Positive cones and bi-orderings on almost-direct products of free groups

[2209.04706] Positive cones and bi-orderings on almost-direct products of free groups Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Group Theory arXiv:2209.04706 (math) [Submitted on 10 Sep 2022 ( v1 ), last revised 9 Apr 2026 (this version, v2)] Title: Positive cones and bi-orderings on almost-direct products of free groups Authors: Oscar Ocampo , Juliana Roberta Theodoro de Lima View a PDF of the paper titled Positive cones and bi-orderings on almost-direct products of free groups, by Oscar Ocampo and Juliana Roberta Theodoro de Lima View PDF HTML (experimental) Abstract: Almost-direct products of free groups arise naturally in braid theory and in the study of automorphism groups of free groups. Although bi-invariant orderings are known to exist for many such groups, their explicit structure is often left implicit. In this paper, we give an explicit description of the positive cones defining bi-invariant orderings on almost-direct products of free groups, using normal forms derived from the almost-direct product decomposition together with Magnus-type orderings on free factors. We establish key structural properties of these cones, including compatibility with natural projections, convexity of canonical subgroups, and invariance under suitable classes of automorphisms. As applications, we show how the construction applies to several families of groups of geometric and algebraic interest, such as pure monomial braid groups and McCool groups. Comments: 14 pages. Title changed. Significant expansion and revision Subjects: Group Theory (math.GR) MSC classes: 20F60, 20F36 Cite as: arXiv:2209.04706 [math.GR] (or arXiv:2209.04706v2 [math.GR] for this version) https://doi.org/10.48550/arXiv.2209.04706 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Oscar Ocampo [ view email ] [v1] Sat, 10 Sep 2022 16:45:49 UTC (13 KB) [v2] Thu, 9 Apr 2026 11:44:27 UTC (16 KB) Full-text links: Access Paper: View a PDF of the paper titled Positive cones and bi-orderings on almost-direct products of free groups, by Oscar Ocampo and Juliana Roberta Theodoro de Lima View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2022-09 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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