  We present an in-depth investigation of the ${\rm SL}(2,\mathbb{R})$ momentum
space describing point particles coupled to Einstein gravity in three
space-time dimensions. We introduce different sets of coordinates on the group
manifold and discuss their properties under Lorentz transformations. In
particular we show how a certain set of coordinates exhibits an upper bound on
the energy under deformed Lorentz boosts which saturate at the Planck energy.
We discuss how this deformed symmetry framework is generally described by a
quantum deformation of the Poincar\'e group: the quantum double of ${\rm
SL}(2,\mathbb{R})$. We then illustrate how the space of functions on the group
manifold momentum space has a dual representation on a non-commutative space of
coordinates via a (quantum) group Fourier transform. In this context we explore
the connection between Weyl maps and different notions of (quantum) group
Fourier transform appeared in the literature in the past years and establish
relations between them.
