2.1 Definition and properties

There are several equivalent definitions of Anosov representations in literature; see [Reference LabourieLab06, Reference Guichard and WienhardGW12, Reference Kapovich, Leeb and PortiKLP17, Reference Guichard, Gueritaud, Kassel and WienhardGGKW17, Reference Bochi, Potrie and SambarinoBPS19, Reference Kassel and PotrieKP22]. Here, we will describe the one that is more suitable for our aims. Let G be a connected semisimple Lie group with finite center and P a parabolic subgroup of G that is conjugate to its opposite parabolic subgroup $P^{\textit {op}}$ . Two points p and q in $G/P$ are called transverse if there exists $g\in G$ such that $g\operatorname {\mathrm {Stab}}_G(p) g^{-1} = P$ and $g\operatorname {\mathrm {Stab}}_G(q)g^{-1} = P^{\textit {op}}$ .

Let now $\Gamma $ be a finitely generated hyperbolic group with Gromov boundary $\partial _\infty \Gamma $ .

Definition 2.1. A representation $\rho \colon \thinspace \Gamma \to G$ is P–Anosov if there exists a continuous, $\rho $ –equivariant map

$$\begin{align*}\xi=\xi_{\rho} \colon\thinspace \partial_\infty \Gamma \longrightarrow G/P\end{align*}$$

that is

• • transverse (i.e., $\xi _\rho (x)$ and $\xi _\rho (y)$ are transverse for all $x \neq y \in \partial _\infty \Gamma $ );

• • strongly dynamics preserving (i.e., for any sequence $(\gamma _n)_{n\in {\mathbb N}} \in \Gamma ^{\mathbb N}$ with $\gamma _n \underset {n\to +\infty }{\longrightarrow } \gamma _+ \in \partial _\infty \Gamma $ and $\gamma _n^{-1} \underset {n\to +\infty }{\longrightarrow } \gamma _- \in \partial _\infty \Gamma $ ,

  $$\begin{align*}\rho(\gamma_n)\cdot p \underset{n\to +\infty}{\longrightarrow} \xi_\rho(\gamma_+)\end{align*}$$
  for all $p\in G/P$ transverse to $\xi _\rho (\gamma _-)$ ).

A subgroup $\Gamma $ of G is called Anosov if it is hyperbolic and the inclusion $\Gamma \hookrightarrow G$ is Anosov with respect to some proper parabolic subgroup P of G.

We denote by $\mathrm {Anosov}_P(\Gamma ,G)$ the subset of $\mathrm {Hom}(\Gamma ,G)$ consisting of P–Anosov representations. Note that P–Anosov representations are discrete and have finite kernel. In this paper, we will only work with groups $\Gamma $ that are torsion-free. For such groups, P–Anosov representations are thus discrete and faithful.

One of the most important properties of Anosov representations is their structural stablility (i.e., $\mathrm {Anosov}_P(\Gamma ,G)$ is open in $\mathrm {Hom}(\Gamma ,G)$ ). Structural stability gives a way to construct several Anosov representations as small deformations of a fixed Anosov representation. This is a major source of examples, as we will discuss in Section 2.2.

Another important property of P–Anosov representations is that they admit cocompact domains of discontinuity in boundaries of G (i.e., in homogeneous spaces $G/Q$ , where Q is a proper parabolic subgroup of G), possibly different from P. We will discuss this property in Section 2.5.

2.2 Construction of Anosov representations via deformation

Let us fix a connected semisimple Lie group H of real rank $1$ with finite center, and let $K \subset H$ be its maximal compact subgroup. The symmetric space $S_H = H/K$ has strictly negative sectional curvature and is thus Gromov hyperbolic. Recall that a uniform lattice $\Gamma < H$ is a discrete cocompact subgroup of H. Any such lattice is quasi-isometric to $S_H$ and is thus a hyperbolic group. Moreover, H has a unique conjugacy class of parabolic subgroups $P_H$ . By Guichard–Wienhard [Reference Guichard and WienhardGW12, Thm 5.15], $\Gamma $ is a $P_H$ –Anosov subgroup of H. We will always assume that $\Gamma $ is torsion-free, which is always virtually true by Selberg’s lemma.

An important case is when H is a compact extensionFootnote ¹ of $\mathrm {PSL}(2,\mathbb R)$ (i.e., H admits a surjective morphism to $\mathrm {PSL}(2,\mathbb R)$ with compact kernel). In that case, $S_H$ is the hyperbolic plane ${\mathbb H}^2$ , and a torsion-free cocompact lattice $\Gamma $ in H is a surface group (i.e., $\Gamma = \pi _1(\Sigma )$ , where $\Sigma $ is a closed orientable surface of genus $g \geq 2$ ). A representation $\rho _0:\pi _1(\Sigma ) \rightarrow \mathrm {PSL}(2,{\mathbb R})$ is called Fuchsian if it is discrete and faithful (in which case $\rho _0(\pi _1(\Sigma )) \backslash {\mathbb H}^2$ is a closed hyperbolic surface diffeomorphic to $\Sigma $ ). Similarly, a discrete and faithful representation into a compact extension H of $\mathrm {PSL}(2,{\mathbb R})$ will be called a twisted Fuchsian representation. It is the case if and only if its projection to $\mathrm {PSL}(2,{\mathbb R})$ is Fuchsian.

Other interesting cases arise when H is (a compact extension of) $\operatorname {PO}_0(1,n)$ or $\mathrm {PU}(1,n)$ , in which cases the symmetric space $S_H$ is respectively the real hyperbolic space ${\mathbb H}^n = {\mathbb H}^n_{{\mathbb R}}$ and the complex hyperbolic space ${\mathbb H}^n_{{{\mathbb C}}}$ . The group $\Gamma $ is then the fundamental group of a closed real hyperbolic or complex hyperbolic manifold. The other Lie groups of real rank $1$ (namely, ${\mathrm {Sp}}(1,n)$ and $\mathrm F_4^{-20}$ ) are slightly less interesting for this paper since their lattices are superrigid (see below). Still, our Theorem A applies also to them.

Let us fix a uniform torsion-free lattice $\Gamma \subset H$ and an embedding $\iota :H \rightarrow G$ , where G is a connected semisimple Lie group G with finite center. By Guichard–Wienhard [Reference Guichard and WienhardGW12, Prop. 4.7], the representation $\iota \circ \rho _0 : \Gamma \rightarrow G$ is P–Anosov for certain parabolic subgroups P of G (depending only on G, H and $\iota $ ). We will call such a representation an $\iota $ –lattice representation of $\Gamma $ in G.

When H is $\mathrm {PSL}(2,{\mathbb R})$ , the representation $\iota \circ \rho _0 : \pi _1(\Sigma ) \rightarrow G$ will be called an $\iota $ –Fuchsian representation in G. Similarly, when H is a compact extension of $\mathrm {PSL}(2,{\mathbb R})$ , $\iota \circ \rho _0$ will be called a twisted $\iota $ –Fuchsian representation in G.

Using the property of structural stability, we can deform the representation $\iota \circ \rho _0$ , obtaining an open subset of $\mathrm {Hom}(\Gamma ,G)$ entirely consisting of P–Anosov representations of $\Gamma $ in G. In the following, we denote by $\mathrm {Anosov}_{P,\iota ,\rho _0}(\Gamma ,G)$ the connected component of $\mathrm {Anosov}_{P}(\Gamma ,G)$ that contains the representation $\iota \circ \rho _0$ . We will say that a representation of $\Gamma $ is a P–Anosov deformation of a lattice representation if it belongs to one of the connected components $\mathrm {Anosov}_{P,\iota ,\rho _0}(\Gamma ,G)$ . In the special case when H is a compact extension of $\mathrm {PSL}(2,{\mathbb R})$ , such representations will be called P–Anosov deformations of a twisted $\iota $ –Fuchsian representation.

For this paper, we are particularly interested in P–Anosov deformations of lattice representations because our Theorem A applies to this special class of Anosov representations. It is a special class, but it is also rather general since most known Anosov representations fall in this class. This is mainly because this is the easiest way to construct Anosov representations.

Now, we want to describe the most important examples of such representations. The main source of examples of interesting deformations of lattice representations come from the case when H is a compact extension of $PSL(2,{\mathbb R})$ (i.e., the case of surface groups). These examples are discussed in Section 2.3. The other important source of examples is the case of uniform lattices in $\operatorname {PO}_0(d,1)$ (i.e., fundamental groups of closed hyperbolic d–manifolds). These examples are discussed in Section 2.4.

Lattices in $\mathrm {PU}(d,1)$ exhibit more rigid behaviour; see [Reference KlinglerKli11] and references therein. Still, some of them admit interesting Zariski dense deformations into higher rank Lie groups, but very few examples are known. When $H \simeq {\mathrm {Sp}}(d,1)$ or $\mathrm F_4^{-20}$ , by a theorem of Corlette [Reference CorletteCor92], $\Gamma $ is superrigid. In particular, there are no nontrivial deformations of $\iota \circ \rho _0 : \Gamma \rightarrow G$ .

2.3 Anosov representations of surface groups

The case of surface groups is the one that is best understood. When H is a compact extension of $\mathrm {PSL}(2,{\mathbb R})$ , Lie theory gives a classification of all representations $\iota : H \rightarrow G$ , for a simple group G. Most of the time, twisted $\iota $ –Fuchsian representations into G admit small deformations with Zariski dense image.

In special cases, for particular twisted $\iota $ –Fuchsian representations into G, all deformations (not just small ones) are Anosov. This phenomenon gives rise to the so-called higher rank Teichmüller components, defined as connected components of the representation variety $\mathrm {Hom}(\pi _1(S),G)/G$ that consist entirely of discrete and faithful representations. They generalize many aspects of classical Teichmüller spaces, which can be seen as connected components of $\mathrm {Hom}(\pi _1(S),\mathrm {PSL}(2,{\mathbb R}))/\mathrm {PSL}(2,{\mathbb R})$ . An interesting feature for us is that most higher Teichmüller components consist of deformations of twisted $\iota $ –Fuchsian representations.

There are four families of higher rank Teichmüller spaces (see Guichard–Wienhard [Reference Guichard and WienhardGW]). The first family consists of Hitchin representations, introduced by Hitchin [Reference HitchinHit87]. In fact, Labourie’s original motivation for defining Anosov representations in [Reference LabourieLab06] was showing that Hitchin representations form higher rank Teichmüller components. Hitchin representations are defined when G is a split real simple Lie group. Then G admits a special conjugacy class of representations $\iota _0:\mathrm {SL}(2,{\mathbb R}) \rightarrow G$ called the principal representation. For this choice, the representation $\iota _0 \circ \rho _0$ is called a principal Fuchsian representation in G, and it is Anosov with respect to the minimal parabolic subgroups $P_{min}$ . Hitchin components are the connected components containing a principal Fuchsian representation. In particular, any Hitchin representation is a deformation of a principal Fuchsian representation.

