Margulis spacetimes via the arc complex

Abstract

We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in an essentially unique way. The infinitesimal version of this result gives a parameterization, by the arc complex, of the moduli space of Margulis spacetimes with fixed convex cocompact linear holonomy. As an application, we provide a new proof of the tameness of such Margulis spacetimes M by establishing the Crooked Plane Conjecture, which states that M admits a fundamental domain bounded by piecewise linear surfaces called crooked planes. The noninfinitesimal version gives an analogous theory for noncompact complete anti-de Sitter 3-manifolds.

Appendix A: Realizing the zero cocycle

This appendix is a complement to the discussion of Sect. 5.2, whose notation and setup we continue with (in particular, we refer to Fig. 6). It is not needed for the proofs of Theorems 1.7 and 1.10.

In Sect. 5.2 we gave a realization, through the map ${\mathscr {L}}$ of (4.4), of the zero cocycle as a linear combination of the infinitesimal strip deformation maps $\psi _{\alpha },\psi _{\alpha '},\psi _{\beta _i}$ by choosing specific geodesic representatives, waists, and widths. In this appendix, we keep the same geodesic representatives (whose extensions to $\mathbb {P}(\mathbb {R}^{2,1})$ intersect in triples as in Fig. 6) but vary the waists; we determine all possible realizations of the zero cocycle supported on the arcs $\alpha , \alpha ', \beta _1, \beta _2, \beta _3, \beta _4$ and discuss their geometric significance in relation with Conjecture 1.8.

1.1 A.1 Generalized infinitesimal strip deformations

Let $\underline{\Delta }$ be a geodesic hyperideal triangulation of S, with set of edges ${\mathscr {E}}$. Recall the notation $\Psi (\pm {\widetilde{\mathscr {E}}},\mathfrak {g})$ from Sect. 4.2.

Definition A.1

A relative motion map $\psi \in \Psi (\pm {\widetilde{\mathscr {E}}},\mathfrak {g})$ is called a generalized infinitesimal strip deformation if $\psi ({\tilde{\alpha }})\in \mathrm {span}({\tilde{\alpha }})$ for any transversely oriented edge ${\tilde{\alpha }} \in \pm {\widetilde{\mathscr {E}}}$. The support of $\psi$ is the set of arcs $\underline{\alpha }\subset S$ such that $\psi$ is nonzero on the lifts to $\mathbb {H}^2$ of $\underline{\alpha }$. We also refer to the cohomology class in $T_{[\rho ]}\mathfrak {F}$ induced by $\psi$ as a generalized infinitesimal strip deformation.

Infinitesimal strip deformations in the sense of Definition 1.4 are generalized infinitesimal strip deformations for which $\psi ({\tilde{\alpha }})$ lies in one particular spacelike quadrant of the timelike plane $\mathrm {span}({\tilde{\alpha }}) \subset \mathbb {R}^{2,1}$. We will call these strip deformations positive (and their opposites negative) to distinguish them among generalized strip deformations. Generalized infinitesimal strip deformations can also be neither positive nor negative: if for instance $\psi ({\tilde{\alpha }})\in \mathfrak {g}$ is timelike, then the relative motion of the two tiles adjacent to ${\tilde{\alpha }}$ is an infinitesimal rotation centered at a point of ${\tilde{\alpha }}$.

Generalized infinitesimal strip deformations supported on a single arc $\underline{\alpha }$ form a linear 2-plane in $H^1_{\rho }(\Gamma ,\mathfrak {g})\simeq T_{[\rho ]}\mathfrak {F}$.

1.2 A.2 The point $w\in \mathbb {R}^{2,1}$ associated with a realization of the zero cocycle

We now work in the setting of Sect. 5.2: the hyperideal triangulations $\underline{\Delta }, \underline{\Delta }'$ differ by a single diagonal exchange and have a common refinement $\underline{\Delta }''$. We have four spacelike vectors $v_1,v_2,v_3,v_4\in \mathbb {R}^{2,1}$, scaled so that

$$\begin{aligned} v_1 + v_3 = v_2 + v_4, \end{aligned}$$

(A.1)

and our chosen geodesic representatives $\underline{\alpha },\underline{\alpha }',\underline{{\beta }}_i\in \underline{\Delta }''$ lift to ${\tilde{\alpha }},{\tilde{\alpha }}',{\tilde{\beta }}_i$ with

