Growth of some transversely homogeneous foliations

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For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of a recent answer given by Breuillard and Gelander to a question of Carrière. Examples of transversely projective foliations satisfying the above condition were constructed by Chihi and ben Ramdane.