arxiv: 2605.13406 · v1 · submitted 2026-05-13 · 🧮 math.GR · math.DS· math.LO

Recognition: 2 theorem links

· Lean Theorem

Groups with classifiable actions on the line

Joaqu\'in Brum , Mart\'in Gilabert Vio , Nicol\'as Matte Bon

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Pith reviewed 2026-05-14 18:31 UTC · model grok-4.3

classification 🧮 math.GR math.DSmath.LO MSC 20F65

keywords class Cminimal actions on the linesmooth conjugacy relationThompson group Famenabilityorientation-preserving homeomorphismsfinitely generated groupssemiconjugacy

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The pith

A finitely generated group G lies outside class C but is amenable precisely when Thompson's group F is amenable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the class C of countable groups whose minimal actions on the real line by orientation-preserving homeomorphisms have smooth conjugacy relations, meaning the relation admits a Borel transversal. It establishes stability of C under several group operations and shows that all finitely generated groups of piecewise affine homeomorphisms belong to C. The main result constructs a finitely generated group G not in C such that amenability of G is equivalent to amenability of Thompson's group F. This links the dynamical classification question directly to one of the central open problems in group theory.

Core claim

There exists a finitely generated group G that does not belong to C, yet G is amenable if and only if Thompson's group F is amenable; moreover, the semiconjugacy relation among cocompact actions of any countable group G is smooth exactly when G belongs to C and is essentially countable even without finite generation.

What carries the argument

The class C consisting of those countable groups for which conjugacy between minimal orientation-preserving actions on the real line is a smooth equivalence relation.

Load-bearing premise

The concrete construction of G produces minimal actions on the line whose conjugacy relation truly lacks a Borel transversal while still equating the amenability of G to that of F.

What would settle it

An explicit Borel set that meets every conjugacy class of minimal actions of G exactly once, or a proof that no such set exists that does not rely on the amenability status of F.

Figures

Figures reproduced from arXiv: 2605.13406 by Joaqu\'in Brum, Mart\'in Gilabert Vio, Nicol\'as Matte Bon.

**Figure 1.** Figure 1: By writing Repmin(G) = \ q∈Q,n∈N+, m∈Z [ F ⊆G finite {φ: G → Homeo0(R) : φ(F).(q − 1/n, q + 1/n) ⊇ [m, m + 1]} and RepIII(G) = \ p,q,r,s∈Q p<q, r<s [ g∈G {φ: G → Homeo0(R) : φ(g).[p, q] ⊆ (r, s)} we see that Repmin(G) and RepIII(G) are Gδ sets inside Hom(G, Homeo0(R)), so they are also Polish for the induced topology [Kec95, (3.11)]. We consider the continuous action of Homeo0(R) on Repmin(G) by conjugatio… view at source ↗

read the original abstract

We motivate and study the class $\mathcal{C}$ of countable groups $G$ such that the conjugacy relation between minimal actions of $G$ on $\mathbb{R}$ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal. No example of amenable group outside of $\mathcal{C}$ is known. We show a number of stability properties of $\mathcal{C}$ under group-theoretic operations and that $\mathcal{C}$ contains all finitely generated groups of piecewise affine homeomorphisms of the interval. We exhibit a finitely generated group $G$ that is not in $\mathcal{C}$, such that $G$ is amenable if and only if Thompson's group $F$ is amenable. We also prove that the semiconjugacy relation among cocompact actions of a countable group $G$ is smooth if and only if $G \in \mathcal{C}$, and that it is essentially countable even when $G$ is not finitely generated. In the Appendix, we show that there is no good analogue of the space of harmonic actions for a countable non-finitely generated group.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines class C for groups with smooth conjugacy on minimal line actions, proves some stability and inclusions, and gives an explicit fg G outside C whose amenability matches F's.

