Hyperbolic actions of Thompson's group $F$ and generalizations

[2406.12982] Hyperbolic actions of Thompson's group $F$ and generalizations Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Group Theory arXiv:2406.12982 (math) [Submitted on 18 Jun 2024 ( v1 ), last revised 14 Apr 2026 (this version, v2)] Title: Hyperbolic actions of Thompson's group $F$ and generalizations Authors: Sahana Balasubramanya , Francesco Fournier-Facio , Matthew C. B. Zaremsky View a PDF of the paper titled Hyperbolic actions of Thompson's group $F$ and generalizations, by Sahana Balasubramanya and 1 other authors View PDF HTML (experimental) Abstract: We study the poset of hyperbolic structures on Thompson's group $F$ and its generalizations $F_n$ for $n \geq 2$. The global structure of this poset is as simple as one would expect, with the maximal non-elementary elements being two quasi-parabolic actions corresponding to well-known ascending HNN-extension expressions of $F_n$. However, the local structure turns out to be incredibly rich, in stark contrast with the situation for the $T$ and $V$ counterparts. We show that the subposet of quasi-parabolic hyperbolic structures consists of two isomorphic posets, each of which contains uncountably many subposets of \emph{lamplike} structures, which can be described combinatorially in terms of certain hyperbolic structures on related lamplighter groups. Moreover, each of these subposets, as well as intersections and complements thereof, is very large, in that it contains a copy of the power set of the natural numbers. We also prove that these uncountably many uncountable subposets are not the entire picture, indeed there exists a copy of the power set of the natural numbers consisting entirely of non-lamplike structures. We also prove that this entire vast array of hyperbolic structures on $F_n$ collapses as soon as one takes a natural semidirect product with $\mathbb{Z}/2\mathbb{Z}$. These results are all proved via a detailed analysis of confining subsets, and along the way we establish a number of fundamental results in the theory of confining subsets of groups. Comments: 60 pages, 4 figures. v2: Final version, to appear in Algebraic Combinatorics Subjects: Group Theory (math.GR) Cite as: arXiv:2406.12982 [math.GR] (or arXiv:2406.12982v2 [math.GR] for this version) https://doi.org/10.48550/arXiv.2406.12982 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Francesco Fournier-Facio [ view email ] [v1] Tue, 18 Jun 2024 18:01:05 UTC (60 KB) [v2] Tue, 14 Apr 2026 11:02:40 UTC (60 KB) Full-text links: Access Paper: View a PDF of the paper titled Hyperbolic actions of Thompson's group $F$ and generalizations, by Sahana Balasubramanya and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2024-06 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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