We prove that a transversely holomorphic foliation, which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not a zero measure subset. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold is finite provided that the set of periodic orbits is not a zero measure subset.

Foliations transverse to fibrations are among the very first and simplest constructible examples of foliations, accompanied by a well-known transverse structure. These foliations are suspensions of groups of diffeomorphisms and their behavior is closely related to the action of the group in the fiber. For these reasons, many results holding for foliations in a more general context are first established for suspensions, that is, foliations transverse to a fibration. In this paper, we pursue this idea, but not restricted to it. We investigate versions of the classical stability theorems of Reeb [

Let

The codimension one case is studied in [

The following stability theorem is proved in [

Let

It is also observed in [

Since a foliation transverse to a fibration is conjugate to a suspension of a group of diffeomorphisms of the fiber, we can rely on the global holonomy of the foliation. As a general fact that holds also for smooth foliations, if the global holonomy group is finite then the foliation is a Seifert fibration. The proof of Theorem

We recall that a subset

In this paper, we improve Theorem

Let

Parallel to this result we have the following version for groups.

Let

As an immediate corollary of the above result, we get that, for a finitely generated subgroup

Let

For a given point

This holonomy group

As it is well known, the fundamental group

From the classical theory [

Let

The holonomy group

Given another intersection point

Suppose that the global holonomy

If

First we recall some facts from the theory of Linear groups. Let

With respect to complex linear groups one has the following.

Burnside, [

Schur, [

Using these results, we obtain in [

About periodic groups of germs of complex diffeomorphisms one has the following.

A finitely generated periodic subgroup

A (not necessarily finitely generated) subgroup

Let

Let

Given a subgroup

Let

If

Assume that there is a point

Assume that

In order to prove (1), we consider the case where

As for (2), since

The following simple remark gives the finiteness of finite exponent groups of holomorphic diffeomorphisms having a periodic orbit.

Let

Fix a point

The following lemma paves the way to Theorems

Let

We have

Fix a base point

The construction of the suspension of a group action gives Theorem

Since