj}^-)_j, (\kappa _m^+)_m, (\kappa _m^-)_m \subset \mathrm {PSp}(1,n) \cap \mathrm {PSp}(n+1)$
such that: -
• The Cartan decomposition of
$g_{mj}$
is
$\kappa _{mj}^+ \mu (g_{mj}) \kappa _{mj}^-$
. -
•
$\kappa _{mj}^\pm \to \kappa _m^\pm $
as
$m \to \infty $
. -
•
$(\kappa _m^\pm )$
converges to
$\kappa ^\pm $
.
-
-
(5) For each
$\ell \in \pi _\beta (x^+)$
,
$\phi (\ell ) = (\kappa ^+ \wedge \kappa ^+) (H \wedge H) (\kappa ^- \wedge \kappa ^-)(\ell )$
.
Theorem 5.6 Let
$G \subset \mathrm {PSp}(1,n)$
be a non-elementary discrete group and
$x^+ \in \Lambda _{CG}(G)$
. Then:
$$\begin{align*}G(x^+) = \{ \phi: \pi_\beta(x^+) \to \pi_\beta(x^+) \mid \phi \text{ is a limit transition map} \} \end{align*}$$
is a non-empty closed Lie subgroup of
$\mathrm {Isom}(x^+, x^+)$
.
Proof We establish the desired properties of
$ G(x^+) $
through four sequential steps:
Stage 1: Non-emptiness of
$\boldsymbol {G(x^+)}$
.
Let
$(x_m^-)_{m \in \mathbb {N}} \subset \Lambda _{CG}(G)$
be a sequence of distinct points converging to
$x^+$
. By Corollary 5.4, for each m, there exists a transition map
$\phi _m \in G(x_m^-, x^+)$
. For every m, select a sequence
$(g_{mj})_{j \in \mathbb {N}} \subset G$
tending simply to infinity (i.e., escaping compact subsets) and associated with
$\phi _m$
. Using the Cartan decomposition, write
$$\begin{align*}g_{mj} = \kappa^+_{mj} \lambda(g_{mj}) \kappa^-_{mj}, \quad \text{where } d_\infty(\kappa^\pm_{mj}, \kappa^\pm_m) < 2^{-j-m}. \end{align*}$$
By compactness of
$\mathrm {PSp}(n+1)$
, there exist limiting elements
$\kappa ^+, \kappa ^- \in \mathrm {PSp}(n+1)\cap \mathrm {PSp}(1,n) $
such that
$d(\kappa _{m}^\pm , \kappa ^\pm ) < 2^{-m}$
for all m. The diagonal subsequence
$(g_{mm})_m$
tends simply to infinity, and its transition map – being a limit of the
$\phi _m$
– lies in
$G(x^+)$
. Hence,
$G(x^+) \neq \emptyset $
.
Stage 2: Closedness in
$\mathbf {\mathrm {Isom}}\boldsymbol{(x^+, x^+)}$
.
Let
$(\phi _m)_m \subset G(x^+)$
converge uniformly to some
$\phi \in \mathrm {Isom}(x^+, x^+)$
. For each m, choose sequences
$(g_{mjl})_{l \in \mathbb {N}} \subset G$
and
$(\kappa ^\pm _{mjl})_{l \in \mathbb {N}}$
associated with
$\phi _m$
as in Definition 5.4. By Lemma 5.5, the action of
$\phi _m$
on any line
$\ell $
is given by
$$\begin{align*}\phi_m(\ell) = (\kappa^+_m \wedge \kappa^+_m) (H \wedge H) (\kappa^-_m \wedge \kappa^-_m)(\ell),\\[-24pt] \end{align*}$$
where H is the scaling component of the Cartan decomposition. As
$m \to \infty $
, the convergence
$\kappa ^\pm _m \to \kappa ^\pm $
ensures
$$\begin{align*}\phi_m(\ell) \longrightarrow (\kappa^+ \wedge \kappa^+) (H \wedge H) (\kappa^- \wedge \kappa^-)(\ell) = \phi(\ell).\\[-24pt] \end{align*}$$
Thus,
$\phi \in G(x^+)$
, proving
$G(x^+)$
is closed.
Stage 3: Monoid Structure Under Composition.
Let
$\phi _1, \phi _2 \in G(x^+)$
. For each j, construct sequences
$(g_{jm1})_m$
and
$(g_{jm2})_m$
associated with
$\phi _1$
and
$\phi _2$
, respectively. Define the composed sequence
$h_{jm} := g_{jm1}g_{jm2}$
, which also tends simply to infinity. The transition map
$\nu $
of
$(h_{jm})$
satisfies
$\nu = \phi _1 \circ \phi _2$
. Since
$\nu \in G(x^+)$
, the set
$G(x^+)$
is closed under composition, forming a monoid.
