부산대학교 도서관

Let $M$ be a manifold and let $G$ be a Lie group. If $\Gamma$ is adiscrete subgroup of $G$, then $G^\Gamma$ is the subgroup of the groupof diffeomorphisms $\text{Diff}(G)$ of $G$ generated by lefttranslation of $G$ and right translation of $\Gamma$. If ${\Cal F}$ and${\Cal E}$ are foliations on $M$, then ${\Cal F}$ is said to be anextension of ${\Cal E}$ if every leaf of ${\Cal E}$ is contained insome leaf of ${\Cal F}$. For example, if $H$ is a closed subgroup of aLie group $G$, then every Lie $G$-foliation on a manifold canonicallyadmits a ${G}/{H}$ transversely homogeneous extension. The problem iswhether a transversely homogeneous foliation necessarily comes from aLie foliation.\parIn this paper, the author studies this problem in the case where $H$ isa discrete subgroup $\Gamma$ of $G$. It is shown that, on $M$, there isan injection between transversally homogeneous${G}/{\Gamma}$-foliations and foliations given by the pseudogroup$\Cal{Loc}(G^\Gamma)$ obtained by localization of the elements of$G^\Gamma$, and that the two corresponding foliations have the same$C^\infty$-transverse structure. Moreover, the author shows that thefoliation given by $\Cal{Loc}(G^\Gamma)$ is a Lie $G$-foliation ifand only if its $\Gamma$-holonomy group is trivial.