  We show that an anomaly-free description of matter in (1+1) dimensions
requires a deformation of the 2d relativity principle, which introduces a
non-trivial center in the 2d Poincare algebra. Then we work out the reduced
phase-space of the anomaly-free 2d relativistic particle, in order to show that
it lives in a noncommutative 2d Minkowski space. Moreover, we build a Gaussian
wave packet to show that a Planck length is well-defined in two dimensions. In
order to provide a gravitational interpretation for this noncommutativity, we
propose to extend the usual 2d generalized dilaton gravity models by a specific
Maxwell component, which gauges the extra symmetry associated with the center
of the 2d Poincare algebra. In addition, we show that this extension is a high
energy correction to the unextended dilaton theories that can affect the
topology of space-time. Further, we couple a test particle to the general
extended dilaton models with the purpose of showing that they predict a
noncommutativity in curved space-time, which is locally described by a Moyal
star product in the low energy limit. We also conjecture a probable
generalization of this result, which provides a strong evidence that the
noncommutativity is described by a certain star product which is not of the
Moyal type at high energies. Finally, we prove that the extended dilaton
theories can be formulated as Poisson-Sigma models based on a nonlinear
deformation of the extended Poincare algebra.
