  The Gemini Multi-conjugate adaptive optics System (GeMS) is a facility
instrument for the Gemini-South telescope. It delivers uniform,
near-diffraction-limited image quality at near-infrared wavelengths over a 2
arcminute field of view. Together with the Gemini South Adaptive Optics Imager
(GSAOI), a near-infrared wide field camera, GeMS/GSAOI's combination of high
spatial resolution and a large field of view will make it a premier facility
for precision astrometry. Potential astrometric science cases cover a broad
range of topics including exo-planets, star formation, stellar evolution, star
clusters, nearby galaxies, black holes and neutron stars, and the Galactic
center. In this paper, we assess the astrometric performance and limitations of
GeMS/GSAOI. In particular, we analyze deep, mono-epoch images, multi-epoch data
and distortion calibration. We find that for single-epoch, un-dithered data, an
astrometric error below 0.2 mas can be achieved for exposure times exceeding
one minute, provided enough stars are available to remove high-order
distortions. We show however that such performance is not reproducible for
multi-epoch observations, and an additional systematic error of ~0.4 mas is
evidenced. This systematic multi-epoch error is the dominant error term in the
GeMS/GSAOI astrometric error budget, and it is thought to be due to
time-variable distortion induced by gravity flexure.
