1 / 76 ページ

Deformations and Products of Polish Groups

by

W. Jake Herndon

A.S., Harold Washington College, 2010

B.S., University of Illinois at Chicago, 2013

THESIS

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Mathematics

in the Graduate College of the

University of Illinois at Chicago, 2019

Chicago, Illinois

Defense Committee:

Christian Rosendal, Chair and Advisor

Daniel Groves

Wouter van Limbeek

Kevin Whyte

Michael Cohen, Carleton College

2 / 76 ページ

Copyright by

W. Jake Herndon

2019

3 / 76 ページ

ACKNOWLEDGMENT

Thank you to everyone who supported and encouraged me throughout this process. I am

especially grateful for the advisorship of Christian Rosendal and Daniel Groves. Thank you

both for your patience, generosity, and expertise. As well, thank you to my defense committee

for helping to strengthen the quality of my research.

Thank you to my parents and family. I feel very fortunate to have been raised by people

that instilled in me a love for science and mathematics.

Finally, I thank my wife, Carol. You have helped so much. Thank you for believing in me.

WJH

iii

4 / 76 ページ

TABLE OF CONTENTS

CHAPTER PAGE

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Uniform structures . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Uniformities on a topological group . . . . . . . . . . . . . . . . 8

2.1.2 Boundedness in a uniformity on a topological group . . . . . . 10

2.2 Coarse structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Coarse structures on topological groups . . . . . . . . . . . . . 12

2.3 Groups of homeomorphisms . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Homeomorphisms of compact 1-manifolds . . . . . . . . . . . . 16

2.3.2 Commuting with integer translations . . . . . . . . . . . . . . . 19

2.4 Deformation retracts . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Deformations of Homeo+(I) . . . . . . . . . . . . . . . . . . . . . 24

2.5 Knit products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1 External definition . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.2 As transformation groups . . . . . . . . . . . . . . . . . . . . . . 28

2.5.3 Topological groups which are knit products . . . . . . . . . . . 30

3 DEFORMATIONS AND BOUNDEDNESS . . . . . . . . . . . . . . 35

3.1 Restrained deformations . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Conjugate moments . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 GEOMETRY OF KNIT PRODUCTS . . . . . . . . . . . . . . . . . . 52

4.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 EXAMPLE: ABSOLUTE CONTINUITY . . . . . . . . . . . . . . . . 60

CITED LITERATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

iv

5 / 76 ページ

LIST OF FIGURES

FIGURE PAGE

1 Three ways to deform a homeomorphism of the interval to the identity

homeomorphism. From left to right, along a path corresponding to the

deformation V, T , and A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 The family of homeomorphisms that appear in the proof of Proposition

3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 The graph of an element of HomeoZ(R) which fixes 0. . . . . . . . . . . . 58

v

6 / 76 ページ

SUMMARY

Polish groups are studied in the general context of large-scale geometry. In this context

many well-known Polish groups can be equipped with a canonically defined quasi-isometry

type which allows for the application of techniques from geometric group theory.

Deformation retracts of Polish groups receive special attention. The main result concerning

deformation retracts is that the existence of a sufficiently tame deformation of a Polish group

implies that the group has a well-defined quasi-isometry type.

Polish groups which are knit products also receive their own study. In a number of examples

it was noticed that a Polish group’s quasi-isometry type was given by its knit product structure.

The main result on knit products provides a way to understand the geometry of all these

examples at once.

As an application of these ideas we study the large-scale geometry of Polish groups whose

elements are absolutely continuous homeomorphisms.

vi