Commensurators of groups and reversible automata (Q2740292)

A partial automorphism \(\alpha\) of a group \(G\) is called virtual if \(|G:\text{Dom}(\alpha)|<\infty\) and \(|G:\text{Im}(\alpha)|<\infty\). On the semigroup \(\text{VAut }G\) of all virtual automorphisms there is a congruence relation \(\simeq\) (commensurability): \(\alpha\simeq\beta\) if their actions on some subgroup of finite index in \(G\) coincide. The quotient \(\text{VAut }G/ \simeq\) is a group which is called the commensurator of \(G\). The authors study basic properties of group commensurators and introduce the notion of a geometric element of the commensurator of the free group \(F_n\). It is shown that a virtual automorphism of the free group \(F_n\) is geometric if and only if it is extendable to an automorphism of the directed Cayley graph of \(F_n\). It is also stated that the subgroup of all geometric elements can be naturally identified with the group of all bireversible automatic permutations. All statements are given without proofs.