Mutual embeddability in groups, trees, and spheres

Two subsets in a group are called twins if each is contained in a left translate of the other, though the two sets themselves are not translates of each other. We show that in the free group $F_{\{\alpha ,\beta \}}$, there exist maximal families of twins of any finite cardinality. This result is used to show that in the context of embeddings of trees, there exist maximal families of twin trees of any finite cardinality. These are counterexamples to the “tree alternative” conjecture, which supplement the first counterexamples published by Kalow, Laflamme, Tateno, and Woodrow. We also investigate twin sets in the sphere $S_2$, where the embeddings considered are isometries of $S_2$. We show that there exist maximal families of twin sets in $S_2$ of any finite cardinality.