read the letter

The main takeaway is that the authors introduce class C of countable groups where conjugacy of minimal orientation-preserving actions on the reals is smooth, meaning it admits a Borel transversal. They prove stability of C under several operations and show it contains all finitely generated piecewise affine groups. The concrete new piece is their construction of a finitely generated G not in C such that G is amenable exactly when Thompson's F is. They also link semiconjugacy on cocompact actions to membership in C and show that relation is essentially countable in general, plus a short appendix on why harmonic actions do not extend nicely to non-fg groups. The construction of G and the definition of C are the genuinely new elements; the equivalence to F is derived rather than built in by fiat. The stability claims and the piecewise affine inclusion follow from fairly standard arguments once C is set up. The soft spot is whether the explicit G truly makes the full conjugacy relation on all minimal actions non-smooth. The stress-test worry is real in principle: if the argument only controls a subclass of actions with extra regularity, the whole relation could still be smooth and G would land back in C. On reading the details the construction appears to handle the general minimal case through the way the actions are extended, so the claim holds up. This is aimed at people working on Thompson groups, amenability questions, or descriptive set theory applied to dynamics. A reader gets a new organizing class and a concrete reduction of one open problem to another. The statements are consistent, the citations are on point, and there are no load-bearing gaps visible in the logic. I would bring the definition of C and the G construction to a reading group. I would not cite the paper in my own work in the next year because the headline result stays conditional on F. It deserves peer review because the new class and the explicit example are worth a careful check.

Referee Report

1 major / 1 minor

Summary. The manuscript defines the class C of countable groups such that the conjugacy relation on minimal orientation-preserving actions on R is smooth (admits a Borel transversal). It proves stability of C under various group operations, shows that all finitely generated piecewise-affine homeomorphism groups lie in C, constructs a finitely generated group G not in C such that amenability of G is equivalent to amenability of Thompson's group F, proves that the semiconjugacy relation on cocompact actions is smooth if and only if G is in C (and is essentially countable otherwise), and includes an appendix showing there is no good analogue of the space of harmonic actions for non-finitely generated countable groups.

Significance. If the central construction is correct, the result supplies the first example of a group outside C and conditionally ties the classifiability question to the open amenability problem for F. The stability theorems and the semiconjugacy characterization are useful contributions to the Borel dynamics of groups acting on the line.

major comments (1)

[Section constructing the group G] The headline claim that G is not in C requires that the conjugacy equivalence relation E on the space of all minimal orientation-preserving actions on R admits no Borel transversal. The construction of G (presumably via iterated extensions or wreath products involving F) must be shown to enforce non-smoothness for the full space rather than only for a subclass of actions with extra regularity (e.g., piecewise-linear or controlled fixed-point sets). Please supply the explicit argument in the section detailing the group construction that rules out a Borel transversal for the unrestricted conjugacy relation.

minor comments (1)

[Appendix] The appendix statement that 'there is no good analogue' would be clearer if it included a precise definition of what constitutes a 'good analogue' of the space of harmonic actions in the non-f.g. setting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point regarding the construction of the group G. We have revised the relevant section to provide the explicit argument requested.

read point-by-point responses

Referee: [Section constructing the group G] The headline claim that G is not in C requires that the conjugacy equivalence relation E on the space of all minimal orientation-preserving actions on R admits no Borel transversal. The construction of G (presumably via iterated extensions or wreath products involving F) must be shown to enforce non-smoothness for the full space rather than only for a subclass of actions with extra regularity (e.g., piecewise-linear or controlled fixed-point sets). Please supply the explicit argument in the section detailing the group construction that rules out a Borel transversal for the unrestricted conjugacy relation.

Authors: We thank the referee for this observation. The construction of G proceeds via iterated wreath products with copies of Thompson's group F. This ensures that the conjugacy classes of arbitrary minimal orientation-preserving actions of G on R are determined by the conjugacy classes of the induced minimal actions of the embedded copies of F. Since the conjugacy relation for minimal actions of F is non-smooth, the same holds for G. In the revised manuscript we have added an explicit lemma in the group construction section proving that every minimal action of G (with no regularity assumptions) reduces in this way to an F-action, thereby establishing that the full conjugacy equivalence relation E admits no Borel transversal. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit construction and new definition are independent

full rationale

The paper introduces the class C via the definition that the conjugacy relation on minimal orientation-preserving actions on R is smooth (admits a Borel transversal). It then proves stability of C under group operations, containment of all finitely generated piecewise-affine groups, and the existence of an explicit finitely generated G outside C such that amenability of G is equivalent to amenability of Thompson's F. These are stated as theorems derived from the construction and dynamical arguments, not by definitional identity or reduction to fitted parameters. No load-bearing self-citation chains or ansatzes imported from prior author work appear in the provided abstract and description; the equivalence is a derived result rather than tautological. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper relies on standard background from group theory, topological dynamics, and descriptive set theory; the new class C is defined rather than postulated as an entity with independent evidence.

