Schottky presentations of positive representations

Abstract

We show that total positivity gives rise to a partial cyclic order on the set of oriented flags in ${\mathbb {R}}^n$. Using the notion of interval given by this partial cyclic order, we construct Schottky groups in ${{\,\mathrm{PSL}\,}}(n,{\mathbb {R}})$ and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct polyhedral fundamental domains for the cocompact domain of discontinuity that these groups admit in the projective space or the sphere, depending on the dimension.

Anti-de Sitter crooked planes

In this appendix we will show that our notion of halfspace in ${{\mathbb {R}}}{{\mathbb {P}}}^{3}$ coincides with that of anti-de Sitter crooked halfspace introduced in [8] and studied in [7, 11]. More precisely, an $\mathsf {AdS}^3$ crooked halfspace is the restriction to the projective model of $\mathsf {AdS}^3$ of an ${{\mathbb {R}}}{{\mathbb {P}}}^{3}$ halfspace as defined in Sect. 5.

Definition A.1

The 3-dimensional anti-de Sitter space $\mathsf {AdS}^3$ is the group ${{\,\mathrm{PSL}\,}}(2,{\mathbb {R}})$ endowed with the Lorentzian metric given by its Killing form.

The neutral component of the isometry group of $\mathsf {AdS}^3$ is ${{\,\mathrm{PSL}\,}}(2,{\mathbb {R}})\times {{\,\mathrm{PSL}\,}}(2,{\mathbb {R}})$ acting by left and right multiplication.

Definition A.2

Given a geodesic $\ell \subset {\mathbb {H}}^2$ in the hyperbolic plane, the associated $\mathsf {AdS}^3$ right crooked plane $C(\ell )$ is the set of isometries $g\in {{\,\mathrm{PSL}\,}}(2,{\mathbb {R}})$ which have a nonattracting fixed point on ${\overline{\ell }}\subset \overline{{\mathbb {H}}^2}$.

We will consider the following embedding of $\mathsf {AdS}^3$ in ${{\mathbb {R}}}{{\mathbb {P}}}^{3}$:

Its image is the projectivization of the set of negative vectors for the signature (2, 2) quadratic form $q(v) = -v_1^2 + v_3^2 - v_2 v_4$.

Proposition A.3

The standard halfspace ${\mathcal {H}}$ in ${{\mathbb {R}}}{{\mathbb {P}}}^{3}$ intersects the image of in one of the two crooked $\mathsf {AdS}^3$ halfspaces bounded by $C(\ell _0)$, where $\ell _0$ is the geodesic represented by the positive imaginary axis in the Poincaré upper half plane.

Proof

The boundary of the standard halfspace is the closure of the projectivization of vectors which have signs $(+,0,+,*),(*,+,0,+),(0,+,*,-)$, or $(+,*,-,0)$.

Let us analyse each of these cases. If $M=\begin{pmatrix} a &{} b\\ c &{} d\end{pmatrix}$ and has signs $(+,0,+,*)$, then $b=0$ and so M fixes 0. Moreover, $a>0$ and $-a<d<a$ which means that 0 is a repelling fixed point.

Similarly, if has signs $(+,*,-,0)$, then $c=0$ and M fixes $\infty$. Now, $d>0$ and $-d<a<d$ so $\infty$ is repelling.

If has signs $(*,+,0,+)$, then $a=d$, $b<0$ and $c>0$. This means that M is an elliptic element fixing the point $i\sqrt{\frac{-b}{c}}\in i {\mathbb {R}}$.

Finally, the set with signs $(0,+,*,-)$ does not intersect the image of since any vector with these signs is positive for the quadratic form q.

Conversely, any $M\in {{\,\mathrm{PSL}\,}}(2,{\mathbb {R}})$ in the crooked plane must fall into one of these categories or their closure (if M is parabolic or the identity).