Involution-free realizations for non-generalized-dihedral infinite groups

Establish whether every infinite group that is not generalized dihedral admits an involution-free realization of the inverse-pair symmetry group \(\Xi(G)\), and hence an oriented regular representation.

Background

An oriented regular representation (ORR) is a directed Cayley graph $Cay^{\to}(G,T)$ with $T\cap T^{-1}=\varnothing$, generating connection set $T$, and automorphism group equal to the right-translation group $R(G)$. The paper observes that an involution-free realization of $\Xi(G)$ would yield an ORR by orienting each inverse pair in the connection set.

The paper establishes such realizations for uncountable abelian groups $A$ with $|2A|=|A|$ and for uncountable generalized dicyclic groups with $|2A|=|A|$, but does not settle the question for all infinite groups outside the generalized-dihedral case. It specifically notes that the regular-case construction does not directly resolve the issue because deleting involutions can create infinitely many error pairs.