Lorentz--Finsler metrics on symplectic and contact transformation groups

It has been noticed a while ago that several fundamental transformation groups of symplectic and contact geometry carry natural causal structures, i.e., fields of tangent convex cones. Our starting point is that quite often the latter come to-gether with Lorentz–Finsler metrics, a notion originated in relativity theory, whichenable one to do geometric measurements with timelike curves. This includes finite-dimensional linear symplectic groups, where these metrics can be seen as Finslergeneralizations of the classical anti-de Sitter spacetime, infinite-dimensional groupsof contact transformations, with the simplest example being the group of circle dif-feomorphisms, and symplectomorphism groups of convex domains. In the first twocases, the Lorentz–Finsler metrics we introduce are bi-invariant. A Lorentz–Finslerperspective on these transformation groups turns out to be unexpectedly rich: somebasic questions about distance, geodesics and their conjugate points, and existenceof time functions, are naturally related to the contact systolic problem, group quasi-morphisms, the Monge–Amp`ere equation, and a subtle interplay between symplecticrigidity and flexibility. We discuss these interrelations, providing necessary prelimi-naries, albeit mostly focusing on new results which have not been published before.Along the way, we formulate a number of open questions.