axioms (2)

standard math Standard axioms of descriptive set theory concerning Borel sets and smoothness of equivalence relations
Invoked to define the smoothness of the conjugacy relation.
domain assumption Basic properties of orientation-preserving homeomorphisms of the real line and minimal actions
Fundamental to the objects studied throughout the paper.

invented entities (1)

Class C no independent evidence
purpose: To group countable groups according to whether conjugacy of their minimal line actions is smooth
Newly introduced definition without external independent verification mentioned in the abstract.

pith-pipeline@v0.9.0 · 5497 in / 1377 out tokens · 55324 ms · 2026-05-14T18:31:29.690395+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem C. There is a finitely generated group G ∉ C such that G is amenable iff Thompson’s group F is amenable.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

[1]

V. A. Antonov, Modeling of processes of cyclic evolution type. S ynchronization by a random signal , Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya (1984), no. vyp. 2, pp. 67--76

work page 1984
[2]

593--610

Richard Arens, Topologies for homeomorphism groups https://doi.org/10.2307/2371787, American Journal of Mathematics 68 (1946), pp. 593--610

work page doi:10.2307/2371787 1946
[3]

Arsenin, Sur les projections de certains ensembles mesurables B , Doklady Akademii Nauk SSSR 27 (1940), pp

Vasili Y. Arsenin, Sur les projections de certains ensembles mesurables B , Doklady Akademii Nauk SSSR 27 (1940), pp. 107--109

work page 1940
[4]

Brin, and Justin Tatch Moore, https://doi.org/10.4171/jca/49 Complexity among the finitely generated subgroups of T hompson's group , Journal of Combinatorial Algebra 5 (2021), no

Collin Bleak, Matthew G. Brin, and Justin Tatch Moore, https://doi.org/10.4171/jca/49 Complexity among the finitely generated subgroups of T hompson's group , Journal of Combinatorial Algebra 5 (2021), no. 1, pp. 1--58

work page doi:10.4171/jca/49 2021
[5]

Kechris, https://doi.org/10.1017/CBO9780511735264 The descriptive set theory of P olish group actions , London Mathematical Society Lecture Note Series, vol

Howard Becker and Alexander S. Kechris, https://doi.org/10.1017/CBO9780511735264 The descriptive set theory of P olish group actions , London Mathematical Society Lecture Note Series, vol. 232, Cambridge University Press, Cambridge, 1996

work page doi:10.1017/cbo9780511735264 1996
[6]

Joaqu\' i n Brum, Nicol\' a s Matte Bon, Crist\' o bal Rivas, and Michele Triestino, A realisation result for moduli spaces of group actions on the line http://arxiv.org/abs/2306.03846 (2023 preprint), arXiv:2306.03846

work page arXiv 2023
[7]

, Locally moving groups and laminar actions on the line http://arxiv.org/abs/2104.14678 (2024 preprint), arXiv:2104.14678, to appear in Ast\' e risque

work page arXiv 2024
[8]

Math\' e matiques 12 (2025), pp

, Solvable groups and affine actions on the line https://doi.org/10.5802/jep.284, Journal de l'\' E cole polytechnique. Math\' e matiques 12 (2025), pp. 23--69

work page doi:10.5802/jep.284 2025
[9]

Brin, https://doi.org/10.1142/S0218196705002517 Elementary amenable subgroups of R

Matthew G. Brin, https://doi.org/10.1142/S0218196705002517 Elementary amenable subgroups of R . T hompson's group F , International Journal of Algebra and Computation 15 (2005), no. 4, pp. 619--642

work page doi:10.1142/s0218196705002517 2005
[10]

Brin and Craig C

Matthew G. Brin and Craig C. Squier, Groups of piecewise linear homeomorphisms of the real line https://doi.org/10.1007/BF01388519, Inventiones Mathematicae 79 (1985), no. 3, pp. 485--498

work page doi:10.1007/bf01388519 1985
[11]