The second family consists of Maximal representations, which are defined when G is a real simple Lie group of Hermitian type [Reference Burger, Iozzi and WienhardBIW10]. Maximal representations are in general only Anosov with respect to a particular maximal parabolic subgroup. Most maximal representations are deformations of twisted $\iota $ –Fuchsian representations, but for G locally isomorphic to ${\mathrm {Sp}}(4,{\mathbb R})$ , there exist connected components in the space of maximal representations where every representation is Zariski dense [Reference GothenGot01, Reference Guichard and WienhardGW10, Reference Bradlow, García-Prada and GothenBGPG12].

The other two families of higher Teichmüller components arise from the notion of $\Theta $ –positivity introduced in [Reference Guichard and WienhardGW18, Reference Guichard and WienhardGW, Reference Guichard, Labourie and WienhardGLW], which leads to the notion of positive representations. Hitchin representations and maximal representations are positive representations, but there are two further families of Lie groups admitting positive representations, Lie groups locally isomorphic to $\mathrm {SO}(p,q)$ , as well as an exceptional family. With a positive structure comes again a special representation $\mathrm {SL}(2,{\mathbb R}) \rightarrow G$ . This representation might have a compact centralizer, so there is a compact extension H of $\mathrm {SL}(2,{\mathbb R})$ that embeds into G; this is the representation $\iota $ . From the classification of the connected components of positive representations, we see that deformations of twisted $\iota $ –Fuchsian representations account for most of the connected components; exceptions occur only for $\mathrm {SO}(p,p+1)$ . See [Reference Guichard and WienhardGW18], Guichard–Labourie–Wienhard [Reference Guichard, Labourie and WienhardGLW], Collier [Reference CollierCol15], Aparicio-Arroyo–Bradlow–Collier–Garcia-Prada–Gothen–Oliveira [Reference Aparicio-Arroyo, Bradlow, Collier, Garcia-Prada, Gothen and OliveiraAABC+19], Bradlow–Collier–Garcia-Prada–Gothen–Oliveira [Reference Bradlow, Collier, García-Prada, Gothen and OliveiraBCGP+], Beyrer–Pozzetti [Reference Beyrer and Beatrice PozzettiBP], Beyrer–Guichard–Labourie–Pozzetti–Wienhard [Reference Beyrer, Guichard, Labourie, Pozzetti and WienhardBGL+24] for more details on positive representations.

Given a representation $\rho \colon \thinspace \pi _1(\Sigma ) \to G$ in a higher Teichmüller space, we can embed G into its complexification $G_{{\mathbb C}}$ . If $\rho $ is Anosov with respect to a parabolic subgroup P, the composition will be Anosov with respect to the parabolic $P_{{\mathbb C}}<G_{{\mathbb C}}$ . In the complex group not every deformation will be discrete and faithful, but we can consider the space of Anosov representations $\mathrm {Anosov}_{P_{{\mathbb C}}}(\pi _1(\Sigma ),G_{{\mathbb C}})$ and the connected component of this space containing $\rho : \pi _1(\Sigma ) \to G < G_{{\mathbb C}}$ . This generalizes the notion of quasi-Fuchsian representation into $\mathrm {PSL}(2,{{\mathbb C}})$ to this higher rank setting. Of particular interest to us will be the connected component of $\mathrm {Anosov}_{P_{{\mathbb C}}}(\pi _1(\Sigma ),G_{{\mathbb C}})$ which contains the principal Fuchsian representation $\iota _0 \circ \rho _0$ . We call this set the quasi-Hitchin space and representations therein quasi-Hitchin representations. Theorem A applies in particular to quasi-Hitchin representations, and Theorem D focuses on quasi-Hitchin representations for $G_{{\mathbb C}} = {\mathrm {Sp}}(4,{{\mathbb C}})$

2.4 Fundamental groups of hyperbolic manifolds

As mentioned in the introduction, applications of Theorem A are not limited to representations of surface groups. There are indeed interesting classes of Anosov deformations of fundamental groups of closed hyperbolic manifolds of higher dimension.

Let $\Gamma = \pi _1(M)$ be the fundamental group of a closed orientable hyperbolic manifold of dimension $d\geq 3$ . Then $\Gamma $ identifies with a uniform torsion-free lattice in $\mathrm {SO}_0(d,1) \simeq \operatorname {Isom}_+(\mathbb H^d)$ via a representation $\rho _0 : \Gamma \to \mathrm {SO}_0(p,1)$ . One can construct in every dimension many examples where M contains an embedded totally geodesic hypersurface. There is then a general ‘bending’ procedure that allows one to deform the representation $\rho _0$ into a higher rank Lie group. We mention two particularly interesting examples.

Representations dividing a convex set. In a series of four papers, Benoist developed the theory of divisible convex sets – that is, proper convex subsets of a real projective space admitting a cocompact action of a discrete group of projective transformations. In particular, taking $\iota : \mathrm {SO}_0(d,1) \to \mathrm {SL}(d+1,\mathbb R)$ to be the standard representation, Benoist proved in [Reference BenoistBen05] that any continous deformation $\rho : \Gamma \to \mathrm {SL}(d+1,{\mathbb R})$ of $\iota \circ \rho _0$ is discrete, faithful and acts properly discontinuously and cocompactly on an open strictly convex domain $\Omega _\rho $ of the projective space $\mathbb{RP}^d$ . It follows (see [Reference Guichard and WienhardGW12, Proposition 6.1]) that any continuous deformation of $\iota \circ \rho _0$ is $P_{1,d}$ -Anosov.

The domain $\Omega _\rho \subset \mathbb R\mathbb {P}^{d}$ is a flag domain of discontinuity whose quotient is easily described: $\Omega _\rho $ is contractible and $\rho (\Gamma )\backslash \Omega _\rho $ is diffeomorphic to M. However, it is not the only flag domain of discontinuity one can associate to a convex divisible representation. There are many flag domains of discontinuity in other flag varieties of $\mathrm {SL}(d+1,{\mathbb R})$ , as well as flag domains of discontinuities in $G'/Q' $ when we embed $\mathrm {SL}(d+1,{\mathbb R})$ into a larger Lie group $G'$ . The topology of these domains of discontinuity can be more complicated. Theorem A applies to all these domains of discontinuity, as long as $\rho $ is in the same connected component (of Anosov representations) as $\iota \circ \rho _0$ .

AdS quasi-Fuchsian and $\mathbf {\mathbb H^{p,q}}$ -convex-cocompact representations. Similarly, consider the embedding $\iota : \mathrm {SO}_0(d,1) \to \mathrm {SO}(d,2)$ . Barbot and Mérigot proved in [Reference Barbot and MérigotBM12] and [Reference BarbotBar15] that any continuous deformation $\rho $ of $\iota \circ \rho _0$ into $\mathrm {SO}(d,2)$ is P-Anosov, where P is the stabilizer of an isotropic line. Moreover, $\rho $ acts properly discontinuously on a convex domain $\Omega _\rho $ contained in the anti-de Sitter space of dimension $d+1$ , and the quotient $\rho (\Gamma )\backslash \Omega _\rho $ is a globally hyperbolic Cauchy compact anti-de Sitter spacetime. Barbot and Mérigot call these representations AdS-quasi-Fuchsian. Our main theorem applies to these representations.

Recently, these results have been generalized to deformations of $\iota \circ \rho _0$ , for $\iota : \mathrm {SO}_0(d,1) \to \mathrm {SO}(d,d')$ the standard embedding: Beyrer and Kassel proved in [Reference Beyrer and KasselBK23] that any continuous deformation into $\mathrm {SO}(d,d')$ is P-Anosov and $\mathbb H^{d,d'-1}$ -convex-cocompact in the sense of [Reference Danciger, Guéritaud and KasselDGK18]. Our theorem applies to these deformations, which again exist in many examples. Such groups have in particular a cocompact flag domain of discontinuity in the space of maximal totally isotropic subspaces of ${\mathbb R}^{d,d'}$ . The topology of this domain is described in the recent paper [Reference SeppiSST23].Footnote ²

Complex deformations. Note finally that whenever $\iota \circ \rho _0$ admits deformations into a real linear algebraic group G, it also admits deformations into its complexification $G_{{\mathbb C}}$ which are not real. For instance, the above examples admit respectively Anosov deformations in $\mathrm {SL}(d+1,{{\mathbb C}})$ and $\mathrm {SO}(d+d',{{\mathbb C}})$ . Theorem A applies to such complex deformations as well.

2.5 Domains of discontinuity

A P–Anosov representation $\rho \colon \thinspace \Gamma \to G$ acts on all homogeneous spaces $G/Q$ , where Q is a proper parabolic subgroup. The theory of domains of discontinuity, introduced by Guichard–Wienhard [Reference Guichard and WienhardGW12] and further developed by Kapovich–Leeb–Porti [Reference Kapovich, Leeb and PortiKLP18], gives conditions for the existence of a $\rho $ –invariant open subset $\Omega \subset G/Q$ where the action is properly discontinuous and/or cocompact. We sketch very briefly this construction here and refer the reader to [Reference Kapovich, Leeb and PortiKLP18] for details.

The action of P on $G/Q$ has finitely many orbits which are labelled by elements of $W_P \backslash W/W_Q$ , where W is the Weyl group of G and $W_P, W_Q$ are the subgroups corresponding to P and Q. A subset I of $W_P \backslash W/W_Q$ corresponds to a P–invariant subset $K_I$ of $G/Q$ (consisting of the union of the orbits labelled by elements of I). The set $K_I$ is closed if and only if I is an ideal for the Bruhat order on W.

Given $x = gP\in G/P$ , set $K_I(x) = g K_I$ (this is well-defined since $K_I$ is P–invariant). The I–thickening of a subset $A\subset G/P$ is the set $K_I(A) = \bigcup _{x\in A} K_I(x)$ . Finally, an ideal is called balanced if $I \cap -I = \emptyset $ and $I\cup -I = W_P\backslash W /W_Q$ .

Now, let $\Gamma $ be a hyperbolic group, $\rho \colon \thinspace \Gamma \to G$ a P–Anosov representation and $\xi _\rho \colon \thinspace \partial _\infty \Gamma \to G/P$ the associated boundary map.

Theorem 2.4 (Kapovich–Leeb–Porti, [Reference Kapovich, Leeb and PortiKLP18]).

If $I \subset W_P\backslash W /W_Q$ is a balanced ideal, then $\Gamma $ acts properly discontinuously and cocompactly on the domain

$$\begin{align*}\Omega_{\rho,I} = (G/Q) \ \setminus \ K_I(\xi_\rho(\partial_\infty \Gamma)).\end{align*}$$

When $\Gamma $ is a uniform lattice in a rank $1$ subgroup H of G, these domains of discontinuity are actually preserved by the group H:

Proof. Now, let C be a compact subset of H such that $\Gamma C = H$ . Let D be any compact subset of $\Omega _{\rho ,I}$ .

Assume $\Gamma $ is properly discontinuous. The set

$$\begin{align*}\{h\in H \mid h\cdot D \cap D \neq \emptyset\}\end{align*}$$

is contained in $FC$ , where

$$\begin{align*}F= \{\gamma \in \Gamma \mid \gamma \cdot (C\cdot D) \cap D \neq \emptyset \}.\end{align*}$$

Since $C\cdot D$ is compact, F is finite by proper discontinuity of $\Gamma $ . Hence, $FC$ is compact, proving the properness of the action of H.

