부산대학교 도서관

Let $M$ be a manifold and let $G$ be a Lie group. If $\Gamma$ is a discrete subgroup of $G$, then $G^\Gamma$ is the subgroup of the group of diffeomorphisms $\text{Diff}(G)$ of $G$ generated by left translation of $G$ and right translation of $\Gamma$. If ${\Cal F}$ and ${\Cal E}$ are foliations on $M$, then ${\Cal F}$ is said to be an extension of ${\Cal E}$ if every leaf of ${\Cal E}$ is contained in some leaf of ${\Cal F}$. For example, if $H$ is a closed subgroup of a Lie group $G$, then every Lie $G$-foliation on a manifold canonically admits a ${G}/{H}$ transversely homogeneous extension. The problem is whether a transversely homogeneous foliation necessarily comes from a Lie foliation. \par In this paper, the author studies this problem in the case where $H$ is a discrete subgroup $\Gamma$ of $G$. It is shown that, on $M$, there is an 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 the foliation given by $\Cal{Loc}(G^\Gamma)$ is a Lie $G$-foliation if and only if its $\Gamma$-holonomy group is trivial.