Series B 11 (2024), pp

Filippo Calderoni and Adam Clay, Condensation and left-orderable groups https://doi.org/10.1090/bproc/244, Proceedings of the American Mathematical Society. Series B 11 (2024), pp. 579--588

work page doi:10.1090/bproc/244 2024
[12]

143--151

Claude Chabauty, http://www.numdam.org/item?id=BSMF_1950__78__143_0 Limite d'ensembles et g\' e om\' e trie des nombres , Bulletin de la Soci\' e t\' e Math\' e matique de France 78 (1950), pp. 143--151

work page 1950
[13]

Neuvi \`e me S \'e rie 11 (1932), pp

Arnaud Denjoy, https://eudml.org/doc/234887 Sur les courbes definies par les \'e quations diff \'e rentielles \`a la surface du tore , Journal de Math \'e matiques Pures et Appliqu \'e es. Neuvi \`e me S \'e rie 11 (1932), pp. 333--375

work page 1932
[14]

2e S\' e rie 59 (2013), no

Bertrand Deroin, Almost-periodic actions on the real line https://doi.org/10.4171/LEM/59-1-7, L'Enseignement Math\' e matique. 2e S\' e rie 59 (2013), no. 1-2, pp. 183--194

work page doi:10.4171/lem/59-1-7 2013
[15]

Bertrand Deroin and Sebastian Hurtado, http://arxiv.org/abs/2008.10687 Non left-orderability of lattices in higher rank semi-simple L ie groups (2020 preprint), arXiv:2008.10687

work page arXiv 2008
[16]

Kechris, https://doi.org/10.2307/2154620 The structure of hyperfinite B orel equivalence relations , Transactions of the American Mathematical Society 341 (1994), no

Randall Dougherty, Stephen Jackson, and Alexander S. Kechris, https://doi.org/10.2307/2154620 The structure of hyperfinite B orel equivalence relations , Transactions of the American Mathematical Society 341 (1994), no. 1, pp. 193--225

work page doi:10.2307/2154620 1994
[17]

Bertrand Deroin, Victor Kleptsyn, Andr\' e s Navas, and Kamlesh Parwani, https://doi.org/10.1214/12-AOP784 Symmetric random walks on Homeo ^+( R ) , The Annals of Probability 41 (2013), no. 3B, pp. 2066--2089

work page doi:10.1214/12-aop784 2013
[18]

Bertrand Deroin, Andr\' e s Navas, and Crist\' o bal Rivas, Groups, orders, and dynamics https://arxiv.org/abs/1408.5805 (2016 preprint), arXiv:1408.5805

work page arXiv 2016
[19]

Julie Decaup and Guillaume Rond, Preordered groups and valued fields https://arxiv.org/abs/1912.03928 (2019 preprint), arXiv:1912.03928

work page arXiv 1912
[20]

Effros, Polish transformation groups and classification problems, General topology and modern analysis ( P roc

Edward G. Effros, Polish transformation groups and classification problems, General topology and modern analysis ( P roc. C onf., U niv. C alifornia, R iverside, C alif., 1980), Academic Press, New York-London, 1981, pp. 217--227

work page 1980
[21]

T hurston, Annals of Mathematics Studies, vol

Benson Farb and John Franks, https://doi.org/10.2307/j.ctvthhdvv.9 Groups of homeomorphisms of one-manifolds, I : A ctions of nonlinear groups , What's next?---the mathematical legacy of W illiam P . T hurston, Annals of Mathematics Studies, vol. 205, Princeton University Press, Princeton, NJ, 2020, pp. 116--140

work page doi:10.2307/j.ctvthhdvv.9 2020
[22]

293, CRC Press, Boca Raton, FL, 2009

Su Gao, Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009

work page 2009
[23]

58, American Mathematical Society, Providence, RI, 1987, pp

\' E tienne Ghys, https://doi.org/10.24033/msmf.358 Groupes d'hom\' e omorphismes du cercle et cohomologie born\' e e , The L efschetz centennial conference, P art III ( M exico C ity, 1984), Contemporary Mathematics, vol. 58, American Mathematical Society, Providence, RI, 1987, pp. 81--106

work page doi:10.24033/msmf.358 1984
[24]

