| Title: | Regular and semi-regular representations of groups by posets |
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| Authors: | ID Barmak, Jonathan A. (Author) |
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| Files: |  AMC_Barmak_2025.pdf (431,19 KB)
MD5: E6C7CF9AF8576811C4658AB76800E6A0
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | ZUP - University of Primorska Press
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| 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 Aut(P) ≃ G such that the action of Aut(P) on the underlying set is free and with three orbits. Among finite groups, only the trivial group and ℤ_2 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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| Keywords: | automorphism group of posets, Cayley graph, Dehn presentation, simple groups, random groups |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 13.03.2025 |
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| Publisher: | Založba Univerze na Primorskem |
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| Year of publishing: | 2025 |
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| Number of pages: | 18 str. |
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| Numbering: | Vol. 25, no. 2, [article no.] P2.06 |
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| PID: | 20.500.12556/RUP-21991  |
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| UDC: | 51 |
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| eISSN: | 1855-3974 |
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| DOI: | https://doi.org/10.26493/1855-3974.3312.bb7 |
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| Publication date in RUP: | 21.10.2025 |
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| Views: | 313 |
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| Downloads: | 0 |
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