Homogeneity in virtually free groups

Co‐Hopfian virtually free groups and elementary equivalence

“…The following proposition claims that $$\mathcal {D}(G)$$ is cocompact under the action of $$\mathrm{Aut}(G)$$. We refer the reader to [1, Proposition 2.9]. Proposition Let $$G$$ be a virtually free group.…”

“…Perin and Sklinos [15], and independently Ould Houcine [13], proved that free groups are homogeneous (and even $$\forall \exists$$‐homogeneous, see 2.1.3). In [1], we proved that virtually free groups satisfy a weaker property, which we called almost‐homogeneity. We also proved that virtually free groups are not $$\forall \exists$$‐homogeneous in general, and conjectured that they are not homogeneous in general.…”