Cross-ratio dynamics on ideal polygons

Abstract

Two ideal polygons, $(p_1,\ldots,p_n)$ and $(q_1,\ldots,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha$-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha$ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of M\"obius-equivalent polygons. We prove that this relation, which is, generically, a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures, and show that these relations, with different values of the constants $\alpha$, commute, in an appropriate sense. We investigate the case of small-gons, describe the exceptional ideal polygons, that possess infinitely many $\alpha$-related polygons, and study the ideal polygons that are $\alpha$-related to themselves (with a cyclic shift of the indices).