Comments on “semi-graphs of Anabelioids”

(4.) Note that in Theorem 5.4, the case where A is trivial [i.e., is equal to the anabelioid associated to the trivial group {1}] is not excluded. Thus, suppose that, in Theorem 5.4, we assume further that A is trivial. Then let us observe that this implies that the underlying graph of G [orH] consists of a single vertex and no edges. [Indeed, if the underlying graph of G has at least one edge, then since G is assumed to be totally elevated, it follows from the assumption that G is totally arithmetically estranged [cf. Definition 5.3, (ii)] that Π G admits a closed subgroup that fails to be arithmetically ample, hence that ΠA = {1} contains a closed subgroup which is not open — a contradiction.] Thus, Π G itself is a verticial subgroup of Π temp G , hence compact. In particular, Π G is the unique maximal compact subgroup of Π G , so assertions (i), (ii), and (iii) of Theorem 5.4 are, in essence, vacuous.