$$\begin{aligned} \left\{ \begin{array}{lcl} {\tilde{\alpha }} &{} = &{} \mathbb {H}^2 \cap (\mathbb {R}_+ v_1 + \mathbb {R}_+ v_3),\\ {\tilde{\alpha }}' &{} = &{} \mathbb {H}^2 \cap (\mathbb {R}_+ v_2 + \mathbb {R}_+ v_4),\\ {\tilde{\beta }}_i &{} = &{} \mathbb {H}^2 \cap (\mathbb {R}_+ v_i + \mathbb {R}_+ v_{i+1}) \end{array}\right. \end{aligned}$$

for all $1\le i\le 4$ (with the convention that $v_5=v_1$). For simplicity, we henceforth assume that $\underline{{\beta }}_1,\ldots ,\underline{{\beta }}_4$ are all distinct in the quotient surface S. Recall that ${\tilde{\alpha }}={e}_1 \cup {e}_3$ and ${\tilde{\alpha }}'={e}_2\cup {e}_4$. All edges $e_i$ and ${\tilde{\beta }}_i$ carry transverse orientations shown in Fig. 6.

For any $w\in \mathbb {R}^{2,1}$, we define a $(\rho , 0)$-equivariant map $\varphi _w : {\widetilde{\mathscr {T}}}''\rightarrow \mathfrak {g}$ supported on the $\rho (\Gamma )$-orbits of the “small” tiles $\delta _1, \delta _2, \delta _3, \delta _4$ by

$$\begin{aligned} \varphi _w (\delta _i) := w\wedge (v_i\wedge v_{i+1}) \end{aligned}$$

(A.2)

for all $1\le i \le 4$, where $\wedge$ is the Minkowski cross product of Sect. 4.1. As in Sect. 4.2, any $(\rho , 0)$-equivariant map $\varphi : {\widetilde{\mathscr {T}}}''\rightarrow \mathfrak {g}$ defines a relative motion map $\psi : {\widetilde{\mathscr {E}}}''\rightarrow \mathfrak {g}$ given by

$$\begin{aligned} \psi ({e})=\varphi (\delta ')-\varphi (\delta ) \end{aligned}$$

for any tiles $\delta ,\delta '$ adjacent to an edge $e\in \pm {\widetilde{\mathscr {E}}}''$ which is transversely oriented from $\delta$ to $\delta '$.

Lemma A.2

Let $\varphi : {\widetilde{\mathscr {T}}}''\rightarrow \mathfrak {g}$ be a $(\rho , 0)$-equivariant map with $\varphi =0$ outside the $\rho (\Gamma )$-orbits of $\delta _1,\delta _2,\delta _3,\delta _4$. The associated map $\psi : {\widetilde{\mathscr {E}}} \rightarrow \mathfrak {g}$ is a generalized infinitesimal strip deformation (Definition A.1) if and only if $\varphi =\varphi _w$ for some (unique) $w\in \mathbb {R}^{2,1}$.

In this case the support of the generalized infinitesimal strip deformation is contained in $\{\underline{\alpha }, \underline{\alpha }', \underline{{\beta }}_1, \underline{{\beta }}_2, \underline{{\beta }}_3, \underline{{\beta }}_4\}$. Note that Lemma A.2 holds regardless of how the other geodesic representatives $\underline{{\eta }}$ for $\eta \in (\Delta \cap \Delta ')\backslash \{ \beta _1,\beta _2,\beta _3,\beta _4\}$ are chosen.

Proof

Since $\varphi =0$ outside the $\rho (\Gamma )$-orbits of the $\delta _i$, we have $\psi =0$ outside the $\rho (\Gamma )$-orbits of the ${\tilde{\beta }}_i$ and $e_i$ for $1\le i\le 4$. By Definition A.1, the fact that $\psi$ is a generalized infinitesimal strip deformations is equivalent to

$$\begin{aligned} \left\{ \begin{array}{l} \psi ({\tilde{\beta }}_i) \in \mathrm {span}(v_i,v_{i+1}),\\ \psi ({e}_i) \in \mathrm {span}(v_i,v_{i+2}) ,\\ \psi ({e}_i) = - \psi ({e}_{i+2}) \end{array}\right. \end{aligned}$$

