Ordered groups of formal series, and a conjugacy problem

2025/09/11 by Bagayoko, Vincent
#FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.2509.09186

Abstract

Given an ordered field \mathbbT of formal series over an ordered field R equipped with a composition law ∘ \colon \mathbbT × \mathbbT^>ℝ \longrightarrow \mathbbT, we give conditions for (\mathbbT^>ℝ,∘) to be a group. We show that classical fields of transseries and hyperseries satisfy these conditions. We then give further conditions on \mathbbT under which (\mathbbT^>ℝ,∘,<) is a linearly ordered group with exactly three conjugacy classes, and solve the open problem of existence of such a group.

Citations

• Taylor expansions over generalised power series
• Groups with infinite linearly ordered products
• On ordered groups of regular growth rates
• Logarithmic Hyperseries
• Unsolved problems in ordered and orderable groups
• Liouville closed H -fields

Related

• Elements of Finite Order in the Group of Formal Power Series Under Composition
• Proof Theory and Ordered Groups
• Groups with infinite linearly ordered products
• On Archimedean Decompositions of Linearly Ordered Fields
• Extrinsic properties of automorphism groups of formal groups