Ordered groups of formal series, and a conjugacy problem

Vincent Bagayoko

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arXiv · 2509.09186

Ordered groups of formal series, and a conjugacy problem

Vincent Bagayoko

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Abstract

Given an ordered field $\mathbb{T}$ of formal series over an ordered field $\mathbf{R}$ equipped with a composition law $\circ \colon \mathbb{T} \times \mathbb{T}^{>\mathbb{R}} \longrightarrow \mathbb{T}$, 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.

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Vincent Bagayoko. 2025-09-11. Ordered groups of formal series, and a conjugacy problem. https://arxiv.org/abs/2509.09186

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