(A.3)

for all i. We first check that the space of $\rho$-equivariant maps $\varphi : {\widetilde{\mathscr {T}}}''\rightarrow \mathfrak {g}$ for which $\psi$ satisfies (A.3) has dimension ${\le } 3$. By construction, $\psi ({\tilde{\beta }}_i)=\varphi (\delta _i)$ and $\psi ({e}_i)=\varphi (\delta _i) - \varphi (\delta _{i-1})$. If $\psi$ satisfies (A.3), then we may write

$$\begin{aligned} \varphi (\delta _i) = a_i v_i - b_{i+1} v_{i+1} \in \mathrm {span}(v_i,v_{i+1}) \end{aligned}$$

for all i, where $a_1,b_1,\ldots ,a_4,b_4\in \mathbb {R}$. Since (A.1) is the only relation between $v_1,v_2,v_3,v_4$, the condition $\varphi (\delta _i)-\varphi (\delta _{i-1})\in \mathrm {span}(v_i,v_{i+2})$ is equivalent to $b_{i+1}=a_{i-1}$, and so we may eliminate the $b_i$ and write

$$\begin{aligned} \varphi (\delta _i) = a_i v_i - a_{i-1} v_{i+1} \end{aligned}$$

where $a_1,a_2,a_3,a_4\in \mathbb {R}$. The condition $\psi ({e}_i)=-\psi ({e}_{i+2})$ is equivalent to $\varphi (\delta _1) + \varphi (\delta _3) = \varphi (\delta _2) + \varphi (\delta _4)$, which amounts to

$$\begin{aligned} (a_1+a_3)\,v_1 - (a_2+a_4)\,v_2 + (a_3+a_1)\,v_3 - (a_4+a_2)\,v_4 = 0, \end{aligned}$$

i.e. $a_1+a_3=a_2+a_4$ by (A.1). The space of quadruples $(a_1,a_2,a_3,a_4)$ satisfying this condition has dimension 3, as announced.

The map $w\mapsto \varphi _w$ is linear and injective, for its kernel in $\mathbb {R}^{2,1}$ is contained in $\bigcap _{i=1}^4 \mathrm {span}(v_i\wedge v_{i+1}) = \{ 0\}$. Therefore, it only remains to see that for any $w\in \mathbb {R}^{2,1}$, the map $\psi$ associated with $\varphi _w$ satisfies (A.3). We have

$$\begin{aligned} \psi ({\tilde{\beta }}_i)=\varphi _w(\delta _i)-0=w\wedge (v_i \wedge v_{i+1}) \in \mathrm {span}(v_i,v_{i+1}). \end{aligned}$$

(A.4)

Using (A.1) and the skew-symmetry of $\wedge$, we also have

$$\begin{aligned} \psi ({e}_i) = \varphi _w(\delta _i)-\varphi _w(\delta _{i-1})=w\wedge (v_i\wedge (v_{i+1}+v_{i-1}))= w\wedge (v_i\wedge v_{i+2}),\nonumber \\ \end{aligned}$$

(A.5)

hence $\psi ({e}_i)\in \mathrm {span}(v_i,v_{i+2})$ and $\psi ({e}_i)=-\psi ({e}_{i+2})$. $\square$

1.3 A.3 The case of timelike w

We now consider the map $\varphi _w$ of (A.2) when $w\in \mathbb {R}^{2,1}$ is timelike. We see $\mathbb {H}^2$ as a hyperboloid in $\mathbb {R}^{2,1}$ as in (4.1).

Lemma A.3

If $w\in \mathbb {R}^{2,1}$ is timelike,  then the map $\psi : {\widetilde{\mathscr {E}}}\rightarrow \mathfrak {g}$ associated with $\varphi _w$ is a generalized infinitesimal strip deformation whose support is exactly $\{\underline{\alpha }, \underline{\alpha }', \underline{{\beta }}_1, \underline{{\beta }}_2, \underline{{\beta }}_3, \underline{{\beta }}_4\}$. The six vectors $\psi ({\tilde{\alpha }}), \psi ({\tilde{\alpha }}'), \psi ({\tilde{\beta }}_i)$ are all nonzero and spacelike,  i.e. correspond to either positive or negative infinitesimal strip deformations. The corresponding waists on ${\tilde{\alpha }}, {\tilde{\alpha }}', {\tilde{\beta }}_i$ are the respective orthogonal projections of $\mathbb {H}^2\cap \mathbb {R}w$ in $\mathbb {H}^2$.

