  In this manuscript we study geometric regularity estimates for problems
driven by fully nonlinear elliptic operators under strong absorption
conditions. We establish improved geometric regularity along the free boundary,
for a sharp value depending only on structural parameters. Non degeneracy among
others measure theoretical properties are also obtained. A sharp Liouville
result for entire solutions with controlled growth at infinity is proved. We
also present a number of applications consequential of our findings.
