Virtually free groups are almost homogeneous

Abstract

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 $\mathbf{u}$ is the union of a finite number of $\mathrm{Aut}(G)$-orbits, and this number is bounded independently from $\mathbf{u}$ and $k$. Moreover, we prove that there exists a virtually free group which is not $\exists$-homogeneous. We also prove that all hyperbolic groups are homogeneous in a probabilistic sense.