Action of $\mathbb{R}$-Fuchsian groups on $\mathbb{P}_{\mathbb{C}}^{n}$

Let $\Gamma$ be a discrete subgroup of SO${}_{+}(m,1)$, the group of isometries that preserves orientation of the hyperbolic space of dimension $m$, we know that the limit set of $\Gamma$ is contained in $\mathbb{S}^{m-1}$, see [4, 9]. Here, we are taking the limit set of a discrete subgroup $\Gamma\subset\mathrm{SO}_{+}(m,1)$ just as the accumulation set in the sphere $\mathbb{S}^{m-1}$ of an orbit $\Gamma\cdot\mathbf{z}$, where $\mathbf{z}\in\mathbb{H}_{\mathbb{R}}^{m}$, see [4, 9]. We are interested in discrete subgroups $\Gamma\subset\mathrm{SO}_{+}(m,1)$ whose limit set is the whole sphere $\mathbb{S}^{m-1}$. In the remainder of this article all the discrete subgroups we will consider are of this kind, examples of such groups are given by lattices. Given $\Gamma$ a discrete subgroup of SO${}_{+}(m,1)$ we can obtain a complex Kleinian group $\iota(\Gamma)$ whose limit set in the sense Chen and Greenberg, see [6], is a $(m-1)$-dimensional real sphere, by the embedding

where $A$ is the left-upper submatrix of size $m\times m$ of $G$, $\mathbf{c}$ is a row matrix of size $m$, $\mathbf{b}$ is a column vector of size $m$, $d$ is a $1\times 1$ entry and $I$ is the identity matrix of size $(n-m)\times(n-m)$. By the work of Cano et al. in [3] we can compute the Kulkarni limit set (see [12]) of $\iota(\Gamma)$ as the union of all complex projective hyperplanes in $\mathbb{P}_{\mathbb{C}}^{n}$ which are tangent to the sphere $\partial\mathbb{H}_{\mathbb{C}}^{n}$ at points in the Chen-Greenberg limit set, in other words, if we denote by $\Lambda_{Kul}(\iota(\Gamma))$ the Kulkarni limit set of $\iota(\Gamma)$, by $\mathcal{S}$ the $(m-1)$-dimensional real sphere contained in $\partial\mathbb{H}_{\mathbb{R}}^{n}$ whose points have the form $[w_{1}:\cdots:w_{m}:0:\cdots:0:w_{n+1}]$ and by $H_{z}$ the complex projective hyperplane tangent to $\partial\mathbb{H}_{\mathbb{C}}^{n}$ at the point $z\in\mathcal{S}$ then

We show that in this case $\Lambda_{Kul}(\iota(\Gamma))$ is a real semi-algebraic set and that the complement $\Omega_{Kul}:=\mathbb{P}_{\mathbb{C}}^{n}\setminus\Lambda_{Kul}(\iota(\Gamma))$ has three connected components when $m=2$, and is connected in other case, the precise statements are given in Theorems 1.1, 1.2, 1.3. The case $m=n=2$ was studied in [5] using as main tool the Hermitian cross-product $\boxtimes$ defined in $\mathbb{C}^{2,1}$ by Goldman, the Hermitian cross-product of two vectors $\mathbf{z}$ and $\mathbf{w}$ is a orthogonal vector to both $\mathbf{z}$ and $\mathbf{w}$. The function $f(\mathbf{z}):=\langle i\mathbf{z}\boxtimes\overline{\mathbf{z}},i\mathbf{z}\boxtimes\overline{\mathbf{z}}\rangle$ allows to characterize $\Lambda$ as a semi-algebraic subset of $\mathbb{P}_{\mathbb{C}}^{2}$, moreover the function $f$ gives a partition of $\mathbb{C}^{2,1}\setminus\{\mathbf{0}\}$ that under projection provides a partition of $\mathbb{P}_{\mathbb{C}}^{2}$ into $\mathrm{SO}_{+}(2,1)$-invariants sets. This partition allows to describe the set $\Lambda$ and the connected components of $\Omega$. When we consider $\Gamma\subset\mathrm{SO}_{+}(m,1)$ acting on $\mathbb{P}_{\mathbb{C}}^{m}$, we find that the Sylvester’s law of Inertia is a good replacement for the Hermitian cross-product because if we define the function