Proof

From (A.4) and (A.5) we know that $\psi ({\tilde{\beta }}_i)$ and $\psi ({e}_i)$ are orthogonal to w, hence are zero or spacelike if w is timelike. To see that they are nonzero, we note that the planes $\mathrm {span}(v_i,v_{i+1})$ and $\mathrm {span}(v_i,v_{i+2})$ are timelike, hence the vectors $v_i \wedge v_{i+1}$ and $v_i \wedge v_{i+2}$ are spacelike; in particular, these vectors are not collinear to w, and we conclude using (A.4) and (A.5).

By Lemma 4.1, the translation axes of the (positive or negative) infinitesimal strip deformation along ${\tilde{\alpha }}$, ${\tilde{\alpha }}'$, ${\tilde{\beta }}_i$ are the intersections of $\mathbb {H}^2$ with the orthogonal planes in $\mathbb {R}^{2,1}$ to $\psi (e_1)$, $\psi (e_2)$, $\psi ({\tilde{\beta }}_i)$, respectively. By (A.4) and (A.5), these all go through $\mathbb {H}^2\cap \mathbb {R}w$. $\square$

1.4 A.4 The map $\varphi$ of Sect. 5.2

Recall that the map $\varphi$ we constructed in Sect. 5.2 is given by $\varphi (\delta _i)=v_{i+1}-v_i$ for all i. By (5.6), the associated map $\psi : {\widetilde{\mathscr {E}}} \rightarrow \mathfrak {g}$ satisfies the hypotheses of Lemma A.2, hence $\varphi =\varphi _{w_0}$ for some $w_0\in \mathbb {R}^{2,1}$. We now determine $w_0$. (This will not be needed afterwards.)

Lemma A.4

The vector $w_0\in \mathbb {R}^{2,1}$ is timelike and satisfies $\mathbb {H}^2\cap \mathbb {R}w_0=\{p\}$, where $p\in \mathbb {H}^2$ is the intersection point of the common perpendicular to ${\tilde{\beta }}_1$ and ${\tilde{\beta }}_3$ in $\mathbb {H}^2$ with the common perpendicular to ${\tilde{\beta }}_2$ and ${\tilde{\beta }}_4$.

Proof

Since $\varphi (\delta _i)=v_{i+1}-v_i$, we have $\varphi (\delta _i)+\varphi (\delta _{i+2})=0$ by (A.1). By (A.4) we have $\varphi (\delta _i)=w_0 \wedge (v_i\wedge v_{i+1})$. Therefore, $w_0\wedge (v_i\wedge v_{i+1}+v_{i+2}\wedge v_{i+3})=0$, and so $w_0$ is a multiple of $v_i\wedge v_{i+1}+v_{i+2}\wedge v_{i+3}$. In particular, $w_0$ belongs to the plane $\mathbb {R}(v_i\wedge v_{i+1})+\mathbb {R}(v_{i+2}\wedge v_{i+3})$, whose intersection with $\mathbb {H}^2$ is the common perpendicular to ${\tilde{\beta }}_i$ and ${\tilde{\beta }}_{i+2}$. This holds for all $1\le i\le 4$. $\square$

In general, $\mathbb {H}^2\cap \mathbb {R}w_0$ is not the point ${\tilde{\alpha }}\cap {\tilde{\alpha }}'$.

1.5 A.5 Link with Conjecture 1.8

The following lemma provides evidence for Conjecture 1.8.

Lemma A.5

Let the infinitesimal strip map ${\varvec{f}} : {\overline{X}}\rightarrow H^1_{\rho }(\Gamma ,\mathfrak {g})$ of Sect. 1.2 be defined with respect to our choice of geodesic representatives of the arcs of $\Delta \cup \Delta ',$ with waists on ${\tilde{\alpha }}, {\tilde{\alpha }}', \tilde{\beta _i}$ that are all orthogonal projections of a common point $p\in \mathbb {H}^2,$ and with infinitesimal widths $m_{\alpha }, m_{\alpha '}, m_{\beta _i}$ all equal to 1. Then the image of ${\varvec{f}}$ looks salient at the codimension-2 face shared by ${\varvec{f}}(\Delta )$ and ${\varvec{f}}(\Delta '),$ when seen from the origin of $H^1_\rho (\Gamma ,\mathfrak {g}).$