Stage 4: Lie Group Structure.
To show
$G(x^+)$
is a group, observe that for any
$a \in G(x^+)$
, the closure
$\overline {\langle a \rangle }$
of the cyclic semigroup generated by a is a compact Abelian semigroup in
$\mathrm {Isom}(x^+, x^+) \cong \mathrm {PSp}(n-1)$
. By Green’s theorem [Reference Green17], every compact Abelian semigroup is a group; hence,
$\overline {\langle a \rangle }$
is a compact Abelian Lie group. In particular,
$a^{-1} \in G(x^+)$
. Since
$G(x^+)$
is a closed subgroup of the Lie group
$\mathrm {Isom}(x^+, x^+)$
, Cartan’s closed subgroup theorem ensures that
$G(x^+)$
is itself a Lie group.
This concludes the proof.
5.1 Proof of Theorem 1.2
Assume for contradiction that there exists
$y \in \Lambda _{CG}(G)$
and
$w \in \Omega (G)$
. Let
$Id_y$
be the identity of
$\mathrm { Isom}(y, y)$
. By Lemma 5.6, there exists a sequence
$(g_m) \subset G$
tending simply to infinity associated with
$Id_y$
. Then,
$\langle \langle w, y \rangle \rangle = \mathcal {D}_{(\gamma _m)}(w) \subset \mathbb {HP}^n - \Omega (G)$
, a contradiction.
6 Uniformization theorems
A quaternionic hyperbolic manifold is defined as the quotient space
$\mathbf {H}_{\mathbb {H}}^n/\Gamma $
, where
$\Gamma $
is a discrete subgroup of
$\mathrm {PSp}(1, n)$
. For finite-volume manifolds endowed with quaternionic hyperbolic structures, Mok and Yeung [Reference Mok and Yeung25] and Klingler [Reference Klingler21] independently established the following rigidity theorem.
Theorem 6.1 [Reference Klingler21, Reference Mok and Yeung25]
For any discrete subgroup
$G \subset \mathrm {PSp}(1, n)$
with
$\mathbf {H}_{\mathbb {H}}^n/G$
of finite volume, the quotient manifold admits a unique projective structure compatible with its natural quaternionic structure.
We now proceed to prove Theorems 1.3 and 1.4.
6.1 Proof of Theorem 1.3 (algebraic version)
Let
$M = \mathbf {H}_{\mathbb {H}}^n/G$
be a compact manifold and assume, by way of contradiction, that:
-
(1) There exists a discrete, purely loxodromic free subgroup
$H \subset \mathrm {PGL}(n + 1, \mathbb {H})$
. -
(2) There exists a G-invariant open set
$U \subset \mathbb {HP}^n$
with
$M \cong U/H$
.
Theorem 6.1 implies that G and H are conjugate. Examining the Cayley graph
$\Delta (G)$
, we observe:
-
•
$\Delta (G)$
is quasi-isometric to
$\mathbf {H}_{\mathbb {H}}^n$
by [Reference Drutu and Kapovich13]. -
• Their Gromov boundaries, see [Reference Drutu and Kapovich13], are consequently homeomorphic.
This leads to a contradiction because:
-
• For free groups,
$\partial \Delta (G)$
is a Cantor set, while -
•
$\partial \mathbf {H}_{\mathbb {H}}^n$
is a topological sphere.
The geometric version follows immediately from these arguments.
6.2 Proof of Theorem 1.4
We argue by contradiction. Let
$M = \mathbf {H}_{\mathbb {H}}^n/G$
be a compact quaternionic manifold with
$G \subset \mathrm {PSp}(n, 1)$
discrete. Suppose the double
$\widetilde {M} = M \sqcup M$
admits a realization as
$U/H$
, where:
-
•
$H \subset \mathrm {PGL}(n + 1, \mathbb {H})$
is a transformation group; -
•
$U \subset \mathbb {HP}^n$
is an H-invariant domain.
Theorem 6.1 establishes that G and H are projectively conjugate. However, this contradicts Theorem 1.2, as
$\mathbf {H}_{\mathbb {H}}^n$
is maximal among domains where G acts properly discontinuously.
Declarations
All authors contributed equally to this work and declare no conflicts of interest.
Acknowledgments
We thank the colleagues at IM-UNAM Cuernavaca and IIT-UACJ for their hospitality and kindness during our work on this article. We are grateful to N. Gusevskii and A. Verjovsky for introducing us to this topic and for their insightful discussions. Special thanks go to S. Dutta, K. Gongopadhyay, and Lohan T. for generously sharing their preprint [Reference Dutta, Gongopadhyay and Tejbir14] with us; their work proved invaluable during the writing process.