then the Sylvester’s law of Inertia allows to describe the set $\Lambda$ as a real semi-algebraic subset of $\mathbb{P}_{\mathbb{C}}^{m}$. The function $f$ defined in terms of the determinant of $\left(\begin{array}[]{cc}\langle\mathbf{z_{1}},\mathbf{z_{1}}\rangle&\langle\mathbf{z_{1}},\mathbf{z_{2}}\rangle\\ \langle\mathbf{z_{2}},\mathbf{z_{1}}\rangle&\langle\mathbf{z_{2}},\mathbf{z_{2}}\rangle\end{array}\right)$ coincides with the Hermitian product $\langle i\mathbf{z}\boxtimes\overline{\mathbf{z}},i\mathbf{z}\boxtimes\overline{\mathbf{z}}\rangle$ when $m=2$. For our purposes we just need to know if the orthogonal complement of the plane spanned by $\mathbf{z}$ and $\overline{\mathbf{z}}$, under the assumption that $\mathbf{z}$ and $\overline{\mathbf{z}}$ are linearly independent, is elliptic (positive definite), parabolic (degenerate) or hyperbolic (non-degenerate and indefinite), this is the main reason to use the Sylvester’s law of Inertia. This information as well as the linearly dependence of $\mathbf{z}$ and $\overline{\mathbf{z}}$ is codified by the function $f$. In the same manner that before the function $f$ determines a partition of $\mathbb{C}^{m,1}\setminus\{\mathbf{0}\}$ which projects to a partition of $\mathbb{P}_{\mathbb{C}}^{m}$ into $\mathrm{SO}_{+}(m,1)$-invariants sets which are useful to describe $\Lambda$ and $\Omega$. For $2<m=n$ we have that the set $\Omega$ is connected, in contrast to the case $m=n=2$, this occurs because $\mathbb{C}^{m,1}$ does not contain negative definite subspaces of codimension 2 while $\mathbb{C}^{2,1}$ does contain such spaces. This phenomenon makes the set $\Lambda$ become “larger” for $m>2$, in fact in this case the set $\Lambda$ has non-empty interior. In order to study the case $m<n$ we consider the projection

which is induced by the map

the set $\Lambda_{0}$ is the projectivization of the kernel of the map $\widetilde{Q}$ and is contained in the Kulkarni limit set $\Lambda$, the projection $Q$ allows to obtain the topological information of the sets $\Lambda$ and $\Omega$ from the information obtained when $m$ and $n$ coincide. Actually, if we denote by $\Lambda_{(m)}$ the Kulkarni limit set of $\Gamma$ when it acts on the copy of $\mathbb{P}_{\mathbb{C}}^{m}$, then the set $\Lambda$ is obtained as the union of $\Lambda_{0}$ and the preimage under $Q$ of $\Lambda_{(m)}$. Also, writing $\Omega_{(m)}$ the complement of $\Lambda_{(m)}$ in $\mathbb{P}_{\mathbb{C}}^{m}$ we obtain that $\Omega$ is the preimage under $Q$ of $\Omega_{(m)}$. Our main results can be summarized in the following theorems.

The map $\Pi$ that appears in Theorem 1.1 assigns to $z\in\Omega$ the class $[\mathbf{z}\overline{\eta}(\mathbf{z})+\mathbf{\overline{z}}\eta(\mathbf{z})]$, where $\eta^{2}(\mathbf{z})\!=\!-<\!\mathbf{z},\mathbf{\overline{z}}\!>$, we can compute explicitly the fibre over the origin and use the equivariance of $\Pi$ to show that it is a smooth fibre bundle. Theorem 1.2 follows from the Lemmas 3.7 and 3.8 which use the Sylvester’s law of Inertia to give a criterion to decide whether or not a point is in $\Lambda$ for $m>2$, the result for $m=2$ was proved in [3]. Theorem 1.3 follows from Lemmas 4.3 and 5.2 which are obtained using the projections defined in (1.2). We have the following consequence of our results:

