Bounded Cohomology of Groups acting on Cantor sets Konstantin Andritsch

2025-04-30 0 0 939.84KB 82 页 10玖币

侵权投诉

Bounded Cohomology of Groups

acting on Cantor sets

Konstantin Andritsch

Thesis presented for the degree

Master of Science ETH in Mathematics

Supervised by Prof. Alessandra Iozzi

Co-Supervised by Francesco Fournier-Facio

Department of Mathematics ETH Zurich

09 August 2022.

arXiv:2210.00459v1 [math.GR] 2 Oct 2022

Abstract

We study the bounded cohomology of certain groups acting on the Cantor set. More

speciﬁcally, we consider the full group of homeomorphisms of the Cantor set as well as

Thompson’s group V. We prove that both of these groups are boundedly acyclic, that

is the bounded cohomology with trivial real coeﬃcients vanishes in positive degrees.

Combining this result with the already established Z-acyclicity of Thompson’s group

V, will make Vthe ﬁrst example of a ﬁnitely generated group, in fact the ﬁrst example

of a group of type F∞, which is universally boundedly acyclic.

We exploit recent results of the theory of bounded cohomology developed in [38, 17,

35] which allow to deduce the aforementioned statements of bounded acyclicity.

Prior to studying these concrete groups we review basic facts of bounded cohomology

and describe how actions of discrete groups on boundedly acyclic modules can verify

the vanishing of the bounded cohomology of these groups [38]. Moreover, we adapt the

idea, developed in [36], that fat points and a generic relation on these points enable to

calculate the bounded cohomology, to our situation.

Before proving bounded acyclicity, we gather various properties of the groups under

consideration and certain subgroups thereof. As a consequence the proofs of bounded

acyclicity will be relatively short.

It will turn out that the approaches to handle these groups are very similar. This sug-

gests that there could be a unifying approach which would imply the bounded acyclicity

of a larger class of groups acting on the Cantor set, including the discussed ones.

Acknowledgements

First and foremost I would like to thank Francesco Fournier-Facio, without his assis-

tance I probably would have never found my way into the fascinating topic of Bounded

Cohomology. Thank you for all the good advice during the process of this thesis and

for suggesting to work on this speciﬁc problem. Moreover, I have to thank you for

recommending to attend the conferences in Bielefeld, Zurich and Regensburg. Thank

you!

Further, I need to thank Alessandra Iozzi. She made it possible for me to work on

this project. Also, I am very grateful for your generosity and kindness throughout this

semester.

A special thanks goes to Benedikt Andritsch for many useful suggestions making my

work more understandable and easier to read.

Last but not least, I want to thank Charlotte Meyer for her never ending support in

any shape or form, for all of her advice and for listening to all of my ideas regardless

of whether they worked out or not.

Contents

1. Introduction 1

1.1. Motivation and Origins of Bounded Cohomology . . . . . . . . . . . . . 1

1.1.1. Applications and Connections to other Research Fields . . . . . 1

1.1.2. Vanishing and Non-Vanishing of Bounded Cohomology . . . . . 3

1.3. Presentwork ................................ 5

1.3.1. Homeomorphisms of the Cantor set . . . . . . . . . . . . . . . . 5

1.3.2. Thompson’s Group V........................ 6

1.4. Organisation................................. 7

1.5. Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Bounded cohomology 9

2.1. Basic Deﬁnitions of Bounded Cohomology . . . . . . . . . . . . . . . . 9

2.2. How to Compute Bounded Cohomology . . . . . . . . . . . . . . . . . . 10

2.2.1. Relatively Injective Resolutions . . . . . . . . . . . . . . . . . . 10

2.3. Amenability and Bounded Acyclicity . . . . . . . . . . . . . . . . . . . 16

2.4. Bounded Cohomology via Boundedly Acyclic Modules . . . . . . . . . . 23

2.5. Dissipated and Binate Groups . . . . . . . . . . . . . . . . . . . . . . . 24

2.6. Generic Relations and Semi-simplicial Sets . . . . . . . . . . . . . . . . 26

3. Homeomorphisms of the Cantor Set 28

3.1. Introduction to the Cantor Set K..................... 28

3.2. Properties of the Cantor Set K...................... 29

3.3. Bounded Cohomology of Subgroups of Homeo(K)............ 35

3.3.1. Bounded Acyclicity of Rigid Stabilizer Subgroups . . . . . . . . 36

3.3.2. Fat Points on the Cantor Set . . . . . . . . . . . . . . . . . . . 40

3.3.3. Generic Relation on Fat Points . . . . . . . . . . . . . . . . . . 43

3.3.4. Transitivity of the G-action .................... 43

3.4. Bounded Cohomology of Homeo(K).................... 44

4. Thompson’s Group V45

4.1. Introduction to Thompson’s Groups F, T, V ............... 45

4.2. Properties of Thompson’s Group V.................... 48

4.3. Bounded Cohomology of Subgroups of V................. 51

4.3.1. Bounded Acyclicity of Point Stabilizers . . . . . . . . . . . . . . 51

4.3.2. Generic Relation on Z1

2..................... 57

4.3.3. Transitivity of the V-action .................... 57

4.4. Bounded Cohomology of V......................... 58

5. Conclusion, Outlook and Further Questions 60

Appendix 62

A. Cantor sets and maps between them 62

A.1. Topological properties of the Cantor set . . . . . . . . . . . . . . . . . . 62

A.2. Cantor sets are homeomorphic and Homeo(K) acts transitively . . . . . 63

iii

B. Ultraﬁlters and Ultralimits 67

B.1. Filters and Ultraﬁlters . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

B.2. Limits along Filters and Ultralimits . . . . . . . . . . . . . . . . . . . . 69

References 75

List of Figures

1. Iterative action of the dissipator γion Xion the top; as above, together

with the action of φ(g)onthebottom................... 25

2. Schematic illustration of the homeomorphism from Kto O. . . . . . . . 30

3. Illustration of the action of γUon the sets {z},P±∞ and Pn,n∈Z. . . 37

4. Schematic illustration of a possible situation in the proof of Corollary 3.3.6

for k=5. .................................. 40

5. Schematic illustration of the homeomorphism g∈Gconstructed in the

proof of Lemma 3.3.16 for k=2. ..................... 44

6. Illustration of an inﬁnite rooted binary tree, where the path 0110 starting

from the root risemphasized........................ 47

7. Illustration of (00,01,1) A

−−→ (0,10,11)................... 48

8. Illustration of (0,10,110,111) B

−−→ (0,100,101,11)............. 48

9. Illustration of (0,10,11) C

−−→ (11,0,10)................... 50

10. Illustration of (0,10,11) D

−−→ (10,0,11)................... 50

11. Illustration of (00,01,1) E

−−→ (0,11,10). We have E=C2◦D◦C◦A. . 51

12. Illustration of (00,010,011,10,110,111) γ

−−→ (00,01,10,1100,1101,111). 55

13. Illustration of h∈Vin the case m=3................... 55

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

BoundedCohomologyofGroupsactingonCantorsetsKonstantinAndritschThesispresentedforthedegreeMasterofScienceETHinMathematicsSupervisedbyProf.AlessandraIozziCo-SupervisedbyFrancescoFournier-FacioDepartmentofMathematicsETHZurich09August2022.AbstractWestudytheboundedcohomologyofcertaingroupsactingontheCant...

展开>> 收起<<

Bounded Cohomology of Groups acting on Cantor sets Konstantin Andritsch.pdf

共82页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Bounded Cohomology of Groups acting on Cantor sets Konstantin Andritsch

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: