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

Balasubramanya, Sahana; Fournier-Facio, Francesco; Zaremsky, Matthew C. B.

doi:10.5802/alco.506

Balasubramanya, Sahana ¹; Fournier-Facio, Francesco ²; Zaremsky, Matthew C. B.³

¹Department of Mathematical Sciences, Lafayette College, Easton, PA (USA)
²Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, England (UK)
³Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY (USA)

Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 901-952

Abstract

We study the poset of hyperbolic structures on Thompson’s group $F$ and its generalizations $F_n$ for $n \ge 2$. 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 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.

Article information
Cite this article

Received: 2025-05-01
Revised: 2026-03-14
Accepted: 2026-04-14
Published online: 2026-09-02

DOI: 10.5802/alco.506

Classification: 20F65, 20F60, 20E08, 20E22
Keywords: Thompson group, group action, hyperbolic metric space, quasi-parabolic actions, poset of hyperbolic structures

Author's affiliations:

Balasubramanya, Sahana ¹ ; Fournier-Facio, Francesco ² ; Zaremsky, Matthew C. B. ³

¹ Department of Mathematical Sciences, Lafayette College, Easton, PA (USA)
² Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, England (UK)
³ Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Balasubramanya, Sahana; Fournier-Facio, Francesco; Zaremsky, Matthew C. B. Hyperbolic actions of Thompson’s group $F$ and generalizations. Algebraic Combinatorics, Volume 9 (2026) no. 4, pp. 901-952. doi: 10.5802/alco.506

@article{ALCO_2026__9_4_901_0,
     author = {Balasubramanya, Sahana and Fournier-Facio, Francesco and Zaremsky, Matthew C. B.},
     title = {Hyperbolic actions of {Thompson{\textquoteright}s} group $F$ and generalizations},
     journal = {Algebraic Combinatorics},
     pages = {901--952},
     year = {2026},
     publisher = {The Combinatorics Consortium},
     volume = {9},
     number = {4},
     doi = {10.5802/alco.506},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.506/}
}

TY  - JOUR
AU  - Balasubramanya, Sahana
AU  - Fournier-Facio, Francesco
AU  - Zaremsky, Matthew C. B.
TI  - Hyperbolic actions of Thompson’s group $F$ and generalizations
JO  - Algebraic Combinatorics
PY  - 2026
SP  - 901
EP  - 952
VL  - 9
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.506/
DO  - 10.5802/alco.506
LA  - en
ID  - ALCO_2026__9_4_901_0
ER  -

%0 Journal Article
%A Balasubramanya, Sahana
%A Fournier-Facio, Francesco
%A Zaremsky, Matthew C. B.
%T Hyperbolic actions of Thompson’s group $F$ and generalizations
%J Algebraic Combinatorics
%D 2026
%P 901-952
%V 9
%N 4
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.506/
%R 10.5802/alco.506
%G en
%F ALCO_2026__9_4_901_0

References
Cited by

[1] Abbott, C.; Balasubramanya, S.; Rasmussen, A. J. Higher rank confining subsets and hyperbolic actions of solvable groups, Adv. Math., Volume 424 (2023), Paper no. 109045, 47 pages | DOI | MR | Zbl

[2] Abbott, C. R.; Balasubramanya, S.; Rasmussen, A. J. Valuations, completions, and hyperbolic actions of metabelian groups, J. Lond. Math. Soc. (2), Volume 109 (2024) no. 6, Paper no. e12916, 72 pages | DOI | MR | Zbl

[3] Abbott, C. R.; Balasubramanya, S. H.; Osin, D. Hyperbolic structures on groups, Algebr. Geom. Topol., Volume 19 (2019) no. 4, pp. 1747-1835 | DOI | MR | Zbl

[4] Abbott, C. R.; Rasmussen, A. J. Actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces, Algebr. Geom. Topol., Volume 23 (2023) no. 4, pp. 1641-1692 | DOI | MR | Zbl

[5] Balasubramanya, S. H. Hyperbolic structures on wreath products, J. Group Theory, Volume 23 (2020) no. 2, pp. 357-383 | DOI | MR | Zbl

[6] Balasubramanya, S. H.; Fournier-Facio, F.; Genevois, A. Property (NL) for group actions on hyperbolic spaces, Groups Geom. Dyn., Volume 19 (2025) no. 4, pp. 1319-1371 (with an appendix by A. Sisto) | DOI | MR | Zbl

[7] Bavard, C. Longueur stable des commutateurs, Enseign. Math. (2), Volume 37 (1991) no. 1-2, pp. 109-150 | MR | Zbl

[8] Belk, J. M. Thompson’s group F, Ph. D. Thesis, Cornell University (2004)

[9] Bieri, R.; Neumann, W. D.; Strebel, R. A geometric invariant of discrete groups, Invent. Math., Volume 90 (1987) no. 3, pp. 451-477 | DOI | MR | Zbl

[10] Bieri, R.; Strebel, R. On groups of PL-homeomorphisms of the real line, Mathematical Surveys and Monographs, 215, American Mathematical Society, Providence, RI, 2016, xvii+174 pages | DOI | MR | Zbl

