Serre showed that, for a discrete valuation field, the group of linear fractional transformations acts on an infinite regular tree with vertex degree determined by the residue degree of the field. Since the p-adics and the polynomials over the finite field of order p act on isomorphic trees, we may ask whether pairs of actions from these two groups are ever conjugate as tree automorphisms. We analyze permutations induced on finite vertex sets, and show a permutation classification result for actions by these linear fractional transformation groups. We prove that actions by specific subgroups of these groups are conjugate only in specific special cases.
Talbott, Henry W.
"Disjointness of Linear Fractional Actions on Serre Trees,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 22:
1, Article 6.
Available at: https://scholar.rose-hulman.edu/rhumj/vol22/iss1/6