Conversely, if H is properly discontinuous, the set

$$\begin{align*}\{\gamma \in \Gamma \mid h\cdot D \cap D \neq \emptyset\}\end{align*}$$

is the intersection of $\Gamma $ with a compact subset of H, which is thus finite since $\Gamma $ is discrete.

Assume some compact subset D of X satisfies $\Gamma \cdot D = X$ . Then obviously $H\cdot D= X$ . Conversely, assume $H\cdot D = X$ . Then $\Gamma \cdot (C\cdot D) = H$ and $C\cdot D$ is compact. This proves that cocompactness of $\Gamma $ and H are equivalent.

In order to illustrate the theory, let us now describe the example that will be studied in detail in Part II of this paper. Consider the case where the group G is ${\mathrm {Sp}}(2n,\mathbb {K})$ , where $\mathbb {K}$ can be ${\mathbb R}$ of ${{\mathbb C}}$ , and the parabolic subgroup P is the stabilizer of a point in $\mathbb {KP}^{2n-1}$ . In this case, $G/P=\mathbb {KP}^{2n-1}$ . Every P–Anosov representation $\rho \colon \thinspace \Gamma \to {\mathrm {Sp}}(2n,\mathbb {K})$ has an associated $\rho $ –equivariant map

$$\begin{align*}\xi_\rho \colon\thinspace \partial_{\infty}\Gamma \to \mathbb{KP}^{2n-1}.\end{align*}$$

We now consider as second parabolic subgroup Q the stabilizer of a Lagrangian subspace in $\mathbb {K}^{2n}$ . Then $G/Q$ is the Lagrangian Grassmannian ${\mathrm {Lag}}(\mathbb {K}^{2n})$ (i.e., the space of all the Lagrangian subspaces of $\mathbb {K}^{2n}$ ). The action of P on $G/Q$ has only two orbits: a closed orbit consisting of Lagrangian subspaces containing the line fixed by P, and its complement which is open. In this case, $W_P \backslash W/W_Q$ has only two elements and admits a unique nontrivial ideal I, for which $K_I$ is the closed P–orbit. This ideal is balanced.

For each line $\ell \in \mathbb {KP}^{2n-1}$ , we have

$$\begin{align*}K_\ell = K_I(\ell) = \{W \in \mathrm{Lag}(\mathbb{K}^{2n}) \mid \ell \subset W\} \subset {\mathrm{Lag}}(\mathbb{K}^{2n}),\end{align*}$$

and we define the subset

$$\begin{align*}K_{\rho,I} = K_I(\xi(\partial_\infty \Gamma)) = \bigcup_{t \in \partial_{\infty}\Gamma}K_{\xi(t)} \subset {\mathrm{Lag}}(\mathbb{K}^{2n}).\end{align*}$$

Guichard and Wienhard [Reference Guichard and WienhardGW12] showed that the complement

$$\begin{align*}\Omega_{\rho,I} = \mathrm{Lag}(\mathbb{K}^{2n}) \ \setminus\ K_{\rho,I}\end{align*}$$

is a cocompact domain of discontinuity for $\rho $ . Recall that $\Gamma $ acts freely since it is torsion-free, so we have that

$$\begin{align*}M_{\rho,I} = \rho(\Gamma) \backslash \Omega_{\rho,I}\end{align*}$$

is a closed manifold endowed with a geometric structure modelled on the parabolic geometry $(G, G/Q) = \left ({\mathrm {Sp}}(2n,\mathbb {K}), {\mathrm {Lag}}(\mathbb {K}^{2n})\right )$ . Determining the topology of this quotient manifold (and more general such constructions) is a main focus of this paper.

2.6 Deformations

Consider one of our spaces $\mathcal {A} = \mathrm {Anosov}_{P,\iota ,\rho _0}(\Gamma ,G)$ , defined in Section 2.2. Let Q be a parabolic subgroup and I a balanced ideal of $W_P \backslash W /W_Q$ . For every representation $\rho \in \mathcal {A}$ , we obtain a closed manifold $M_{\rho ,I} = \rho (\Gamma ) \backslash \Omega _{\rho ,I}$ endowed with a geometric structure locally modelled on the parabolic geometry $(G, G/Q)$ , and whose holonomy factors throughFootnote ³ the representation $\rho $ .

For a $\rho \in \mathcal {A} = \mathrm {Anosov}_{P,\iota ,\rho _0}(\Gamma ,G)$ , the topology of $M_{\rho ,I}$ does not depend on $\rho $ ; hence, we can denote this smooth manifold by $M_{\rho _0,\iota ,I}$ . Thus, the space $\mathcal {A}$ can be seen as a deformation space for a family of $(G,G/Q)$ –structures on a the fixed closed manifold $M_{\rho _0,\iota ,I}$ . This is particularly interesting for higher rank Teichmüller spaces because it gives a nice geometric interpretation of these spaces. It is also interesting for the theory of geometric structures on manifolds because it gives several interesting examples of closed manifolds with a large deformation space of geometric structures.

3.1 General statement

We can now rephrase Theorem A, which describes the topology of $M_{\rho _0,\iota ,I}$ constructed from an Anosov deformation of an $\iota $ –lattice representation.

Let us fix a connected semisimple Lie group H of real rank $1$ with finite center, a uniform torsion-free lattice $\Gamma \subset H$ and a representation $\iota $ of H into some connected semisimple Lie group G with finite center. Denote by $\rho _0$ the inclusion of $\Gamma $ into H, let P be a parabolic subgroup of G such that $\iota \circ \rho _0$ is P–Anosov, and let $\rho $ be a P–Anosov deformation of $\iota \circ \rho _0$ . Finally, let ${S_H}$ denote the symmetric space of H.

In fact, one can say a bit more on the structure of this bundle. For this, let us recall the notion of fiber bundle associated to a principal bundle. Let K be a Lie group, $T\to B$ a principal K-bundle and $\mathfrak F$ a smooth manifold equipped with a smooth action of K.

Theorem 3.3. In the setting of Theorem 3.1, the manifold $\mathfrak F$ admits a smooth action of the compact subgroup K, and the manifold $M_{\rho _0,\iota ,I}$ is the $\mathfrak F$ –bundle over $\Gamma \backslash {S_H}$ associated to the principal K–bundle

$$\begin{align*}\Gamma \backslash H \to \Gamma \backslash {S_H}.\end{align*}$$

In the previous theorem, the bundle $\Gamma \backslash H \to \Gamma \backslash {S_H}$ must be thought of as an explicit object that depends only on the lattice $\Gamma $ . For example, when $H = PO_0(n,1)$ , $\Gamma \backslash {S_H}$ is a closed hyperbolic manifold, $\Gamma $ is its fundamental group, and the bundle $\Gamma \backslash H \to \Gamma \backslash {S_H}$ is its frame bundle.

In the special case when H is a compact extension of $\mathrm {PSL}(2,{\mathbb R})$ , these theorems take an even more explicit form. In this case, $S_H = \mathbb {H}^2$ is the hyperbolic plane, the group $\Gamma = \pi _1(\Sigma )$ is a surface group, and $\Gamma \backslash \mathbb {H}^2$ is the surface $\Sigma $ . The principal bundle

$$\begin{align*}\Gamma \left(\Gamma\right) \backslash H \to \Sigma\end{align*}$$

depends on the extension H. For example, when $H = \mathrm {PSL}(2,{\mathbb R})$ , this bundle is a circle bundle isomorphic to the unit tangent bundle of $\Sigma $ (i.e., a circle bundle with Euler class $2g-2$ ). When $H = \mathrm {SL}(2,{\mathbb R})$ , this bundle is the double cover of the unit tangent bundle of $\Sigma $ (i.e., a circle bundle with Euler class $g-1$ ). For all the interesting groups H, it is possible to understand this bundle explicitly. We will now restate the previous theorems in the case when $H = \mathrm {SL}(2,{\mathbb R})$ .

One of the main applications of this corollary is for (quasi)-Hitchin representations. Recall that Hitchin representations are deformations of $\iota _0 \circ \rho _0$ where $\rho _0: \pi _1(\Sigma )\to \mathrm {SL}(2,{\mathbb R})$ is a Fuchsian representation and $\iota _0$ is the principal representation of $\mathrm {SL}(2,{\mathbb R})$ into a real split semisimple Lie group G, and quasi-Hitchin representations are their $P_{\min }$ –Anosov deformations into its complexification $G_{{\mathbb C}}$ .

We can also apply Theorems 3.1 and 3.3 to the positive representations that are Anosov deformations of twisted $\iota $ –Fuchsian representations. As discussed in Section 2.3, almost all the positive representations in the classical groups are of this type, with the only exception of the exceptional components in ${\mathrm {Sp}}(4,{\mathbb R})$ and $\mathrm {SO}(p,p+1)$ . In order to apply our results to positive representations, we need to consider the group $H = \mathrm {SL}(2,{\mathbb R}) \times C$ for a certain compact subgroup C. The statement is similar to Corollary 3.4, except that the structure group of the bundle is now $\mathrm {SO}(2)\times C$ . The invariants that characterize the bundle are the Euler class and the characteristic classes of the C component of $\rho _0$ .

3.2 Proof of the theorems

A key hypothesis in Theorem 3.1 is the assumption that $\rho $ is a P–Anosov deformation of an $\iota $ –Fuchsian representation $\iota \circ \rho _0$ . Indeed, by Guichard–Wienhard’s Theorem 2.10, the topology of $M_{\rho ,I}$ does not change, and we only have to determine it for $\rho = \iota \circ \rho _0$ . The key result for the proof is thus the following.

Lemma 3.5. Let H be a semisimple Lie group with finite center, and $K \subset H$ be its maximal compact subgroup. Let X be a manifold with a proper action of H. Then there exists an H–equivariant smooth fibration

$$\begin{align*}p:X \rightarrow S_H,\end{align*}$$

where $S_H$ denotes the symmetric space of H.

For the proof, we use the fact that the symmetric space $S_H = H/K$ has non-positive curvature. We need the notion of barycenter: Given a finite measure of compact support $\nu $ on $S_H$ , we consider the function

$$\begin{align*}b:S_H \rightarrow {\mathbb R} \end{align*}$$

defined by

$$\begin{align*}b(y) = \int d(y,z)^2 \mathrm d \nu(z).\end{align*}$$

Since $S_H$ has non-positive curvature, the squared distance function is smooth (as push-forward of the squared norm under the exponential map), the distance function is convex (see [Reference Ballmann, Gromov and SchroederBGS85, Thm 1.3]), and hence the squared distance function is strongly convex. This implies that the function b is strongly convex, and since it is also proper, it has a unique critical point, which is a global minimum. Moreover, the Hessian of b at the global minimum is positive definite. The barycenter $\mathrm {Bar}\{\nu \}$ of $\nu $ is defined to be the unique critical point of b.

Proof of Lemma 3.5.

