Liftings of automorphisms of hypermaps (Q1974535)

A hypermap \( {\mathcal H} = (B,\sigma, \alpha) \) is a non-empty set of bits \( B \) together with permutations \( \sigma \) and \( \alpha \) acting on the right of \( B \) such that each of the cycles of the permutations \( \sigma, \alpha \) and \( \alpha^{-1} \sigma \) is of finite length. A voltage assignment \( z : B \rightarrow G \) from \( B \) into a group \( G \) is defined to be finitely ramified if each of the products \( z(b)z(b\gamma)z(b \gamma^2) \ldots z(b \gamma^{k-1}) \), where \( b \in B \), \( \gamma \in\{\sigma, \alpha\}\), and \( k \) is the size of the cycle of \( \gamma \) containing \( b \), is an element of finite order in \( G \). Given a finitely ramified voltage assignment \( z : B \rightarrow G \) into a group \( G \) acting on the right of a set \( X \), the author defines a derived hypermap of \( {\mathcal H} \) to be the hypermap \( {\mathcal H}_z(X) = (B \times X, \sigma_z, \alpha_z) \) with the two permutations \( \sigma_z \) and \( \alpha_z \) of the Cartesian product \( B \times X \) defined by setting \( (b,x) \gamma_z = (b \gamma, xz(b)) \), for all \( b \in B \), \( x \in X \), and \( \gamma \in\{\sigma, \alpha\}\). The derived hypermaps obtained from this construction differ from the maps constructed from voltage assignments of maps by other authors; see \textit{J. L. Gross} [Discrete Math. 9, 239-246 (1974; Zbl 0286.05106)], \textit{P. Gvozdjak} and \textit{J. Širáň} [Contemp. Math. 147, 441-454 (1993; Zbl 0791.05025)] and \textit{H. Hofmeister} [Discrete Math. 98, No.~3, 175-183 (1991; Zbl 0760.05071)]. The present author successfully argues for the usefulness of his new definition by developing a simple sufficient condition for lifting automorphisms of \( {\mathcal H} \) to automorphisms of \( {\mathcal H}_z(X) \) based on the existence of certain automorphisms of \( G \), and by reproving the Accola's theorem for hypermaps in a new and simpler way. The paper concludes by considering ramified coverings closer to those of the other authors stated in a cohomological setting.