[11] Brady, T.; Burillo, J.; Cleary, S.; Stein, M. Pure braid subgroups of braided Thompson’s groups, Publ. Mat., Volume 52 (2008) no. 1, pp. 57-89 | DOI | MR | Zbl

[12] Brin, M. G. The algebra of strand splitting. I. A braided version of Thompson’s group ${V}$, J. Group Theory, Volume 10 (2007) no. 6, pp. 757-788 | DOI | MR | Zbl

[13] Brin, M. G.; Squier, C. C. Groups of piecewise linear homeomorphisms of the real line, Invent. Math., Volume 79 (1985) no. 3, pp. 485-498 | DOI | MR | Zbl

[14] Brown, K. S. Trees, valuations, and the Bieri-Neumann-Strebel invariant, Invent. Math., Volume 90 (1987) no. 3, pp. 479-504 | DOI | MR | Zbl

[15] Brown, K. S.; Geoghegan, R. An infinite-dimensional torsion-free ${FP}_\infty $ group, Invent. Math., Volume 77 (1984) no. 2, pp. 367-381 | DOI | MR | Zbl

[16] Brown, Kenneth S. Finiteness properties of groups, J. Pure Appl. Algebra, Volume 44 (1987) no. 1-3, pp. 45-75 Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985) | DOI | MR | Zbl

[17] Burillo, J.; Cleary, S.; Stein, M. I. Metrics and embeddings of generalizations of Thompson’s group ${F}$, Trans. Amer. Math. Soc., Volume 353 (2001) no. 4, pp. 1677-1689 | DOI | MR | Zbl

[18] Burillo, J.; Nucinkis, B.; Reeves, L. An irrational-slope Thompson’s group, Publ. Mat., Volume 65 (2021) no. 2, pp. 809-839 | DOI | MR | Zbl

[19] Calegari, D. Stable commutator length in subgroups of ${\rm {PL}}^+({I})$, Pacific J. Math., Volume 232 (2007) no. 2, pp. 257-262 | DOI | MR | Zbl

[20] Calegari, D. scl, MSJ Memoirs, 20, Mathematical Society of Japan, Tokyo, 2009, xii+209 pages | DOI | MR | Zbl

[21] Cannon, J. W.; Floyd, W. J.; Parry, W. R. Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), Volume 42 (1996) no. 3-4, pp. 215-256 | MR | Zbl

[22] Caprace, P.-E.; Cornulier, Y.; Monod, N.; Tessera, R. Amenable hyperbolic groups, J. Eur. Math. Soc. (JEMS), Volume 17 (2015) no. 11, pp. 2903-2947 | DOI | MR | Zbl

[23] Cleary, S. Regular subdivision in $\mathbf{Z}[\frac{1+\sqrt{5}}{2}]$, Illinois J. Math., Volume 44 (2000) no. 3, pp. 453-464 | MR | DOI | Zbl

[24] Dehornoy, P. The group of parenthesized braids, Adv. Math., Volume 205 (2006) no. 2, pp. 354-409 | DOI | MR | Zbl

[25] Fournier-Facio, F.; Lodha, Y. Second bounded cohomology of groups acting on 1-manifolds and applications to spectrum problems, Adv. Math., Volume 428 (2023), Paper no. 109162, 42 pages | DOI | MR | Zbl

[26] Fournier-Facio, F.; Lodha, Y.; Zaremsky, M. C. B. Braided Thompson groups with and without quasimorphisms, Algebr. Geom. Topol., Volume 24 (2024) no. 3, pp. 1601-1622 | DOI | MR | Zbl

[27] Genevois, A. Hyperbolic and cubical rigidities of Thompson’s group ${V}$, J. Group Theory, Volume 22 (2019) no. 2, pp. 313-345 | DOI | MR | Zbl

[28] Genevois, A.; Tournier, G. Hyperbolic structures on Houghton groups, Geom. Dedicata, Volume 220 (2026) no. 1, Paper no. 1, 31 pages | DOI | MR | Zbl

[29] Geoghegan, R.; Mihalik, M. L.; Sapir, M.; Wise, D. T. Ascending HNN extensions of finitely generated free groups are Hopfian, Bull. London Math. Soc., Volume 33 (2001) no. 3, pp. 292-298 | DOI | MR | Zbl

[30] Golan, G.; Sapir, M. On the stabilizers of finite sets of numbers in the R. Thompson group ${F}$, Algebra i Analiz, Volume 29 (2017) no. 1, pp. 70-110 | DOI | MR | Zbl

[31] Golan, G.; Sapir, M. Divergence functions of Thompson groups, Geom. Dedicata, Volume 201 (2019), pp. 227-242 | DOI | MR | Zbl

[32] Gromov, M. Hyperbolic groups, Essays in group theory (Math. Sci. Res. Inst. Publ.), Volume 8, Springer, New York, 1987, pp. 75-263 | DOI | MR | Zbl

[33] Haettel, T. Hyperbolic rigidity of higher rank lattices, Ann. Sci. Éc. Norm. Supér. (4), Volume 53 (2020) no. 2, pp. 439-468 | DOI | MR | Zbl