We choose a torsion-free uniform lattice $\Gamma $ in H. This always exists. See, for example, Borel and Harish–Chandra [Reference Borel and ChandraBHC62]. Then $\Gamma $ acts freely and properly discontinuously on X (see Proposition 2.8); hence, the quotient $\Gamma \backslash X$ is a manifold. Then $\Gamma $ is isomorphic to the quotient $\pi _1(\Gamma \backslash X)/\pi _1(X)$ , and we have a homomorphism $\psi \colon \thinspace \pi _1(\Gamma \backslash X) \to \Gamma = \pi _1(\Gamma \backslash S_H)$ . Since $S_H$ is contractible, there exists a map $\Gamma \backslash X \to \Gamma \backslash S_H$ inducing $\psi $ (for the details, [Reference AlessandriniAle03, Prop. 13]). A priori this map is only continuous, but since smooth maps are dense in the space of continuous maps between compact manifolds, we can assume that the map is smooth. Lifting this map, we obtain a smooth $\Gamma $ -equivariant map

$$\begin{align*}f\colon\thinspace X \to S_H.\end{align*}$$

This allows us to define, for all $x\in X$ , a smooth map $F^x\colon \thinspace H \to S_H$ by

$$\begin{align*}F^x(g) = g\cdot f(g^{-1}\cdot x).\end{align*}$$

Since f is $\Gamma $ –equivariant, we have that $F^x(g\gamma )= F^x(g)$ for all $\gamma \in \Gamma $ . Hence, $F^x$ descends to a continuous map $\overline {F}^x: H/\Gamma \to S_H$ .

Note that we have

(1) $$ \begin{align} \overline{F}^{h\cdot x}(g) = g \cdot f(g^{-1} h \cdot x) = h \cdot \overline{F}^{x}(h^{-1} g). \end{align} $$

We can finally define a map $\bar f\colon \thinspace X \to S_H$ as follows:

$$\begin{align*}\bar f(x) = \mathrm{Bar} \{\overline{F}^x_* \mu\}\,,\end{align*}$$

where $\mu $ is the Haar measure on $H/\Gamma $ , $\overline {F}^x_*\mu $ its push-forward by $\overline {F}^x$ , and $\mathrm {Bar}$ is the barycenter of a finite measure with compact support, as defined above. Note that the image of $\overline {F}^x$ is compact since $H/\Gamma $ is compact; hence, $\overline {F}^x_*\mu $ does have compact support.

We claim that $\bar f$ is H–equivariant. Indeed, for $x\in X$ and $g\in H$ , we have

$$ \begin{align*} \bar f(gx) &= \mathrm{Bar} \{ \overline{F}^{gx}_*\mu\} \\ &= \mathrm{Bar}\{g_* \overline{F}^x_* g^{-1}_* \mu\} \quad \textrm{by (1)}\\ &= \mathrm{Bar}\{g_* \overline{F}^x_* \mu \} \quad \textrm{by left invariance of the Haar measure}\\ &= g\cdot \mathrm{Bar}\{\overline{F}^x_* \mu \} \quad \textrm{by equivariance of the barycenter map}\\ &= g\cdot \bar f(x). \end{align*} $$

The rest follows from the two lemmas below. Lemma 3.6 guarantees that the map $\bar {f}$ is smooth, and Lemma 3.7 shows that it is an Ehresmann fibration.

Proof. For every $x\in X$ , $\bar {f}(x)$ is the unique critical point of the function

$$\begin{align*}b(y) = \int_{S_H} d(y,z)^2 \mathrm d \nu(z)\,,\end{align*}$$

where the measure $\nu $ is the push-forward $\nu = \overline {F}^x_* \mu $ . By the change-of-variable formula, this can be written as

$$\begin{align*}b(y) = \int_{\Gamma \backslash H} d(y,\overline{F}^x(h))^2 \mathrm d \mu(h) = \int_{\Gamma \backslash H} d(y,h f(h^{-1}x))^2 \mathrm d \mu(h).\end{align*}$$

In this setting, the function b depends on the parameter x. To make this more explicit, we write it as $b(x,y)$ , a smooth function of two variables $x \in X$ and $y \in S_H$ . We consider the differential of b with respect to y:

$$\begin{align*}\beta(x,y) = d_y b(x,y) : X \times S_H \rightarrow T^* S_H. \end{align*}$$

Let us fix an $x_0 \in X$ and the corresponding $y_0 = \bar {f}(x_0) \in S_H$ . We choose local coordinates on a small neighborhood U of $y_0$ in $S_H$ . This trivializes the cotangent bundle on U: $T^* U \simeq U \times {\mathbb R}^k$ , where $k = \dim (S_H)$ . Let $\pi _2: T^* U \rightarrow {\mathbb R}^k$ denote the projection onto the second factor. Now we consider the composition

$$\begin{align*}\pi_2 \circ \beta(x,y) = d_y b(x,y) : X \times U \rightarrow {\mathbb R}^k . \end{align*}$$

Now, for all x close enough to $x_0$ , the pairs $(x,\bar {f}(x))$ are precisely the solutions to the equation

$$\begin{align*}\pi_2 \circ \beta(x,y) = 0.\end{align*}$$

We can now apply the implicit function theorem to the function $\pi _2 \circ \beta (x,y)$ . Consider a tangent vector to $X\times U$ of the form $(0,u)$ at a point $(x,\bar {f}(x))$ . Then we have

$$\begin{align*}d \pi_2(0,u) : v\mapsto \mathrm{Hess}_y b_{\bar f(x)} (u,v),\end{align*}$$

where $\mathrm {Hess}_y b{\bar f(x)}(u,v)$ denotes the Hessian of the function $b(x,\cdot )$ at the point $\bar f(x)$ (recall that the Hessian of a function at critical point is well-defined independently of the local coordinate). Since $b(x,\cdot )$ is strongly convex, $\mathrm {Hess}_y b{\bar f(x)}$ is positive definite, and the differential of $\pi _2$ in the y direction is an isomorphism from $T_{\bar f(x)} U$ to $T^*_{\bar f(x)} U = {\mathbb R}^k$ . Hence, $\pi _2$ is a submersion, and the set $\{(x,\bar f(x))\} = \pi _2^{-1}(0)$ is the graph of a smooth function.

Proof. First note that, by homogeneity of Y, the map $\phi $ needs to be onto. We will construct a local trivialization around every point $y \in Y$ . We choose the subgroup L as the stabilizer of y. Hence, L is acting on the fiber $F = \phi ^{-1}(y)$ . Let U be a neighborhood of y in Y that trivializes the bundle $H \rightarrow H/L$ . The trivialization is a map $t:U \times L \rightarrow H$ .

A trivialization of $\phi $ over U is given by the map

$$\begin{align*}T: U \times F \ni (u,f) \rightarrow t(u,e)f \in X\,,\end{align*}$$

where e is the identity of H. Clearly, $\phi (T(u,f)) = u$ , because $t(u,e)$ sends y to u. The map T is $1$ – $1$ because if $t(u,e)f = t(u',e)f'$ , then $u = u'$ because $\phi (T(u,f))=u$ , and then by multiplying by $t(u,e)^{-1}$ , we see that $f=f'$ . We can also see that the map T is onto $\phi ^{-1}(U)$ because given $x \in \phi ^{-1}(U)$ , let $f = t(\phi (x),e)^{-1}x \in F$ , and then $x = T(\phi (x),f)$ .

The construction above shows that every atlas for the bundle $H \rightarrow H/L$ induces an atlas for the bundle $\phi : X \rightarrow Y$ . It is easy to check that the two atlases have the same transition functions, and hence, the two bundles are associated.

The following Proposition 3.8 is similar to Theorems 3.1 and 3.3, but the difference is that it can be applied to domains of discontinuity that are not necessarily cocompact. Anyway, if the domain is not cocompact, our conclusion only holds for $\iota $ –lattice representations, but does not automatically extend to their deformations. After that, we will add the hypothesis that the domain of discontinuity is cocompact and prove the full Theorems 3.1 and 3.3 for their deformations.

Part II Quasi-Hitchin representations into $\mathbf {{\mathrm {Sp}}(4,{{\mathbb C}})}$

In the second part of the paper, we will focus on quasi-Hitchin representations into $G = {\mathrm {Sp}}(4,{{\mathbb C}})$ . We fix the principal representation $\iota _0 \colon \thinspace \mathrm {SL}(2, {\mathbb R}) \to G$ and a Fuchsian representation $\rho _0 \colon \thinspace \pi _1(\Sigma ) \to \mathrm {SL}(2, {\mathbb R})$ . The principal Fuchsian representation $\iota _0\circ \rho _0$ is Anosov with reference to every parabolic subgroup P of G. Recall that a P–quasi-Hitchin representation is a P–Anosov deformation of $\iota _0\circ \rho _0$ . Our aim is to determine the topology of the quotient manifolds of the cocompact domains of discontinuity for these representations.

The group ${\mathrm {Sp}}(4,{{\mathbb C}})$ has (up to conjugation) three different proper parabolic subgroups, so there are three flag varieties for us to consider: the projective space $\mathbb {CP}^3$ , the Lagrangian Grassmannian ${\mathrm {Lag}}({{\mathbb C}}^4)$ , and the full flag variety, which consists of full isotropic flags (i.e., pairs consisting of a line in ${{\mathbb C}}^4$ and a Lagrangian subspace of ${{\mathbb C}}^4$ containing that line). The principal Fuchsian representation $\iota _0\circ \rho _0$ admits four cocompact domains of discontinuity constructed by a balanced thickening: one in the projective space $\mathbb {CP}^3$ , one in the Lagrangian Grassmannian ${\mathrm {Lag}}({{\mathbb C}}^4)$ (whose construction is described in Section 2.5), and two in the isotropic full flag variety. The two domains of discontinuity in the isotropic full flags variety are in fact the pullback of the two domains in $\mathbb {CP}^3$ and in ${\mathrm {Lag}}({{\mathbb C}}^4)$ under the natural projection from the full flag variety to the partial flag varieties; hence, they can be understood from a description of the latter two. The domain of discontinuity in $\mathbb {CP}^3$ was described in Alessandrini–Davalo–Li [Reference Alessandrini, Davalo and LiADL24, Corol. 10.2], where it is proved that the quotient manifold M is diffeomorphic to a fiber bundle over the surface $\Sigma $ with fiber $\mathbb {S}^2 \times \mathbb {S}^2$ .

The only cocompact domain of discontinuity that is not yet understood is the one in ${\mathrm {Lag}}({{\mathbb C}}^4)$ . The second part of this paper is devoted to the description of this domain and its quotient manifold. In fact, the domain of discontinuity in the Lagrangian Grassmannian is of particular interest because it contains two copies of the symmetric space associated to ${\mathrm {Sp}}(4,{\mathbb R})$ : the Siegel upper half space and the Siegel lower half space; see [Reference WienhardWie16]. This is very reminiscent of the situation for quasi-Fuchsian representations, and we hope that a good understanding of the domain of discontinuity and its quotient manifold might help to shed some light on possible generalizations of the Bers’ double uniformization theorem for quasi-Hitchin representations.

The construction of the domain of discontinuity in ${\mathrm {Lag}}({{\mathbb C}}^4)$ , described in Section 2.5, only uses the fact that the representation is Anosov with respect to P, where P is the stabilizer of a point in $\mathbb {CP}^3$ . We will thus consider a representation $\rho $ in the quasi-Hitchin space $\mathrm {QHit}_{P}(\Sigma , {\mathrm {Sp}}(4,{{\mathbb C}})) := \mathrm {Anosov}_{P,\iota _0,\rho _0}(\pi _1(\Sigma ),{\mathrm {Sp}}(4,{{\mathbb C}}))$ .

