\(p\)-adic actions on Peano continua (Q2862917)

The main aim of paper is to investigate \(p\)-adic actions on Peano continua. In the specific case of a \(p\)-adic group acting on a Peano continuum the author provides a means to construct equivariant partitions and he shows that a \(p\)-adic action on a Peano continuum can be approximated by finite permutations. Every effective \(p\)-adic action on a Peano continuum admits a refining sequence of partitions each with a corresponding finite action on its members (Theorem 3.4, Lemma 3.3, Corollary 3.5).NEWLINENEWLINEIt is proved that any map from a simply connected continuum can be lifted from the orbit space of such an action to the top space.NEWLINENEWLINEFrom the useful Lemmata 5.1 and 5.2 it follows that, between any two points, one can construct an arc that can be made to not only avoid the remainder of the orbits of its end points, but also to avoid its own orbit.NEWLINENEWLINEBesides, the paper concludes with a construction of a \(p\)-adic invariant Menger curve \(\mu\) from these arcs. In particular, the Menger curve can be constructed so that the inherited free \(A_p\) action on it is precisely the one described by \textit{A. N. Dranishnikov} in the paper [Math. USSR, Izv. 32, No. 1, 217--232 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, No. 1, 212--228 (1988; Zbl 0689.57025)].