  We present a purely geometrical method for probing the expansion history of
the Universe from the observation of the shape of stacked voids in
spectroscopic redshift surveys. Our method is an Alcock-Paczynski (AP) test
based on the average sphericity of voids posited on the local isotropy of the
Universe. It works by comparing the temporal extent of cosmic voids along the
line of sight with their angular, spatial extent. We describe the algorithm
that we use to detect and stack voids in redshift shells on the light cone and
test it on mock light cones produced from N-body simulations. We establish a
robust statistical model for estimating the average stretching of voids in
redshift space and quantify the contamination by peculiar velocities. Finally,
assuming that the void statistics that we derive from N-body simulations is
preserved when considering galaxy surveys, we assess the capability of this
approach to constrain dark energy parameters. We report this assessment in
terms of the figure of merit (FoM) of the dark energy task force and in
particular of the proposed EUCLID mission which is particularly suited for this
technique since it is a spectroscopic survey. The FoM due to stacked voids from
the EUCLID wide survey may double that of all other dark energy probes derived
from EUCLID data alone (combined with Planck priors). In particular, voids seem
to outperform Baryon Acoustic Oscillations by an order of magnitude. This
result is consistent with simple estimates based on mode-counting. The AP test
based on stacked voids may be a significant addition to the portfolio of major
dark energy probes and its potentialities must be studied in detail.