There is a unique nontrivial ideal I, which allows us to define a cocompact domain of discontinuity $\Omega _{\rho ,I}$ , with quotient manifold $M_{\rho ,I}$ ; see Section 2.5. Since, by Theorem 2.10, the topology of the quotient manifold does not depend on $\rho $ , we can restrict our attention to the case when $\rho = \iota _0\circ \rho _0$ . With the representation fixed once for all, we will denote the domain of discontinuity and the quotient manifold simply by $\Omega $ and M, instead of $\Omega _{\rho ,I}$ and $M_{\rho ,I}$ .

Our Theorem A gives smooth fibrations

$$\begin{align*}p\colon\thinspace\Omega \to {\mathbb H}^2 \quad\text{and}\quad\widehat{p}\colon\thinspace M \to \Sigma.\end{align*}$$

In this second part of the paper, we will study the fiber $\mathfrak {F} = \mathfrak {F}_p$ of these maps and prove Theorem D, which states that $\mathfrak {F}$ is homeomorphic to the $4$ –manifold ${{\mathbb C}}\mathbb {P}^2 \# \overline {{{\mathbb C}}\mathbb {P}}^2$ .

Our strategy will be to describe a new $\rho $ –equivariant fibration

$$\begin{align*}q\colon\thinspace\Omega \to {\mathbb H}^2 \quad\text{and}\quad\widehat{q}\colon\thinspace M \to \Sigma,\end{align*}$$

which is not smooth, but has a more geometric definition and for which the fiber $F = F_q$ is easier to understand. This new fiber F is not a manifold, but it is homotopically equivalent to the domain of discontinuity $\Omega $ ; hence, F is homotopically equivalent to the smooth fiber $\mathfrak {F}$ . We will see that F is simply connected, and hence $\mathfrak {F}$ is a simply connected $4$ –manifold. A smooth simply connected $4$ –manifold is determined up to homeomorphism by its homotopy type; hence, we can determine $\mathfrak {F}$ by computing the homotopy invariants of F.

In order to define the fibration q, we will study the action of $\mathrm {SL}(2,{{\mathbb C}})$ on the Lagrangian Grassmannian ${\mathrm {Lag}}({{\mathbb C}}^4)$ induced by the principal representation $\iota _0$ . In Section 4, we will discuss the $\mathrm {SL}(2,{{\mathbb C}})$ –orbits in $\mathrm {Lag}({{\mathbb C}}^{4})$ , and this discussion will allow us to define q at the end of the section. In Section 5, we will study the fiber F, and in Section 6, we will use our results on the topology of F to understand $\mathfrak {F}$ .

4.1 Lagrangian subspaces in ${{\mathbb C}}^{2n}$

Let $\mathbb {K}$ be ${\mathbb R}$ or ${{\mathbb C}}$ , and let $V_{n, \mathbb {K}} = \mathbb {K}^{(2n-1)}[X, Y]$ be the space of homogeneous polynomials of degree $2n-1$ in the variables X and Y. We fix an explicit basis of $V_{n, \mathbb {K}}$ given by the polynomials $P_k = X^{2n-1-k}Y^k$ for $k = 0, \ldots , 2n-1$ . In particular, $\dim (V_{n, \mathbb {K}}) = 2n$ . We remark that $V_{n, \mathbb {K}} = \mathrm {Sym}^{2n-1}(V_{1, \mathbb {K}})$ , or, in words, that $V_{n, \mathbb {K}}$ is the $(2n-1)$ -st symmetric power of $V_{1, \mathbb {K}}$ .

We endow $V_{1, \mathbb {K}}$ with the symplectic form $\omega _{1, \mathbb {K}}$ determined by $\omega _{1, \mathbb {K}}(X,Y) = 1$ . This induces a symplectic form $\omega _{n, \mathbb {K}} = \mathrm {Sym}^{2n-1}\omega _{1, \mathbb {K}}$ on $V_{n, \mathbb {K}}$ . Explicitly, this symplectic form is determined by the following formulae:

$$\begin{align*}\left\{ \begin{array}{ll} \omega_{n, \mathbb{K}}(P_k, P_l) = 0 & \mbox{if } k+l \neq 2n-1 \\ \omega_{n, \mathbb{K}}(P_k, P_{2n-1-k}) = (-1)^k \ \frac{k! (2n-1-k)!}{(2n-1)!}. & \end{array} \right. \end{align*}$$

Here, we are mainly interested in the case $n=2$ , where these formulae can be written more explicitly:

(2) $$ \begin{align} \left\{ \begin{array}{ll} \omega_{2, \mathbb{K}}(X^3, Y^3) = 1 \,,\\ \omega_{2, \mathbb{K}}(X^2 Y, X Y^2) = -\frac{1}{3} \,, & \end{array} \right. \end{align} $$

all other pairings being zero.

The group ${\mathrm {Sp}}(V_{1, \mathbb {K}}, \omega _{1, \mathbb {K}}) \cong {\mathrm {Sp}}(2, \mathbb {K}) \cong \mathrm {SL}(2, \mathbb {K})$ acts, via symmetric power, on $V_{n, \mathbb {K}}$ , and this action preserves the symplectic form $\omega _{n, \mathbb {K}}$ . This defines a representation

$$\begin{align*}\iota_0: \mathrm{SL}(2, \mathbb{K}) \rightarrow {\mathrm{Sp}}(V_{n, \mathbb{K}}, \omega_{n, \mathbb{K}}) \cong {\mathrm{Sp}}(2n,\mathbb{K})\, \end{align*}$$

that is irreducible: the principal representation in ${\mathrm {Sp}}(2n,\mathbb {K})$ . This representation induces an action of $\mathrm {SL}(2, \mathbb {K})$ on the projective space $\mathbb {P}(\mathbb {K}^{(2n-1)}[X, Y]) \cong \mathbb {K}\mathbb {P}^{2n-1}$ . This action can be described explicitly by considering how $\mathrm {SL}(2, \mathbb {K})$ moves the $2n-1$ roots of a polynomial.

A vector subspace $L \subset V_{n, \mathbb {K}}$ is called isotropic if $L \subset L^{\perp _{\omega _{n, \mathbb {K}}}}$ , where $L^{\perp _{\omega _{n, \mathbb {K}}}}$ is the orthogonal complement with respect to $\omega _{n, \mathbb {K}}$ . An isotropic subspace $L \subset V_{n,\mathbb {K}}$ is maximal if it has dimension n, or equivalently if $L = L^{\perp _{\omega _{n, \mathbb {K}}}}$ . In this case, L is called a Lagrangian subspace. Using the fact that $\omega _{n, \mathbb {K}}$ is skew-symmetric, we can see that all the subspaces of $V_{n, \mathbb {K}}$ of dimension one are isotropic, and hence, the space of $1$ –dimensional isotropic subspaces can be identified with the projective space $\mathbb {P}(V_{n, \mathbb {K}})$ . We denote the space of n–dimensional Lagrangian subspaces of $V_{n, \mathbb {K}}$ by $\mathrm {Lag}(V_{n,\mathbb {K}})$ , and we will call it the Lagrangian Grassmannian. We can think of it as a subspace of the Grassmannian of n–dimensional subspaces in $V_{n, \mathbb {K}}$ . The representation $\iota _0$ induces an action of $\mathrm {SL}(2, \mathbb {K})$ on $\mathrm {Lag}(V_{n, \mathbb {K}})$ .

One last thing we want to recall about this Lagrangian Grassmanian is its topology. There are different ways to describe the topology, but the way we will mostly use in this paper is via the subspace topology inherited from the Grassmanian space, whose topology can be described using the Plücker map or Plücker coordinates. In our case, the Plücker map is an embedding of the Grassmanian of n–planes in $V_{n, \mathbb {K}}$ into the projectivization of the n-th exterior power of $V_{n, \mathbb {K}}$ :

$$\begin{align*}\mathrm{Gr}(n, V_{n, \mathbb{K}}) \to \mathbb{P}\left(\Lambda^n(V_{n, \mathbb{K}})\right)\,,\end{align*}$$

which realizes $\mathrm {Gr}(n,V_{n, \mathbb {K}})$ as an algebraic variety, since the image consists of the intersection of a number of quadrics defined by the Plücker relations. To write these relations, we need to be more precise. Given $W \in \mathrm {Gr}(n,V_{n, \mathbb {K}})$ , choose a basis $W_1, W_2, \ldots , W_n$ of W consisting of column vectors. Let $\widehat {W}$ be the $(2n) \times n$ matrix whose columns are $W_1, W_2, \ldots , W_n$ . For any ordered sequence $1 \leq i_1 < i_2 < \cdots < i_n\leq 2n$ of n integers, let $W_{i_1, \ldots , i_n}$ be the determinant of the $n \times n$ matrix given by the rows $i_1, \ldots , i_n$ of $\widehat {W}$ . The numbers $W_{i_1, \ldots , i_n}$ are projective coordinates for W, and they satisfy the following relations:

$$\begin{align*}\sum_{l=1}^{n+1} W_{i_1, \ldots, i_{n-1}, j_l} W_{j_1, \ldots,\hat{j}_l, \ldots j_{n+1}} = 0\,,\end{align*}$$

for $1 \leq i_1 < i_2 < \cdots < i_{n-1}\leq 2n$ and $1 \leq j_1 < j_2 < \cdots < j_{n+1}\leq 2n$ and where $\hat {j}_l$ denotes the fact that the $j_l$ term is omitted. For example, in the case $n = 2$ , which will be the main focus of the article, we will have coordinates $W_{1, 2}$ , $W_{1, 3}$ , $W_{1, 4}$ , $W_{2, 3}$ , $W_{2, 4}$ and $W_{3, 4}$ , with the relation

$$\begin{align*}W_{1, 2}W_{3, 4} - W_{1, 3}W_{2, 4}+W_{1, 4}W_{2, 3}= 0.\end{align*}$$

The condition that the space W is Lagrangian gives one additional polynomial equation, so that ${\mathrm {Lag}}({{\mathbb C}}^4)$ is a projective variety of complex dimension $3$ .

