finite group

Let G a sub-group of the group of homeomorphisms of a metric space E,
Homéo(E), such that each orbit of G is finite (the cardinality of
orbits is not uniformly limited i.e for all n \in IN there exists x\in
E such that card(G(x)) > n (G(x) is the orbit of x by G).

A point x in E is regular if there exists an open neighboorhod U_x and
an integer k_x such that for all y\in U_x we have card(G(y)) < k_x.

if E is connected and G is finitely generated and if each point x \in
E is regular is it true that G is finite?

In particular we have if E is compact (or each orbit meets a compact K
\subset E) then G is finite.

cordially.