Regular and semi-regular representations of groups by posets

2023/07/06 by Jonathan Ariel Barmak, Barmak, Jonathan Ariel
Mathematics · #05E18 #06A11 #20B25 #20B27 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2307.03106

openalex publication_date 2023/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By a result of Babai, with finitely many exceptions, every group G admits a semi-regular poset representation with three orbits, that is, a poset P with automorphism group \textrmAut(P) ≃ G such that the action of \textrmAut(P) on the underlying set is free and with three orbits. Among finite groups, only the trivial group and ℤ₂ have a regular poset representation (i.e. semi-regular with one orbit), however many infinite groups admit such a representation. In this paper we study non-necessarily finite groups which have a regular representation or a semi-regular representation with two orbits. We prove that if G admits a Cayley graph which is locally the Cayley graph of a free group, then it has a semi-regular representation of height 1 with two orbits. In this case we will see that any extension of the integers by G admits a regular representation. Applications are given to finite simple groups, hyperbolic groups, random groups and indicable groups.

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