4.2 $\mathrm {SL}(2,{{\mathbb C}})$ –orbits of $\mathrm {Lag}({{\mathbb C}}^{4})$

From now on, we focus on dimension $4$ . So, let $V_{\mathbb {K}} = \mathbb {K}^{(3)}[X, Y] \cong \mathbb {K}^4$ be the symplectic space of homogeneous polynomials of degree $3$ in X and Y, equipped with the symplectic form $\omega _{\mathbb {K}} = \omega _{2, \mathbb {K}}$ defined above. We define the Veronese embeddings

$$ \begin{align*}\xi^1_{{{\mathbb C}}} \colon\thinspace {{\mathbb C}}\mathbb{P}^1 \to {{\mathbb C}}\mathbb{P}^3 \quad\text{and}\quad\xi^2_{{{\mathbb C}}} \colon\thinspace {{\mathbb C}}\mathbb{P}^1 \to \mathrm{Lag}({{\mathbb C}}^4)\end{align*} $$

by

$$ \begin{align*} \begin{aligned} \xi^1_{{\mathbb C}}([a:b]) &:= \langle (bX-aY)^3 \rangle\in {{\mathbb C}}\mathbb{P}^3\\ \xi^2_{{\mathbb C}}([a:b]) &:= \langle (bX-aY)^3, (dX-cY)(bX-aY)^2\rangle \in \mathrm{Lag}({{\mathbb C}}^4), \end{aligned} \end{align*} $$

where $[c:d]$ is any point chosen in ${{\mathbb C}}\mathbb {P}^1 \setminus \{[a:b]\}$ . The Lagrangian $\xi ^2_{{\mathbb C}}([a:b])$ does not depend on the choice of $[c:d]$ . Let

$$ \begin{align*}\xi^1 = \xi^1_{{\mathbb R}} = \xi^1_{{{\mathbb C}}} \mid_{{\mathbb R}\mathbb{P}^1} \quad\text{and}\quad\xi^2 = \xi^2_{{\mathbb R}} = \xi^2_{{{\mathbb C}}} \mid_{{\mathbb R}\mathbb{P}^1}. \end{align*} $$

Similar to Section 2.5, for a line $\ell \in \mathbb {CP}^{3}$ , we define

$$\begin{align*}K_\ell = \{W \in \mathrm{Lag}({{\mathbb C}}^{4}) \mid \ell \subset W\} \subset {\mathrm{Lag}}({{\mathbb C}}^{4}).\end{align*}$$

We now introduce the set

$$\begin{align*}K_{{\mathbb C}} = \bigcup_{t \in {{\mathbb C}}\mathbb{P}^1} K_{\xi^1_{{\mathbb C}}(t)} \subset {\mathrm{Lag}}({{\mathbb C}}^{4}).\end{align*}$$

More explicitly, we can write

$$\begin{align*}K_{{{\mathbb C}}} = \{W \in \mathrm{Lag}({{\mathbb C}}^{4}) \ \mid\ \exists\;[a:b]\in {{\mathbb C}}\mathbb{P}^1 \ \text{ s.t. } (b X- a Y)^3 \in W\};\end{align*}$$

that is, $K_{{{\mathbb C}}}$ is the set of Lagrangian subspaces that contain a polynomial with a triple complex root.

Lemma 4.1. $K_{{{\mathbb C}}}$ is the set of Lagrangians $W \in \mathrm {Lag}({{\mathbb C}}^{4})$ with a common root; that is,

$$\begin{align*}K_{{{\mathbb C}}} = \{W \in \mathrm{Lag}({{\mathbb C}}^{4}) \ \mid\ \exists\; [a:b]\in {{\mathbb C}}\mathbb{P}^1,\; \forall p \in W,\; p(X,Y) = (b X- aY) q(X,Y)\}.\end{align*}$$

Proof. Let $W \in K_{{{\mathbb C}}}$ . We know that it contains an element with a triple root. By acting with $\mathrm {SL}(2,{{\mathbb C}})$ , we can assume that the triple root is zero – in other words, that $X^3 \in W$ . Let $p\in W$ . We can write $p = a X^3 + bX^2Y + c XY^2 + d Y^3$ . Since W is isotropic, we know that $\omega _{2,{{\mathbb C}}}(X^3,p) = 0$ , and hence, by (2), we see that $d=0$ . Hence, zero is a common root of every element of W.

Conversely, assume that all elements of W have a common root. By acting with $\mathrm {SL}(2,{{\mathbb C}})$ , we can assume that the common root is zero, and hence, all elements of W are of the form $p = a X^3 + bX^2Y + c XY^2$ . By (2),

$$\begin{align*}\omega_{2,{{\mathbb C}}}(a_1 X^3 + b_1 X^2Y + c_1 XY^2, a_2 X^3 + b_2 X^2Y + c_2 XY^2) = \frac{1}{3}(c_1 b_2 - b_1 c_2). \end{align*}$$

Hence, all polynomials in W have the same ratio $\frac {b}{c}$ (which can be infinite if $c=0$ ). Given a basis $p_1, p_2$ of W, we can multiply one of them by a scalar to make sure they have the same coefficients $b,c$ . Then $p_1 - p_2$ is a multiple of $X^3$ , and hence, W has an element with a triple root.

We can now prove the following:

Proof. We construct an explicit bijection

$$\begin{align*}g\colon\thinspace{{\mathbb C}}\mathbb{P}^1 \times {{\mathbb C}}\mathbb{P}^1 \stackrel{\cong}{\longrightarrow} K_{{{\mathbb C}}}\end{align*}$$

defined by

$$\begin{align*}g(([a:b], [c:d])) = \left\{ \begin{array}{ll} \xi^2_{{\mathbb C}}([a:b]) & \mbox{if } [a:b] = [c:d]\\ \langle (bX-aY)^3, (dX-cY)^2 (bX-aY) \rangle & \mbox{if } [a:b] \neq [c:d] \end{array} \right.\end{align*}$$

Easy calculations show that the map g is well-defined and bijective.

Remark 4.3 (The space $K_{{\mathbb R}}$ ).

The space

$$\begin{align*}K_{\mathbb R} = \bigcup_{t \in {\mathbb R}\mathbb{P}^1} K_{\xi^1_{\mathbb R}(t)} \subset {\mathrm{Lag}}({{\mathbb C}}^{4}) \end{align*}$$

is precisely the space $K_{\rho ,I}$ in Section 2.5; hence, $\Omega = {\mathrm {Lag}}({{\mathbb C}}^{4}) \setminus K_{\mathbb R}$ . Note that

$$\begin{align*}K_{{\mathbb R}} = \{W \in \mathrm{Lag}({{\mathbb C}}^{4})\ \mid\ \exists\; [a:b]\in{\mathbb R}\mathbb{P}^1 \text{ s.t. } ( bX- a Y)^3 \in W \}\end{align*}$$

corresponds to the set of Lagrangian subspaces with a triple real root $[a:b] \in {\mathbb R}\mathbb {P}^1$ . We thus see that

$$\begin{align*}K_{{\mathbb R}}:= \cup_{t \in {\mathbb R}\mathbb{P}^1} K_{\xi^1_{\mathbb R}(t)}\cong {\mathbb R}\mathbb{P}^1 \times {{\mathbb C}}\mathbb{P}^1 \cong {\mathbb R}\mathbb{P}^1 \times \mathrm{Lag}({{\mathbb C}}^2)\,,\end{align*}$$

or, more precisely, $K_{{\mathbb R}} = g({\mathbb R}\mathbb {P}^1 \times \mathrm {Lag}({{\mathbb C}}^2))$ .

Lemma 4.5. Every Lagrangian W contains a polynomial

$$\begin{align*}p(X,Y) = (b X - a Y)^2 (d X-c Y)\end{align*}$$

with a double root $[a:b] \in {{\mathbb C}}\mathbb {P}^1$ and a single root $[c:d]\neq [a:b]\in {{\mathbb C}}\mathbb {P}^1$ .

Proof. If W has a polynomial with a triple root, then $W\in K_{{\mathbb C}}$ , and we saw above that these Lagrangians also contain a polynomial with a double root and a single root. If W does not contain a polynomial with a triple root but has a polynomial with a double root, the third root must be distinct, and hence, we are done.

Assume now that W has a polynomial with three distinct roots. Acting with $\mathrm {SL}(2,{{\mathbb C}})$ , we can assume that the three roots are $[-1:1], [0:1], [1:0]$ , and hence that $XY(X+Y)=X^2Y + XY^2 \in W$ . Let $p\in W$ , $p = a X^3 + bX^2Y + c XY^2 + d Y^3$ . By (2), we have

$$\begin{align*}\omega_{2,{{\mathbb C}}}(XY(X+Y), p) = \frac{1}{3}(b-c)\,, \end{align*}$$

which implies that $b=c$ . Hence, $W = \left <XY(X+Y), a X^3 + d Y^3\right>$ , and W is determined by $[a:d]$ . The other elements of W are scalar multiples of the ones of the form

$$\begin{align*}q = aX^3 + \beta X^2 Y + \beta X Y^2 + d Y^3.\end{align*}$$

We want to find elements with a double root. We can find them using the discriminant. The discriminant $\Delta $ of a degree– $3$ polynomial $\alpha x^3+\beta x^2y +\gamma xy^2 +\delta y^3$ is defined by

$$\begin{align*}\Delta(\alpha x^3+\beta x^2y +\gamma xy^2 +\delta y^3) := \beta^2\gamma^2-4\alpha\gamma^3-4\delta\beta^3-27\alpha^2\delta^2+18\alpha\beta\gamma\delta\,,\end{align*}$$

and polynomials with a double root correspond to the zeros of the discriminant. In our case, we have

$$\begin{align*}\Delta(q) = \beta^4 - 4(a+d)\beta^3 + 18ad \beta^2 - 27 a^2d^2. \end{align*}$$

A non-constant polynomial has always at least one solution over the complex numbers; hence, for every value of a and d, we can find a $\beta $ such that q has a double root.

The next statement summarizes the results of this subsection, describing the three orbits of the $\mathrm {SL}(2, {{\mathbb C}})$ –action.

Proof. We have already discussed above the bijection $g\colon \thinspace {{\mathbb C}}\mathbb {P}^1 \times {{\mathbb C}}\mathbb {P}^1 \stackrel {\cong }{\longrightarrow } K_{{{\mathbb C}}}$ . From the discussion above, you can see that

• • $\xi ^2_{{\mathbb C}}({{\mathbb C}}\mathbb {P}^1) = \mathrm {SL}(2, {{\mathbb C}}) \cdot \langle X^3, X^2 Y \rangle $ corresponds to Lagrangians all of whose polynomials share a common double root;

• • $K_{{{\mathbb C}}} \setminus \xi ^2_{{\mathbb C}}({{\mathbb C}}\mathbb {P}^1) = \mathrm {SL}(2, {{\mathbb C}}) \cdot \langle X^3, X Y^2 \rangle $ corresponds to Lagrangians all of whose polynomials share a common single root.

Since $\xi ^2_{{\mathbb C}}$ is an embedding, $\xi ^2_{{\mathbb C}}({{\mathbb C}}\mathbb {P}^1) \cong {{\mathbb C}}\mathbb {P}^1$ is closed.

