Group actions on one-manifolds, II: Extensions of Hölder's Theorem

This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on $\mathbf R$ with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in $\mathrm{Diff}^2(\mathbf R)$ as those groups whose elements have at most one fixed point.