Free lattice-ordered groups and the space of left orderings

Clay, Adam

doi:10.1007/s00605-011-0305-5

Cited by 18 publications

(16 citation statements)

References 19 publications

(31 reference statements)

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“…We also show how to build a very special representation of

π_{1} false(normalΣ false)

into

H o m e o_{+} false(double-struckR false)

, whose existence can be thought of as a strong form of flexibility. Actually, results such as Theorems and below were only known to hold for the non‐Abelian free groups .…”

Section: Introductionmentioning

confidence: 99%

Orderings and flexibility of some subgroups of Homeo+(R)

Alonso

Brum

Rivas

2017

J. London Math. Soc.

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In this work we exhibit flexibility phenomena for some (countable) groups acting by order preserving homeomorphisms of the line. More precisely, we show that if a left orderable group admits an amalgam decomposition of the form G=double-struckFn∗double-struckZdouble-struckFm where n+m⩾3, then every faithful action of G on the line by order preserving homeomorphisms can be approximated by another action (without global fixed points) that is not semi‐conjugated to the initial action. We deduce that LO(G), the space of left orders of G, is a Cantor set. In the special case where G=π1false(normalΣfalse) is the fundamental group of a closed hyperbolic surface, we found finer techniques of perturbation. For instance, we exhibit a single representation whose conjugacy class in dense in the space of representations. This entails that the space of representations without global fixed points of π1false(normalΣfalse) into Homeo+false(double-struckRfalse) is connected, and also that the natural conjugation action of π1false(normalΣfalse) on LO(π1false(normalΣfalse)) has a dense orbit.

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“…We also show how to build a very special representation of

π_{1} false(normalΣ false)

into

H o m e o_{+} false(double-struckR false)

, whose existence can be thought of as a strong form of flexibility. Actually, results such as Theorems and below were only known to hold for the non‐Abelian free groups .…”

Section: Introductionmentioning

confidence: 99%

Orderings and flexibility of some subgroups of Homeo+(R)

Alonso

Brum

Rivas

2017

J. London Math. Soc.

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“…Problem 3. 20. Show that f is an embedding (it is here that we need the map t : π 1 (B) → R to have discrete image).…”

Section: A Theorem Of Farrellmentioning

confidence: 99%

“…That the space LO(F n ) admits no isolated points has, to date, been proved in many different ways [20,87,76,58], although it was first proved by McCleary [69]. In fact, more is known:…”

Section: The Space Of Orderings Of a Free Productmentioning

confidence: 99%

Ordered Groups and Topology

Clay

Rolfsen

2016

Graduate Studies in Mathematics

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Preface ix Chapter 1. Orderable groups and their algebraic properties 1.1. Invariant orderings 1.2. Examples 1.3. Bi-orderable groups 1.4. Positive cone 1.5. Topology and the spaces of orderings 1.6. Testing for orderability 1.7. Characterization of left-orderable groups 1.8. Group rings and zero divisors 1.9. Torsion-free groups which are not left-orderable Chapter 2. Hölder's theorem, convex subgroups and dynamics 2.1. Hölder's Theorem 2.2. Convex subgroups 2.3. Bi-orderable groups are locally indicable 2.4. The dynamic realization of a left-ordering Chapter 3. Free groups, surface groups and covering spaces 3.1. Surfaces 3.2. Ordering free groups 3.3. Ordering surface groups 3.4. A theorem of Farrell Chapter 4. Knots 4.1. Review of classical knot theory 4.2. The Wirtinger presentation 4.3. Knot groups are locally indicable 4.4. Bi-ordering certain knot groups 4.5. Crossing changes: a theorem of Smythe Chapter 5. Three-dimensional manifolds 5.1. Ordering 3-manifold groups 5.2. Surgery 5.3. Branched Coverings 5.4. Bi-orderability and surgery Chapter 6. Foliations 6.1. Examples 6.2. The leaf space vii viii CONTENTS 6.3. Seifert fibred spaces 6.4. R-covered foliations 6.5. The universal circle Chapter 7. Left-orderings of the braid groups 7.1. Orderability properties of the braid groups 7.2. The Dehornoy ordering of B n 7.3. Thurston's orderings of B n 7.4. Applications of the Dehornoy ordering to knot theory Chapter 8. Groups of Homeomorphisms 8.1. Homeomorphisms of a space 8.2. PL homeomorphisms of the cube 8.3. Proof of Theorem 8.6 8.4. Generalizations 8.5. Homeomorphisms of the cube Chapter 9. Conradian left-orderings and local indicability 9.1. The defining property of a Conradian ordering 9.2. Characterizations of Conradian left-orderability Chapter 10. Spaces of orderings 10.1. The natural actions on LO(G) 10.2. Orderings of Z n and Sikora's theorem 10.3. Examples of groups without isolated orderings 10.4. The space of orderings of a free product 10.5. Examples of groups with isolated orderings 10.6. The number of orderings of a group 10.7. Recurrent orderings and a theorem of Witte-Morris Bibliography Index PrefaceThe inspiration for this book is the remarkable interplay, expecially in the past few decades, between topology and the theory of orderable groups. Applications go in both directions. For example, orderability of the fundamental group of a 3manifold is related to the existence of certain foliations. On the other hand, one can apply topology to study the space of all orderings of a given group, providing strong algebraic applications. Many groups of special topological interest are now known to have invariant orderings, for example braid groups, knot groups, fundamental groups of (almost all) surfaces and many interesting manifolds in higher dimensions.There are several excellent books on orderable groups, and even more so for topology. The current book emphasizes the connections between these subjects, leaving out some details that are available elsewhere, although we have tried to include enough...

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“…Notice that given a word W , the subset of W -orders is preserved under the conjugacy action. Based on the work of McCleary [34], Clay [12] and, independently, Rivas [46], proved that F 2 carries left-orders whose orbits under the conjugacy action are dense. (This easily yields a new proof of the fact that the space of left-orders of F 2 is a Cantor set [34,38].)…”

Section: Genericity Of Non-verbal Ordersmentioning

confidence: 99%

Groups, orders, and laws

Navas¹

2014

Groups Geom. Dyn.

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Free lattice-ordered groups and the space of left orderings

Cited by 18 publications

References 19 publications

Orderings and flexibility of some subgroups of Homeo+(R)

Orderings and flexibility of some subgroups of Homeo+(R)

Ordered Groups and Topology

Groups, orders, and laws

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