To complete the proof, we need to show that $\mathrm {Lag}({{\mathbb C}}^{4}) \setminus K_{{{\mathbb C}}}$ is one $\mathrm {SL}(2, {{\mathbb C}})$ orbit. Given $W \not \in K_{{{\mathbb C}}}$ , by Lemma 4.5, W contains a polynomial with a double root and a single root. Acting with a matrix in $\mathrm {SL}(2,{{\mathbb C}})$ , we can assume that the roots are $[0:1]$ and $[1:0]$ (i.e., that $X^2 Y \in W$ ). Let $p\in W$ , $p = a X^3 + bX^2Y + c XY^2 + d Y^3$ . By (2), we have

$$\begin{align*}\omega_{2,{{\mathbb C}}}(X^2Y, p) = -\frac{1}{3}c\,, \end{align*}$$

which implies that $c=0$ . Hence, $W = \left <X^2Y, a X^3 + d Y^3\right>$ . Now we act again with a matrix $A \in \mathrm {SL}(2,{{\mathbb C}})$ . We choose $\alpha = \sqrt [6]{\tfrac {d}{a}}$ , one of the 6 complex roots, and we consider the matrix

$$\begin{align*}A = \begin{pmatrix} \alpha & 0 \\ 0 & \alpha^{-1} \end{pmatrix} \end{align*}$$

When acting on ${{\mathbb C}}\mathbb {P}^1$ , A fixes $[0:1]$ and $[1:0]$ and hence sends $X^2 Y$ to one of its multiples. Moreover, A sends $a X^3 + d Y^3$ to a multiple of $X^3 + Y^3$ . Hence, $A \cdot W = \left <X^2Y, X^3 + Y^3\right>$ , and this proves that

$$\begin{align*}\mathrm{Lag}({{\mathbb C}}^{4}) \setminus K_{{{\mathbb C}}} = \mathrm{SL}(2, {{\mathbb C}}) \cdot \langle X^2Y, X^3 + Y^3\rangle .\end{align*}$$

4.3 The space of regular ideal tetrahedra

Recall that an ideal hyperbolic tetrahedron in $\mathbb {H}^3$ is called regular when all the dihedral angles are equal to each other (and equal to $\frac {\pi }{3}$ ). These tetrahedra can also be characterized by their volume or their cross-ratio, since a tetrahedron is regular if and only if it has maximal volume, if and only if the cross-ratio of its vertices is $\frac {1-\sqrt {3}i}{2}.$ Recall that given $4$ points $z_1, z_2, z_3, z_4$ in ${{\mathbb C}}\mathbb {P}^1$ , here seen as the boundary at infinity of $\mathbb {H}^3$ , we define their cross-ratio as

$$\begin{align*}[z_1, z_2, z_3, z_4] = \frac{(z_3 - z_1) (z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)}.\end{align*}$$

Equivalently, $[z_1, z_2, z_3, z_4] = A z_4$ , where $A \in \mathrm {PSL}(2,{{\mathbb C}})$ is defined by $A z_1 = \infty , Az_2 = 0$ , and $A z_3 = 1$ . When computing the cross-ratio of the vertices of a tetrahedron, we use the orientation of $\mathbb {H}^3$ , and we choose the order of the vertices in a way that is compatible with the orientation. More precisely, if $z_1$ is a vertex, the other three vertices lie on the boundary of a copy of the hyperbolic plane. When watched from $z_1$ , we require that the vertices $z_2, z_3, z_4$ rotate in the clockwise direction. Note that if you change the order of the points, in a way that is still compatible with the orientation of $\mathbb {H}^3$ as explained, then the cross-ratio z can become $1-\frac {1}{z}$ or $\frac {1}{1-z}$ . However, if $z_0 = \frac {1-\sqrt {3}i}{2}$ , then $z_0 = 1-\frac {1}{z_0} = \frac {1}{1-z_0}$ , so the characterization of regular tetrahedra in terms of cross-ratio does not depend on the chosen order of the vertices when calculating the cross-ratio, as it should be. We will discuss more properties of regular ideal hyperbolic tetrahedra in Section 5.1.

We will denote by $\mathfrak {T}_{{\mathbb H}^3}$ the space of regular ideal hyperbolic tetrahedra in $\mathbb {H}^3$ . In order to define a topology on $\mathfrak {T}_{{\mathbb H}^3}$ , we embed it in the symmetric product

$$\begin{align*}\mathrm{Sym}^4\left({{\mathbb C}}\mathbb{P}^1\right) = {\left({{\mathbb C}}\mathbb{P}^1\right)}^4/S_4 .\end{align*}$$

The group $\mathrm {SL}(2,{{\mathbb C}})$ acts transitively on $\mathfrak {T}_{{\mathbb H}^3}$ via hyperbolic isometries. Moreover, the action of $\mathrm {SL}(2,{{\mathbb C}})$ on ${{\mathbb C}}\mathbb {P}^1$ gives a way to extend this action to the whole symmetric product.

The homogeneous space $\mathfrak {T}_{{\mathbb H}^3}$ is open but not closed in the symmetric product. We will now describe its closure; this gives an $\mathrm {SL}(2,{{\mathbb C}})$ –invariant compactification of $\mathfrak {T}_{{\mathbb H}^3}$ .

Proof. Given a regular ideal tetrahedron T, denote by $b\in {\mathbb H}^3$ its barycenter. The 4 lines from b to the 4 vertices meet at b, each pair of lines with the same angle, that we will denote here by $\alpha $ . Note that $\tfrac {\pi }{2} < \alpha < \pi $ . Denote by $\beta = \frac {1}{2}(\pi -\alpha )$ , where $0 < \beta < \tfrac {\pi }{4}$ .

We construct the distance d as in Figure 1: Given a plane P passing through b, we construct the cone with angle $\beta $ from P, and we consider the plane $P'$ tangent to the cone at infinity. The distance d is defined as the distance between the two planes P and $P'$ . The quantities $\alpha , \beta $ and d can be made explicit, but we do not need their explicit values here.

Figure 1

The distance d between the two geodesics.

We claim that given a regular ideal tetrahedron with barycenter contained in the open half-space below P, then at most one of its vertices can be in the closed half-space above $P'$ , and the other three vertices must lie in the open half-space below $P'$ . This is because of the way we have chosen the distance d between P and $P'$ . Assume that a regular ideal tetrahedron has barycenter at O, in the open half-space below P. Assume by contradiction that at least two of its vertices are in the closed half-space above $P'$ . Then the line between the two vertices lies entirely in the closed half-space above $P'$ , but it must be at distance d from O, a contradiction.

Now consider a sequence of regular ideal tetrahedra that converges to a point of the symmetric product that does not represent a regular ideal tetrahedron. The barycenters of the tetrahedra of the sequence are unbounded in ${\mathbb H}^3$ ; otherwise, a subsequence would converge to a regular ideal tetrahedron. Up to subsequences, we can assume that the barycenters of the tetrahedra converge to a point $x \in \mathbb {CP}^1$ .

We consider a small ball in $\mathbb {CP}^1$ containing x. This small ball bounds a plane $P'$ . Let P denote a plane whose boundary lies in the ball and whose distance from $P'$ is at least d. For all the tetrahedra in our subsequence whose barycenter lies in the half-space bounded by P, we know from the considerations above that at least three of their vertices lie in the half-space bounded by $P'$ .

This shows that for our subsequence, at least three of the vertices of the tetrahedra converge to the same point x, while the fourth vertices are free to go anywhere. Again up to subsequences, we can assume that the fourth vertices are also converging to some point in $\mathbb {CP}^1$ .

This implies that the limit of our original sequence has an element with multiplicity at least $3$ .

We will call the points of the frontier degenerate tetrahedra. We will denote the compactified space by $\mathfrak {T}_{\overline {{\mathbb H}^3}}$ . This space carries a natural $\mathrm {SL}(2,{{\mathbb C}})$ –action and can be described as

$$\begin{align*}\mathfrak{T}_{\overline{{\mathbb H}^3}} = \mathfrak{T}_{{\mathbb H}^3} \cup {{\mathbb C}}\mathbb{P}^1\times {{\mathbb C}}\mathbb{P}^1.\end{align*}$$

We can now state the main result for this section, which identifies the $\mathrm {SL}(2,{{\mathbb C}})$ –action on $\mathrm {Lag}({{\mathbb C}}^{4})$ with the one on $\mathfrak {T}_{\overline {{\mathbb H}^3}}$ . This will be a key step in the proof of Theorem D.

In order to do this, we will use Theorem 4.6, but we still need to identify the unique open $\mathrm {SL}(2, {{\mathbb C}})$ –orbit in $\mathrm {Lag}({{\mathbb C}}^{4})$ with the space $\mathfrak {T}_{{\mathbb H}^3}$ , and verify the the map between the two spaces is continuous.

Theorem 4.8. The space $\mathrm {Lag}({{\mathbb C}}^{4})$ is $\mathrm {SL}(2,{{\mathbb C}})$ –equivariantly homeomorphic to the space

$$\begin{align*}\mathfrak{T}_{\overline{{\mathbb H}^3}} := \mathfrak{T}_{{\mathbb H}^3} \cup {{\mathbb C}}\mathbb{P}^1\times {{\mathbb C}}\mathbb{P}^1.\end{align*}$$

Proof. We will construct an equivariant homeomorphism between the two spaces. Since both spaces are compact Hausdorff, it will be enough to check that the equivariant map is a continuous bijection. As a first step, we will show that $\mathrm {Lag}({{\mathbb C}}^{4}) \setminus K_{{{\mathbb C}}} = \mathrm {SL}(2, {{\mathbb C}}) \cdot \langle X^2Y, X^3 + Y^3\rangle $ is in bijection with the space $\mathfrak {T}_{{\mathbb H}^3}$ of regular ideal hyperbolic tetrahedra in $\mathbb {H}^3$ .

Consider the Lagrangian subspace $W=\langle X^2Y, X^3 + Y^3\rangle $ . We will see that it contains exactly $4$ ‘special’ polynomials which have a double root. To find them, we use the discriminant as in the proof of Lemma 4.5. The elements of W are of the form

$$\begin{align*}q = \alpha X^3 + \beta X^2 Y + \alpha Y^3\,;\end{align*}$$

hence, the discriminant is

$$\begin{align*}\Delta(q) = -\alpha (4\beta^3+27\alpha^3).\end{align*}$$

We have that $\Delta (q)= 0$ if and only if

1. 1. $[\alpha :\beta ]=[0:1]$ ;

2. 2. $[\alpha :\beta ] = \left [-\frac {\sqrt [3]{4}}{3}:1\right ]$ ;

3. 3. $[\alpha :\beta ] = \left [\frac {1}{3\sqrt [3]{2}}- i \frac {1}{\sqrt [3]{2}\sqrt {3}}:1\right ]$ ;

4. 4. $[\alpha :\beta ] = \left [\frac {1}{3\sqrt [3]{2}}+ i \frac {1}{\sqrt [3]{2}\sqrt {3}}:1\right ]$ .

The associated polynomials have double and single roots, respectively, given by

1. 1. $0$ and $\infty $ ;

2. 2. $\sqrt [3]{2}$ and $-\frac {1}{\sqrt [3]{4}}$ ;

3. 3. $\frac {-1-i\sqrt {3}}{\sqrt [3]{4}}$ and $\frac {1+i\sqrt {3}}{2\sqrt [3]{4}}$ ;

4. 4. $\frac {-1+i\sqrt {3}}{\sqrt [3]{4}}$ and $\frac {1-i\sqrt {3}}{2\sqrt [3]{4}}$ .

