  We control the thickness of GaSe on the level of individual layers and study
the corresponding optical absorption via highly sensitive differential
transmission measurements. Suppression of excitonic transitions is observed
when the number of layers is smaller than a critical value of 8. Through
ab-initio modelling we are able to link this behavior to a fundamental change
in the band structure that leads to the formation of a valence band shaped as
an inverted Mexican hat in thin GaSe. The thickness-controlled modulation of
the optical properties provides attractive resources for the development of
functional optoelectronic devices based on a single material.
