Crooked surfaces and anti-de Sitter geometry

Abstract

Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is the interior of the quotient of Einstein space by an involution fixing an Einstein plane. The purpose of this note is to show that the crooked planes defined in anti-de Sitter space recently by Danciger-Guéritaud-Kassel lift to restrictions of crooked surfaces in Einstein space which are adapted under the involution of Einstein space defining anti-de Sitter space.

Appendix: Einstein hyperspheres bounding \({{\mathsf {AdS}}}^3\)

Because $\mathsf {PSL}(2,{{\mathbb {R}}})$ is homeomorphic to an open solid torus $D^2\times S^1$, its boundary is a $2$-torus $S^1\times S^1$.

Since this theory seems not to be so well-known, we begin with a brief exposition of singular projective transformations. This is a simple case of the theory of wonderful compactifications developed by de Concini-Procesi [7].

A natural compactification of the group $\mathsf {PGL}(2,{{\mathbb {R}}})$ is the projective space ${{\mathbb {R}}}\mathsf {P}^3$ obtained by projectivizing the embedding

$$\begin{aligned} \mathsf {GL}(2,{{\mathbb {R}}})\hookrightarrow \mathsf {Mat}_2({{\mathbb {R}}}){\setminus }\{ {\mathbf {0}}\} \end{aligned}$$

of the group of invertible $2\times 2$ real matrices in the set of nonzero $2\times 2$ real matrices. The induced map

$$\begin{aligned} \mathsf {PGL}(2,{{\mathbb {R}}})\hookrightarrow {{\mathbb {R}}}\mathsf {P}^3\end{aligned}$$

embeds $\mathsf {PGL}(2,{{\mathbb {R}}})$ as an open dense subset of ${{\mathbb {R}}}\mathsf {P}^3$. The complement ${{\mathbb {R}}}\mathsf {P}^3{\setminus }\mathsf {PGL}(2,{{\mathbb {R}}})$ consists of singular projective transformations of ${{\mathbb {R}}}\mathsf {P}^1$. It naturally identifies with ${{\mathbb {R}}}\mathsf {P}^1\times {{\mathbb {R}}}\mathsf {P}^1$ and is homeomorphic to a torus.

A point in the the complement ${{\mathbb {R}}}\mathsf {P}^3{\setminus }\mathsf {PGL}(2,{{\mathbb {R}}})$ is the projective equivalence class of a $2\times 2$ real matrix $M$ which is nonzero but singular. In other words, $M$ is a $2\times 2$ matrix of rank $1$. The corresponding linear endomorphism of ${{\mathbb {R}}}^2$ has a $1$-dimensional kernel and a $1$-dimensional range, which may coincide. This pair of lines in ${{\mathbb {R}}}^2$ determines a point

$$\begin{aligned} \big (\mathsf {Kernel}(M), \mathsf {Image}(M)\big )\ \in \ {{\mathbb {R}}}\mathsf {P}^1\times {{\mathbb {R}}}\mathsf {P}^1. \end{aligned}$$

Two matrices giving the same pair are projectively equivalent. The corresponding singular projective transformation is only defined on ${{\mathbb {R}}}\mathsf {P}^1{\setminus } \{\mathsf {Kernel}(M)\}$ where it is the constant mapping

$$\begin{aligned} {{\mathbb {R}}}\mathsf {P}^1{\setminus } \{\mathsf {Kernel}(M)\} \longrightarrow \{\mathsf {Image}(M)\}. \end{aligned}$$

Here are two basic examples. The one-parameter semigroup of projective transformations defined by the diagonal matrix

converges, as $t\longrightarrow + \infty$, to the singular projective transformation defined by the diagonal matrix

which corresponds to the ordered pair

$$\begin{aligned} \big ([0:1],[1:0] \big ) \ \in \ {{\mathbb {R}}}\mathsf {P}^1\times {{\mathbb {R}}}\mathsf {P}^1. \end{aligned}$$

Likewise the one-parameter semigroup of projective transformations

converges, as $t\longrightarrow + \infty$, to the singular projective transformation defined by

which corresponds to the ordered pair

$$\begin{aligned} \big ([1:0],[1:0]) \big ) \ \in \ {{\mathbb {R}}}\mathsf {P}^1\times {{\mathbb {R}}}\mathsf {P}^1. \end{aligned}$$

For more details see [13] and the references cited there.

The group $\mathsf {PSL}(2,{{\mathbb {R}}})$ is one component of $\mathsf {PGL}(2,{{\mathbb {R}}})$; the other component corresponds to matrices with negative determinant. Denote the closure of $\mathsf {PSL}(2,{{\mathbb {R}}})\subset {{\mathbb {R}}}\mathsf {P}^3$ by $\overline{\mathsf {PSL}(2,{{\mathbb {R}}})}$. Since the inclusion

$$\begin{aligned} \mathsf {PSL}(2,{{\mathbb {R}}})\hookrightarrow \overline{\mathsf {PSL}(2,{{\mathbb {R}}})} \end{aligned}$$

is a homotopy-equivalence, the compactification of $\mathsf {PSL}(2,{{\mathbb {R}}})$ lifts to a compactification of the double covering

$$\begin{aligned} {{\mathsf {SL}}}(2,{{\mathbb {R}}})\longrightarrow \mathsf {PSL}(2,{{\mathbb {R}}}). \end{aligned}$$

Points on the boundary $\partial {{\mathsf {SL}}}(2,{{\mathbb {R}}})\approx S^1\times S^1$ can be geometrically interpreted as follows. Choose an orientation $O_{{{\mathbb {R}}}^2}$ on ${{\mathbb {R}}}^2$. The double covering $\widehat{{{\mathbb {R}}}\mathsf {P}^1}$ of ${{\mathbb {R}}}\mathsf {P}^1$ consists of oriented lines, that is, pairs $(\ell ,O_\ell )$, where $\ell \in {{\mathbb {R}}}\mathsf {P}^1$ is a line in ${{\mathbb {R}}}^2$ and $O_\ell$ is an orientation on $\ell$. Then the boundary $\partial {{\mathsf {SL}}}(2,{{\mathbb {R}}})$ consists of ordered pairs of oriented lines

$$\begin{aligned} \big ( (\ell _1,O_1),\ (\ell _2,O_2)\big ) \ \in \ \widehat{{{\mathbb {R}}}\mathsf {P}^1}\times \widehat{{{\mathbb {R}}}\mathsf {P}^1} \end{aligned}$$

such that:

• $O_1 = O_2$ if $\ell _1 = \ell _2$;

• $O_{{{\mathbb {R}}}^2} = O_1 \oplus O_2$ with respect to the decomposition ${{\mathbb {R}}}^2 = \ell _1 \oplus \ell _2$ if $\ell _1 \ne \ell _2$.

Alternatively, the double covering

$$\begin{aligned} \partial {{\mathsf {SL}}}(2,{{\mathbb {R}}})\longrightarrow \partial \mathsf {PSL}(2,{{\mathbb {R}}})\approx {{\mathbb {R}}}\mathsf {P}^1\times {{\mathbb {R}}}\mathsf {P}^1\end{aligned}$$

is defined by the homomorphism $\pi _1({{\mathbb {R}}}\mathsf {P}^1\times {{\mathbb {R}}}\mathsf {P}^1) \longrightarrow {{\mathbb {Z}}}/2$ which is nontrivial on each Cartesian factor.

These two projections correspond to the rulings of $\partial \widehat{{{\mathsf {AdS}}}}$, as described in Sect. 3.2.4.