Homogeneity in virtually free groups

Free groups are known to be homogeneous, meaning that finite tuples of elements which satisfy the same first-order properties are in the same orbit under the action of the automorphism group. We show that virtually free groups have a slightly weaker property, which we call uniform almost-homogeneity: the set of k-tuples which satisfy the same first-order properties as a given k-tuple u is the union of a finite number of Aut(G)-orbits, and this number is bounded independently from u and k. Moreover, we prove that there exists a virtually free group which is not ∃∀∃-homogeneous and conjecture that this group is not homogeneous. We also prove that all hyperbolic groups are homogeneous in a probabilistic sense.