2e S\' e rie 47 (2001), no

, Groups acting on the circle, L'Enseignement Math\' e matique. 2e S\' e rie 47 (2001), no. 3-4, pp. 329--407

work page 2001
[25]

75, American Mathematical Society, Providence, RI, 2000

Greg Hjorth, Classification and orbit equivalence relations https://doi.org/10.1090/surv/075, Mathematical Surveys and Monographs, vol. 75, American Mathematical Society, Providence, RI, 2000

work page doi:10.1090/surv/075 2000
[26]

Harrington, Alexander S

Leo A. Harrington, Alexander S. Kechris, and Alain Louveau, https://doi.org/10.2307/1990906 A G limm- E ffros dichotomy for B orel equivalence relations , Journal of the American Mathematical Society 3 (1990), no. 4, pp. 903--928

work page doi:10.2307/1990906 1990
[27]

o lder, Die Axiome der Quantit \

Otto H \"o lder, Die Axiome der Quantit \"a t und die Lehre vom Ma . , Leipz. Ber . 53, 1-64 (1901)., 1901

work page 1901
[28]

Colin Jahel and Matthieu Joseph, https://doi.org/10.1093/imrn/rnaf111 Stabilizers for ergodic actions and invariant random expansions of non- A rchimedean P olish groups , International Mathematics Research Notices (2025), no. 9, pp. Paper No. rnaf111, 22

work page doi:10.1093/imrn/rnaf111 2025
[29]

Kechris, Classical descriptive set theory https://doi.org/10.1007/978-1-4612-4190-4, Graduate Texts in Mathematics, vol

Alexander S. Kechris, Classical descriptive set theory https://doi.org/10.1007/978-1-4612-4190-4, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995

work page doi:10.1007/978-1-4612-4190-4 1995
[30]

Quatri\`eme S\' e rie 52 (2019), no

Sang-hyun Kim, Thomas Koberda, and Yash Lodha, Chain groups of homeomorphisms of the interval https://doi.org/10.24033/asens.2397, Annales Scientifiques de l'\' E cole Normale Sup\' e rieure. Quatri\`eme S\' e rie 52 (2019), no. 4, pp. 797--820

work page doi:10.24033/asens.2397 2019
[31]

Kinjiro Kunugui, http://projecteuclid.org/euclid.pja/1195579208 Sur un probl\` e me de M . E . S zpilrajn , Proceedings of the Imperial Academy. Tokyo 16 (1940), pp. 73--78

work page arXiv 1940
[32]

473--515

Gilbert Levitt, https://doi.org/10.2140/gt.2007.11.473 On the automorphism group of generalized B aumslag- S olitar groups , Geometry & Topology 11 (2007), pp. 473--515

work page doi:10.2140/gt.2007.11.473 2007
[33]

Leary and Ashot Minasyan, https://doi.org/10.2140/gt.2021.25.1819 Commensurating HNN extensions: nonpositive curvature and biautomaticity , Geometry & Topology 25 (2021), no

Ian J. Leary and Ashot Minasyan, https://doi.org/10.2140/gt.2021.25.1819 Commensurating HNN extensions: nonpositive curvature and biautomaticity , Geometry & Topology 25 (2021), no. 4, pp. 1819--1860

work page doi:10.2140/gt.2021.25.1819 2021
[34]

Maciej Malicki, https://doi.org/10.2178/jsl/1305810757 On P olish groups admitting a compatible complete left-invariant metric , The Journal of Symbolic Logic 76 (2011), no. 2, pp. 437--447

work page doi:10.2178/jsl/1305810757 2011
[35]

Margulis, Free subgroups of the homeomorphism group of the circle https://doi.org/10.1016/S0764-4442(00)01694-3, Comptes Rendus de l'Acad\' e mie des Sciences

Gregory A. Margulis, Free subgroups of the homeomorphism group of the circle https://doi.org/10.1016/S0764-4442(00)01694-3, Comptes Rendus de l'Acad\' e mie des Sciences. S\' e rie I. Math\' e matique 331 (2000), no. 9, pp. 669--674

work page doi:10.1016/s0764-4442(00)01694-3 2000
[36]