The main interest in counting connected components comes from Kleinian groups in $\mathrm{PSL}(2,\mathbb{C})$ where it is known that the number of components of the discontinuity region can only be $0$, $1$, $2$ or infinity. For higher dimensional Kleinian groups there is no result on the number of components, our results provide examples where the Kulkarni discontinuity region is connected or has three components. In particular, we calculate the number of components for lattices in $\mathrm{SO}_{+}(m,1)$ for $m\geq 2$. The paper is organized as follows: Section 2 is devoted to generalities about complex hyperbolic geometry and projective geometry. The case of a discrete subgroup of $\mathrm{SO}_{+}(m,1)$ acting on $\mathbb{P}_{\mathbb{C}}^{m}$ viewed as a subgroup of $\mathrm{SU}(m,1)$ via the embedding $\iota$ is discussed in Section 3. In Section 4, we treated the case when the subgroup $\Gamma$ of $\mathrm{SO}_{+}(2,1)$ has as limit set a $\mathbb{R}$-circle, and $n$ is greater than 2. Finally, the remaining cases are studied in Section 5.

We extend the work of Cano et al. [5], for $n>2$, in the sense that given $S$ a $\mathbb{R}-$sphere of highest dimension in $\mathbb{P}_{\mathbb{C}}^{n}$, we describe the set $\Lambda$ defined as the union of all complex hyperplanes tangent to $\partial\mathbb{H}_{\mathbb{C}}^{n}$ at points in $S$, also we show that $\Omega:=\mathbb{P}_{\mathbb{C}}^{n}\setminus\Lambda$ is connected.

Consider the real $(n-1)$-sphere

where $\mathbb{P}_{\mathbb{R}}^{n}=\{[r_{1}:r_{2}:\cdots:r_{n+1}]\in\mathbb{P}_{\mathbb{C}}^{n}:r_{j}\in\mathbb{R},\;1\leq j\leq n+1\}$. It is a well known fact that all the other real $(n-1)$-spheres are obtained by the usual action of $\mathrm{PU}(n,1)$ on $\partial\mathbb{H}_{\mathbb{C}}^{n}$, see [8].
Given $p\in\mathbb{P}_{\mathbb{C}}^{n}$, we denote by $H_{p}$ the projective hyperplane

Then we can write the set $\Lambda$ as the set:

The next Lemma is just a restatement in the $n$-dimensional case of the part $(i)$ of Proposition 4.1 in Barrera et al. [1].

We use this characterization to determine when the projectivization of a vector $\mathbf{z}=(z_{1},\ldots,z_{n+1})\in\mathbb{C}^{n,1}\setminus\{\mathbf{0}\}$ lies in $\Lambda$.

Lemmas 3.1 and 3.2 imply that:

The first two conditions above give a system of equations that can be interpreted geometrically. Actually, we can write the condition 1 as,

and using the condition 2, we can rewrite it in the form,

additionally, taking the conjugate of Equation (3.1) we obtain,

Equations (3.2) and (3.3) imply that $\mathbf{w}$ belongs to the orthogonal hyperplane to $\mathbf{z}$ and to the orthogonal hyperplane to $\overline{\mathbf{z}}$, respectively. If we denote by $\mathbf{H_{p}}$ the hyperplane orthogonal to a vector $\mathbf{p}$ in $\mathbb{C}^{n,1}$, then $\mathbf{w}$ belongs to the intersection $\mathbf{H_{z}}\cap\mathbf{H}_{\overline{\mathbf{z}}}$.

In either case, the intersection $\mathbf{H_{z}}\cap\mathbf{H}_{\overline{\mathbf{z}}}$ is preserved by conjugation, so, $\mathbf{H_{z}}\cap\mathbf{H}_{\overline{\mathbf{z}}}$ is either the complexification of a hyperplane in $\mathbb{R}^{n,1}$ or the complexification of a codimension 2 subspace in $\mathbb{R}^{n,1}$.