Ordered groups of formal series, and a conjugacy problem

Abstract

Given an ordered field $\mathbb{T}$ of formal series over an ordered field $\mathbf{R}$ equipped with a composition law $\circ :\mathbb{T}\times {\mathbb{T}}^{>\mathbb{R}}⟶\mathbb{T}$, we give conditions for $({\mathbb{T}}^{>\mathbb{R}},\circ )$ to be a group. We show that classical fields of transseries and hyperseries satisfy these conditions. We then give further conditions on $\mathbb{T}$ under which $({\mathbb{T}}^{>\mathbb{R}},\circ ,<)$ is a linearly ordered group with exactly three conjugacy classes, and solve the open problem of existence of such a group.

27 pages