Groups of piecewise linear homeomorphisms of the real line

Brin, Matthew G.; Squier, Craig C.

doi:10.1007/bf01388519

Cited by 186 publications

(160 citation statements)

References 7 publications

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“…Prove that Thompson's group F satisfies no nontrivial law, that is, for every nontrivial word W in two letters, there exist f, g in F such that W (f, g) is different from the identity. (This result holds for most groups of piecewise homeomorphisms of the interval, as was first shown in [34]. )…”

Section: Lemma 2310 No Group Can Be Written As a Finite Union Of Lsupporting

confidence: 70%

“…According to Exercise B.0.27, one of the main difficulties for this is the fact that F does not contain free subgroups on two generators (see however Exercise 2.3.13). This is a corollary of a much more general and nice result due to Brin and Squier [34] which we reproduce below. We remark that if F is non-amenable, then this would lead to the first example of a finitely presented, torsion-free, non-amenable group which does not contain F 2 .…”

Section: Thompson's Groupssupporting

confidence: 63%

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Groups of Circle Diffeomorphisms

Navas¹

2011

150

171

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The original version (in Spanish) of this text was prepared for a mini-course in Antofagasta, Chile. Subsequently, enlarged and revised versions were published in the series Monografías del IMCA (Perú) and Ensaios Matemáticos (Brasil). This translation arose from the necessity of making this text accessible to a larger audience. I would like to thank Juan Rivera-Letelier for his invitation to the II Workshop on Dynamical Systems ( 2001), for which the original version was prepared, and Roger Metzger for his invitation to IMCA (2006), where part of this material was presented. I would also like to thank both Étienne Ghys and Maria Eulália Vares for motivating me and allowing me to publish the revised Spanish version in Brasil, as well as all the people who strongly encouraged me to conclude this English version.This work was partially supported by CONICYT and PBCT via the Research Network on Low Dimensional Dynamics. I would also to acknowledge the support of both the UMPA Department of the École Normale Supérieure de Lyon, where the idea of writing this text was born during my PhD thesis, and the Institut des Hautes Études Scientifiques, where the original notes started taking its definite form while I benefited from a one-year postdoctoral position.This work owes a lot to many of my colleges. Several remarks spread throughout the text and the content of some examples and exercises were born during fruitful and quite stimulating discussions. It is then a pleasure to thank Sylvain Crovisier (Proposition 4.2.25),

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Section: Lemma 2310 No Group Can Be Written As a Finite Union Of Lsupporting

confidence: 70%

Section: Thompson's Groupssupporting

confidence: 63%

Groups of Circle Diffeomorphisms

Navas¹

2011

150

171

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“…It is not elementary amenable and does not contain a subgroup which is free on two generators [3], [7]. Hence it is a very interesting question whether F is amenable.…”

Section: Suppose That γ Is Z Choose T ∈ C[z] Such That the Principalmentioning

confidence: 99%

Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$ -invariants

Lück

1997

Mathematische Annalen

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“…It was shown in [3] that any subgroup of F is either metabelian or contains an infinite direct power of the group Z. (In fact, one can replace the word "metabelian" by "abelian", see [6].)…”

Section: Corollary 20 For Any N the Group H N = (· · · (Z Wr Z) Wr ·mentioning

confidence: 99%

On subgroups of R. Thompson's group $ F$ and other diagram groups

Guba¹,

Sapir²

1999

Sb. Math.

123

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In this paper, we continue our study of the class of diagram groups. Simply speaking, a diagram is a labelled plane graph bounded by a pair of paths (the top path and the bottom path). To multiply two diagrams, one simply identifies the top path of one diagram with the bottom path of the other diagram, and removes pairs of "reducible" cells. Each diagram group is determined by an alphabet X, containing all possible labels of edges, a set of relationscontaining all possible labels of cells, and a word w over X -the label of the top and bottom paths of diagrams. Diagrams can be considered as 2-dimensional words, and diagram groups can be considered as 2-dimensional analogue of free groups. In our previous paper, we showed that the class of diagram groups contains many interesting groups including the famous R. Thompson group F (it corresponds to the simplest set of relations { x = x 2 }), closed under direct and free products and some other constructions. In this paper we study mainly subgroups of diagram groups. We show that not every subgroup of a diagram group is itself a diagram group (this answers a question from the previous paper). We prove that every nilpotent subgroup of a diagram group is abelian, every abelian subgroup is free, but even the Thompson group contains solvable subgroups of any degree. We also study distortion of subgroups in diagram groups, including the Thompson group. It turnes out that centralizers of elements and abelian subgroups are always undistorted, but the Thompson group contains distorted soluble subgroups.

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Groups of piecewise linear homeomorphisms of the real line

Cited by 186 publications

References 7 publications

Groups of Circle Diffeomorphisms

Groups of Circle Diffeomorphisms

Hilbert modules and modules over finite von Neumann algebras and applications to $L^2$ -invariants

On subgroups of R. Thompson's group $ F$ and other diagram groups

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