Proof

Recall the linear relation (5.3):

$$\begin{aligned} c_{\alpha }\, \varvec{f}(\alpha ) + c_{\alpha '} \varvec{f}(\alpha ') + \sum _{\begin{array}{c} \beta \text { arc of both}\\ \Delta \text { and }\Delta ' \end{array}} c_{\beta }\,{\varvec{f}}(\beta ) \, = \, 0 \, \in H^1_{\rho }(\Gamma ,\mathfrak {g}). \end{aligned}$$

By Claim 3.2(0) the coefficients $c_{\alpha }, c_{\alpha '}, c_{\beta }$ are unique up to scale, and by Claim 3.2(1) we may take $c_{\alpha }$ and $c_{\alpha '}$ to be positive. The fact that the image of ${\varvec{f}}$ looks salient at the codimension-2 face shared by ${\varvec{f}}(\Delta )$ and ${\varvec{f}}(\Delta ')$ is then expressed by

$$\begin{aligned} c_{\alpha } + c_{\alpha '} + \sum _{\begin{array}{c} \beta \text { arc of both}\\ \Delta \text { and }\Delta ' \end{array}} c_{\beta } <0. \end{aligned}$$

(A.6)

Let us prove that this inequality holds.

By definition, the coefficients $c_{\alpha }, c_{\alpha '}, c_{\beta }$ encode a realization of the zero cocycle by (positive and negative) infinitesimal strip deformations with the given geodesic representatives and waists. By Lemma A.3 and uniqueness of $c_{\alpha }, c_{\alpha '}, c_{\beta }$, this realization is of the form $\varphi _w$ for some timelike $w\in \mathbb {R}^{2,1}$ with $\mathbb {H}^2\cap \mathbb {R}w=\{ p\}$. (In particular, $c_{\beta }=0$ for $\beta \notin \{\alpha , \alpha ', \beta _1, \beta _2, \beta _3, \beta _4\}$.) Moreover, if p varies continuously in $\mathbb {H}^2$ while w remains in the same component of the timelike cone of $\mathbb {R}^{2,1}$, then the signs (positive or negative) of the infinitesimal strip deformations defining $\varphi _w$ remain constant, similar to (5.4). Therefore, since the infinitesimal widths $m_{\alpha }, m_{\alpha '}, m_{\beta _i}$ are all equal to 1, the relation (5.3) has the form

$$\begin{aligned} \Vert x_1 - x_2 \Vert \, {\varvec{f}}(\alpha ') + \Vert x_2 - x_3 \Vert \, {\varvec{f}}(\alpha ) - \sum _{i=1}^4 \Vert x_i\Vert \, {\varvec{f}}(\beta _i) = 0 , \end{aligned}$$

where we set $x_i:=\varphi (\delta _i)$, and $\Vert x \Vert :=\sqrt{\langle x, x \rangle } >0$ for spacelike $x\in \mathbb {R}^{2,1}$. Note that the points $v_i\wedge v_{i+1}$, for $1\le i\le 4$, form a parallelogram in $\mathbb {R}^{2,1}$: indeed,

$$\begin{aligned} v_1\wedge v_2 + v_3\wedge v_4 - v_2\wedge v_3 - v_4\wedge v_1 = (v_1+v_3)\wedge (v_2+v_4)=0 \end{aligned}$$

by (A.1). By Lemma A.2, it follows that the $x_i=w\wedge (v_i\wedge v_{i+1})$, for $1\le i\le 4$, form a parallelogram in the spacelike plane $w^{\perp } \subset \mathbb {R}^{2,1}$. As a consequence,

$$\begin{aligned} \Vert x_1 - x_2 \Vert + \Vert x_2 - x_3 \Vert - \sum _{i=1}^4 \Vert x_i\Vert <0, \end{aligned}$$

and so (A.6) holds. $\square$

Note, however, that if the waists are chosen arbitrarily, so that they are not all projections of a common point $p\in \mathbb {H}^2$, then typically none of the terms $c_{\beta }$ of (5.3) vanish, and the signs of the terms other than $c_{\alpha }$ and $c_{\alpha '}$ may vary: the conclusion of Lemma A.5 might then fail.