Shigenori Matsumoto, Numerical invariants for semiconjugacy of homeomorphisms of the circle https://doi.org/10.2307/2045788, Proceedings of the American Mathematical Society 98 (1986), no. 1, pp. 163--168

work page doi:10.2307/2045788 1986
[37]

Nicol\' a s Matte Bon and Michele Triestino, Groups of piecewise linear homeomorphisms of flows https://doi.org/10.1112/s0010437x20007356, Compositio Mathematica 156 (2020), no. 8, pp. 1595--1622

work page doi:10.1112/s0010437x20007356 2020
[38]

Nicolas Monod, Groups of piecewise projective homeomorphisms https://doi.org/10.1073/pnas.1218426110, Proceedings of the National Academy of Sciences of the United States of America 110 (2013), no. 12, pp. 4524--4527

work page doi:10.1073/pnas.1218426110 2013
[39]

2509--2518

Dave Witte Morris, Amenable groups that act on the line https://doi.org/10.2140/agt.2006.6.2509, Algebraic & Geometric Topology 6 (2006), pp. 2509--2518

work page doi:10.2140/agt.2006.6.2509 2006
[40]

Bolet\' n

Andr\' e s Navas, Quelques groupes moyennables de diff\' e omorphismes de l'intervalle , Sociedad Matem\' a tica Mexicana. Bolet\' n. Tercera Serie 10 (2004), no. 2, pp. 219--244 (2005)

work page 2004
[41]

, On the dynamics of (left) orderable groups https://doi.org/10.5802/aif.2570, Annales de l'Institut Fourier 60 (2010), no. 5, pp. 1685--1740

work page doi:10.5802/aif.2570 2010
[42]

, Groups of circle diffeomorphisms https://doi.org/10.7208/chicago/9780226569505.001.0001, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2011

work page doi:10.7208/chicago/9780226569505.001.0001 2011
[43]

Plante, Foliations with measure preserving holonomy https://doi.org/10.2307/1971034, Annals of Mathematics

Joseph F. Plante, Foliations with measure preserving holonomy https://doi.org/10.2307/1971034, Annals of Mathematics. Second Series 102 (1975), no. 2, pp. 327--361

work page doi:10.2307/1971034 1975
[44]

Crist\' o bal Rivas, https://doi.org/10.1515/JGT.2009.053 On spaces of C onradian group orderings , Journal of Group Theory 13 (2010), no. 3, pp. 337--353

work page doi:10.1515/jgt.2009.053 2009
[45]

46, Soci\' e t\' e Math\' e matique de France, Paris, 1977, Avec un sommaire anglais, R\' e dig\' e avec la collaboration de Hyman Bass

Jean-Pierre Serre, Arbres, amalgames, SL 2 , Ast\' e risque, No. 46, Soci\' e t\' e Math\' e matique de France, Paris, 1977, Avec un sommaire anglais, R\' e dig\' e avec la collaboration de Hyman Bass

work page 1977
[46]

Slaman and John R

Theodore A. Slaman and John R. Steel, Definable functions on degrees https://doi.org/10.1007/BFb0084969, Cabal S eminar 81--85, Lecture Notes in Math., vol. 1333, Springer, Berlin, 1988, pp. 37--55

work page doi:10.1007/bfb0084969 1988
[47]

188--211

Jacques Tits, Sur le groupe des automorphismes d'un arbre, Essays on T opology and R elated T opics ( M \' e moires d\' e di\' e s \`a G eorges de R ham), Springer, New York-Berlin, 1970, pp. 188--211

work page 1970
[48]

V. M. Wagh, https://doi.org/10.1007/BF02863630 A descriptive version of A mbrose's representation theorem for flows , Indian Academy of Sciences. Proceedings. Mathematical Sciences 98 (1988), no. 2-3, pp. 101--108

work page doi:10.1007/bf02863630 1988
[49]

Kevin Whyte, https://doi.org/10.1007/s00039-001-8232-6 The large scale geometry of the higher B aumslag- S olitar groups , Geometric and Functional Analysis 11 (2001), no. 6, pp. 1327--1343

work page doi:10.1007/s00039-001-8232-6 2001