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The graph of a foliation $\mathcal{F}$ of a manifold $M$ is the space of all triple $(x,y,[\gamma])$ where $x,y$ lies on the same leaf $F$ and $[\gamma]$ is the equivalent class of a path in $F$ joining $x$ to $y$. The equivalent class is defined as follows: Two paths are equivalents if they generate the same local holonomy.

The graph of foliations generates some special $C^*$ algebras associated to the foliation and this $C^*$ algebra contains useful information about the foliation.

In this post we try to consider two kinds of generalization of the graph of a foliation. Our generalization is only based on a generalization of the equivalent class of paths $\gamma$ joining $x,y$ on the same leaf. We do not change other axioms of the definition of the graph of a foliation. Then we ask whether these generalizations introduce some new objects associated to a foliation?

  1. Let we have a foliation of $\mathbb{R}^n$. We consider affine transversal sections for our foliation. Two path $\gamma_1, \gamma_2$ are equivalent if $\gamma_1 \circ \gamma_2^{-1}$ is an affine map.

2.Let we have a foliation of a manifold. Two path $\gamma_1, \gamma_2$ are equivalent if they have the same linear part.

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