  Let $\mathcal F$ be a Lie foliation on a closed manifold $M$ with structural
Lie group $G$. Its transverse Lie structure can be considered as a transverse
action $\Phi$ of $G$ on $(M,\mathcal F)$; i.e., an ``action'' which is defined
up to leafwise homotopies. This $\Phi$ induces an action $\Phi^*$ of $G$ on the
reduced leafwise cohomology $\bar H(\mathcal F)$. By using leafwise Hodge
theory, the supertrace of $\Phi^*$ can be defined as a distribution
$L_{dis}(\mathcal F)$ on $G$ called the Lefschetz distribution of $\mathcal F$.
A distributional version of the Gauss-Bonett theorem is proved, which describes
$L_{dis}(\mathcal F)$ around the identity element. On any small enough open
subset of $G$, $L_{dis}(\mathcal F)$ is described by a distributional version
of the Lefschetz trace formula.