The vertices corresponding to the $4$ double roots define an ideal hyperbolic tetrahedron $T = \left \{ 0,\sqrt [3]{2},\frac {-1-i\sqrt {3}}{\sqrt [3]{4}}, \frac {-1+i\sqrt {3}}{\sqrt [3]{4}}\right \}$ , and the vertices corresponding to the $4$ single roots define a ‘dual’ ideal hyperbolic tetrahedron $T_{dual} = \left \{\infty ,-\frac {1}{\sqrt [3]{4}}, \frac {1+i\sqrt {3}}{2\sqrt [3]{4}}, \frac {1-i\sqrt {3}}{2\sqrt [3]{4}} \right \}$ . The tetrahedron $T_{dual}$ is the image of T by the central symmetry centered at the barycenter of T. A simple calculation shows that the cross-ratio of the vertices of T and $T_{dual}$ are equal to $\frac {1-\sqrt {3}i}{2}$ :

$$\begin{align*}\left[ 0, \sqrt[3]{2}, \frac{-1+i\sqrt{3}}{\sqrt[3]{4}}, \frac{-1-i\sqrt{3}}{\sqrt[3]{4}} \right] = \left[\infty, -\frac{1}{\sqrt[3]{4}}, \frac{1+i\sqrt{3}}{2\sqrt[3]{4}}, \frac{1-i\sqrt{3}}{2\sqrt[3]{4}} \right] = \frac{1-\sqrt{3}i}{2}.\end{align*}$$

Hence, T and $T_{dual}$ are regular ideal hyperbolic tetrahedra, as we wanted to prove. Notice that in the computation of the cross-ratio for T, we changed the order of the vertices to make sure it is compatible with the orientation of $\mathbb {H}^3$ .

Now we want to see that the bijections described above are actually homeomorphisms. Remember also that the topology of $\mathrm {Lag}({{\mathbb C}}^{4})$ can be described by considering $\mathrm {Lag}({{\mathbb C}}^{4}) \subset \mathrm {Gr}(2, {{\mathbb C}}^{4})$ and using the Plücker coordinates for the topology of $ \mathrm {Gr}(2, {{\mathbb C}}^{4})$ .

To prove this result, we will extend the map g, defined above, to a map

$$\begin{align*}g\colon\thinspace \mathfrak{T}_{{\mathbb H}^3} \cup {{\mathbb C}}\mathbb{P}^1\times {{\mathbb C}}\mathbb{P}^1 \to \mathrm{Lag}({{\mathbb C}}^{4})\,,\end{align*}$$

and check that the map is a homeomorphism.

Given a tetrahedron $T = \{v_1, \ldots , v_4\} \in \mathfrak {T}_{{\mathbb H}^3}$ , we define the dual tetrahedron $T_{dual} = \{v_1^{dual}, \ldots , v_4^{dual}\}$ as the tetrahedron (also in $\mathfrak {T}_{{\mathbb H}^3}$ ) with vertices $v_i^{dual}$ such that $v_i$ , the barycenter b of T and $v_i^{dual}$ lie on the same geodesic for $i = 1, \ldots , 4$ . Let $v_i$ and $v_j$ be two distinct vertices of T. If we let $v_i = [a_1: b_1] \in {{\mathbb C}}\mathbb {P}^1$ , $v_i^{dual} = [c_1: d_1] \in {{\mathbb C}}\mathbb {P}^1$ , $v_j = [a_2: b_2] \in {{\mathbb C}}\mathbb {P}^1$ , and $v_j^{dual} = [c_2: d_2] \in {{\mathbb C}}\mathbb {P}^1$ , then the Lagrangian subspace associated to T is

$$\begin{align*}g(T) = W := \langle (b_1 X - a_1 Y)^2 (d_1 X - c_1 Y), (b_2 X - a_2 Y)^2 (d_2 X - c_2 Y) \rangle \in \mathrm{Lag}({{\mathbb C}}^{4}).\end{align*}$$

Its Plücker coordinates are

• • $W_{1,2} = -b_1^2d_1(b_2^2c_2+2a_2b_2d_2) + b_2^2d_2 (b_1^2c_1+2a_1b_1d_1)$ ;

• • $W_{1,3} = b_1^2d_1(a_2^2d_2+2a_2b_2c_2) - b_2^2d_2 (a_1^2d_1+2a_1b_1c_1)$ ;

• • $W_{1,4} = -b_1^2d_1a_2^2c_2 + b_2^2d_2a_1^2c_1$ ;

• • $W_{2,3} = -(b_1^2c_1+2a_1b_1d_1)(a_2^2d_2+2a_2b_2c_2) + (b_2^2c_2+2a_2b_2d_2)(a_1^2d_1+2a_1b_1c_1)$ ;

• • $W_{2,4} = a_2^2c_2 (b_1^2c_1+2a_1b_1d_1) - a_1^2c_1 (b_2^2c_2+2a_2b_2d_2)$ ;

• • $W_{3,4} = -a_2^2c_2 (a_1^2d_1+2a_1b_1c_1) + a_1^2c_1(a_2^2d_2+2a_2b_2c_2)$ .

Similarly, given a point $([a:b], [c:d]) \in {{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1 \setminus \Delta $ , where $\Delta = \{([a:b], [a:b]) \in {{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1\}$ is the diagonal, then the associated Lagrangian subspace is

$$\begin{align*}U = \langle (b X - a Y)^3, (b X - a Y)(d X - c Y)^2 \rangle \in \mathrm{Lag}({{\mathbb C}}^{4})\,,\end{align*}$$

which has Plücker coordinates

• • $U_{1,2} = 2b^3 d (bc-ad)$ ;

• • $U_{1,3} = b^2(3ad+bc)(bc-ad)$ ;

• • $U_{1,4} = -ba(bc+ad)(bc-ad)$ ;

• • $U_{2,3} = -3ba(bc+ad)(bc-ad)$ ;

• • $U_{2,4} = a^2 (ad+3bc)(bc-ad)$ ;

• • $U_{3,4} = -2a^3c (bc-ad)$ .

Lastly, given a point $([a:b], [a:b]) \in \Delta \subset {{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1$ , the associated Lagrangian subspace is

$$\begin{align*}Z = \langle (b X - a Y)^3, (b X - a Y)^2 (d X - c Y) \rangle \in \mathrm{Lag}({{\mathbb C}}^{4})\,,\end{align*}$$

for $[c:d]\neq [a:b]\in {{\mathbb C}}\mathbb {P}^1$ , and Z has Plücker coordinates

• • $Z_{1,2} = - 2b^4 (bc-ad)$ ;

• • $Z_{1,3} = 2ab^3(bc-ad)$ ;

• • $Z_{1,4} = -a^2b^2(bc-ad)$ ;

• • $Z_{2,3} = -3a^2b^2(bc-ad)$ ;

• • $Z_{2,4} = 2a^3b (bc-ad)$ ;

• • $Z_{3,4} = -a^4 (bc-ad)$ .

We saw in Proposition 4.7 that when a sequence of tetrahedra in $\mathfrak {T}_{{\mathbb H}^3}$ degenerates, their barycenters converge to a point in ${{\mathbb C}}\mathbb {P}^1$ . In that case, at least three vertices will converge to the same point in ${{\mathbb C}}\mathbb {P}^1$ , so tetrahedra can only degenerate to points in ${{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1$ , that is ${{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1 = \partial \mathfrak {T}_{{\mathbb H}^3}$ .

We can now see that $g|_{{{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1}$ and $g|_{\mathfrak {T}_{{\mathbb H}^3}}$ are continuous. For the first case, we only have to consider the expression of the Plücker coordinates and look at the case $([a_n:b_n], [c_n:d_n]) \in {{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1 \setminus \Delta $ such that $([a_n:b_n], [c_n:d_n]) \to ([a:b], [a:b]) \in \Delta $ . In particular, we can do the calculations in the case that $[a:b]= [0:1]\in {{\mathbb C}}\mathbb {P}^1$ . If we denote $U_{i, j}^n$ the Plücker coordinates associated to $F\left (([a_n:b_n], [c_n:d_n])\right )$ , we can see that the only nonzero coordinate in the limit is $U_{1,2}^n$ , as we wanted. For the second case $g|_{\mathfrak {T}_{{\mathbb H}^3}}$ , again, we only have to consider the expression of the Plücker coordinates in term of the vertices of the tetrahedra. Hence, we are left with the discussion of converging sequences $\{T_n\}$ of tetrahedra in $\mathfrak {T}_{{\mathbb H}^3}$ such that $T_n \to T_\infty \in {{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1$ . We have two possibilities:

• • $T_\infty = ([a:b], [c:d])\in {{\mathbb C}}\mathbb {P}^1\times {{\mathbb C}}\mathbb {P}^1 \setminus \Delta $ .

• • $T_\infty = ([a:b], [a:b])\in \Delta $ .

In the first case, three vertices of the tetrahedron $\{T_n\}$ and all the dual vertices of the tetrahedron $\{T_n^{dual}\}$ converge to $[a:b] \in {{\mathbb C}}\mathbb {P}^1$ , while in the second case, all the four vertices of the tetrahedron $\{T_n\}$ and at least three dual vertices of the tetrahedron $\{T_n^{dual}\}$ converge to $[a:b] \in {{\mathbb C}}\mathbb {P}^1$ . In particular, in the first case we, choose vertices $v_i^n = [a_1^n: b_1^n] \in {{\mathbb C}}\mathbb {P}^1$ and $v_j^n = [a_2^n: b_2^n] \in {{\mathbb C}}\mathbb {P}^1$ of $T_n$ with dual vertices $(v_i^n)^{dual} = [c_1^n: d_1^n] \in {{\mathbb C}}\mathbb {P}^1$ and $(v_j^n)^{dual} = [c_2^n: d_2^n] \in {{\mathbb C}}\mathbb {P}^1$ , such that

• • $[a_1^n:b_1^n]\to [a:b]$ ;

• • $[c_1^n:d_1^n]\to [a:b]$ ;

• • $[a_2^n:b_2^n]\to [c:d]$ ;

• • $[c_2^n:d_2^n]\to [a:b]$ .

We can also assume $[a:b]= [0:1]$ and $[c:d]= [1:0]$ . If we denote $W_{i, j}^n$ the Plücker coordinates associated to $g\left (T_n\right )$ , we can see that the only nonzero coordinate in the limit is $W_{1,3}^n$ , as we wanted.

In the second case, we choose vertices $v_i^n = [a_1^n: b_1^n] \in {{\mathbb C}}\mathbb {P}^1$ and $v_j^n = [a_2^n: b_2^n] \in {{\mathbb C}}\mathbb {P}^1$ of $T_n$ with dual vertices $(v_i^n)^{dual} = [c_1^n: d_1^n] \in {{\mathbb C}}\mathbb {P}^1$ and $(v_j^n)^{dual} = [c_2^n: d_2^n] \in {{\mathbb C}}\mathbb {P}^1$ , such that

• • $[a_1^n:b_1^n]\to [a:b]$ ;

• • $[a_2^n:b_2^n]\to [a:b]$ .

Again, we can assume $[a:b]= [0:1]$ . Let’s denote $W_{i, j}^n$ the Plücker coordinates associated to $F\left (T_n\right )$ . We have two cases:

• • At least one of the sequences $[c_1^n:d_1^n]$ or $[c_2^n:d_2^n]$ do not converge to $[a:b]$ . Then we can see that the only nonzero coordinate in the limit is $W_{1,2}^n$ , as we wanted.

• • If both $[c_1^n:d_1^n], [c_2^n:d_2^n]$ converge to $[a:b]$ , then we need to be more careful and analyze the rate of convergence, but after diving all coordinates by $c_1^n$ or $c_2^n$ , we can see that the only nonzero coordinate in the limit is $W_{1,2}^n$ , as we wanted.