We investigate expressiveness of a fragment of the ambient calculus, a formalism for describing distributed and mobile computations. More precisely, we study expressiveness of the pure and public ambient calculus from which the capability £ ¥ ¤ § ¦ © has been removed, in terms of the reachability problem of the reduction relation. Surprisingly, we show that even for this very restricted fragment, the reachability problem is not decidable. At a second step, for a slightly weaker reduction relation, we prove that reachability can be decided by reducing this problem to markings reachability for Petri nets. Finally, we show that the name-convergence problem as well as the model-checking problem turn out to be undecidable for both the original and the weaker reduction relation.