[34] Hamann, M. Group actions on metric spaces: fixed points and free subgroups, Abh. Math. Semin. Univ. Hambg., Volume 87 (2017) no. 2, pp. 245-263 | DOI | MR | Zbl

[35] Hartley, B.; Zalesskiĭ, A. E. Confined subgroups of simple locally finite groups and ideals of their group rings, J. London Math. Soc. (2), Volume 55 (1997) no. 2, pp. 210-230 | DOI | MR | Zbl

[36] Higman, G. Finitely presented infinite simple groups, Notes on Pure Mathematics, No. 8, Australian National University, Department of Pure Mathematics, Department of Mathematics, I.A.S., Canberra, 1974, vii+82 pages | MR | Zbl

[37] Houghton, C. H. The first cohomology of a group with permutation module coefficients, Arch. Math. (Basel), Volume 31 (1978/79) no. 3, pp. 254-258 | DOI | MR | Zbl

[38] Hyde, J.; Lodha, Y. Finitely presented simple left-orderable groups in the landscape of Richard Thompson’s groups, Ann. Sci. Éc. Norm. Supér. (4), Volume 58 (2025) no. 2, pp. 419-432 | MR | Zbl

[39] Hyde, J.; Moore, J. T. Subgroups of ${\rm {PL}}_+{I}$ which do not embed into Thompson’s group ${F}$, Groups Geom. Dyn., Volume 17 (2023) no. 2, pp. 533-554 | DOI | MR | Zbl

[40] Lehnert, J. Gruppen von quasi-Automorphismen, Ph. D. Thesis, Goethe-Universität, Frankfurt am Main (2008) | Zbl

[41] Lodha, Y. Coherent actions by homeomorphisms on the real line or an interval, Israel J. Math., Volume 235 (2020) no. 1, pp. 183-212 | DOI | MR | Zbl

[42] Lodha, Y. A nonamenable type $\rm {F}_\infty $ group of piecewise projective homeomorphisms, J. Topol., Volume 13 (2020) no. 4, pp. 1767-1838 | DOI | MR | Zbl

[43] Lodha, Y.; Moore, J. T. A nonamenable finitely presented group of piecewise projective homeomorphisms, Groups Geom. Dyn., Volume 10 (2016) no. 1, pp. 177-200 | DOI | MR | Zbl

[44] Lodha, Y.; Zaremsky, M. C. B. The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type ${\rm {F}}_\infty $, Math. Proc. Cambridge Philos. Soc., Volume 174 (2023) no. 1, pp. 25-48 | DOI | MR | Zbl

[45] Manning, J. F. Actions of certain arithmetic groups on Gromov hyperbolic spaces, Algebr. Geom. Topol., Volume 8 (2008) no. 3, pp. 1371-1402 | DOI | MR | Zbl

[46] Minasyan, A.; Osin, D. Correction to: Acylindrical hyperbolicity of groups acting on trees, Math. Ann., Volume 373 (2019) no. 1-2, pp. 895-900 | DOI | MR | Zbl

[47] Molyneux, L.; Nucinkis, B.; Santos Rego, Y. The sigma invariants for the golden mean Thompson group, New York J. Math., Volume 30 (2024), pp. 532-549 | MR | Zbl

[48] Nucinkis, B. E. A.; St. John-Green, S. Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree, Groups Geom. Dyn., Volume 12 (2018) no. 2, pp. 529-570 | DOI | MR | Zbl

[49] Serre, J.-P. Arbres, amalgames, ${\rm {SL}}_{2}$, Société Mathématique de France, Paris, 1977, 189 pages | MR | Numdam | Zbl

[50] Sheng, X. Divergence property of the Brown-Thompson groups and braided Thompson groups, Transform. Groups, Volume 30 (2025) no. 1, pp. 413-446 | DOI | MR | Zbl

[51] Spahn, R.; Zaremsky, M. C. B. The BNSR-invariants of the Stein group ${F}_{2,3}$, J. Group Theory, Volume 24 (2021) no. 6, pp. 1149-1162 | DOI | MR | Zbl

[52] Stein, M. Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc., Volume 332 (1992) no. 2, pp. 477-514 | DOI | MR | Zbl

[53] Winstone, N. Bieri-Strebel Groups With Irrational Slopes, Ph. D. Thesis, Royal Holloway, University of London (2022)

[54] Zaremsky, M. C. B. HNN decompositions of the Lodha-Moore groups, and topological applications, J. Topol. Anal., Volume 8 (2016) no. 4, pp. 627-653 | DOI | MR | Zbl

[55] Zaremsky, M. C. B. On the $\Sigma $-invariants of generalized Thompson groups and Houghton groups, Int. Math. Res. Not. IMRN (2017) no. 19, pp. 5861-5896 | MR | Zbl

[56] Zaremsky, M. C. B. The BNSR-invariants of the Houghton groups, concluded, Proc. Edinb. Math. Soc. (2), Volume 63 (2020) no. 1, pp. 1-11 | DOI | MR

Cited by Sources: