Group Colorings and Bernoulli Subflows

2.6 MB·English

by  Su Gao

Group Colorings and Bernoulli Subflows
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0 Su Gao
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Steve Jackson
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Brandon Seward
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a Author address:
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DepartmentofMathematics, UniversityofNorthTexas,1155Union
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Circle #311430, Denton, TX 76203-5017

1 E-mail address: [email protected]
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3 DepartmentofMathematics, UniversityofNorthTexas,1155Union
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Circle #311430, Denton, TX 76203-5017
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E-mail address: [email protected]
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1 Department of Mathematics, University of Michigan, 530 Church
0 Street, Ann Arbor, MI 48109-1043
2
E-mail address: [email protected]
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Contents
Chapter 1. Introduction 1
1.1. Bernoulli flows and subflows 1
1.2. Basic notions 3
1.3. Existence of free subflows 4
1.4. Hyper aperiodic points and k-colorings 6
1.5. Complexity of sets and equivalence relations 9
1.6. Tilings of groups 11
1.7. The almost equality relation 13
1.8. The fundamental method 14
1.9. Brief outline 16
Chapter 2. Preliminaries 19
2.1. Bernoulli flows 19
2.2. 2-colorings 21
2.3. Orthogonality 23
2.4. Minimality 24
2.5. Strengthening and weakening of 2-colorings 27
2.6. Other variations of 2-colorings 30
2.7. Subflows of (2N)G 32
Chapter 3. Basic Constructions of 2-Colorings 35
3.1. 2-Colorings on supergroups of finite index 35
3.2. 2-Colorings on group extensions 39
3.3. 2-Colorings on Z 43
3.4. 2-Colorings on nonabelian free groups 46
3.5. 2-Colorings on solvable groups 48
3.6. 2-Colorings on residually finite groups 51
Chapter 4. Marker Structures and Tilings 53
4.1. Marker structures on groups 53
4.2. 2-Colorings on abelian and FC groups by markers 57
4.3. Some properties of ccc groups 61
4.4. Abelian, nilpotent, and polycyclic groups are ccc 63
4.5. Residually finite and locally finite groups and free products are ccc 70
Chapter 5. Blueprints and Fundamental Functions 77
5.1. Blueprints 77
5.2. Fundamental functions 84
5.3. Existence of blueprints 91
5.4. Growth of blueprints 98
iii
iv CONTENTS
Chapter 6. Basic Applications of the Fundamental Method 103
6.1. The uniform 2-coloring property 103
6.2. Density of 2-colorings 107
6.3. Characterization of the ACP 109
Chapter 7. Further Study of Fundamental Functions 117
7.1. Subflows generated by fundamental functions 117
7.2. Pre-minimality 122
7.3. ∆-minimality 127
7.4. Minimality constructions 131
7.5. Rigidity constructions for topological conjugacy 138
Chapter 8. The Descriptive Complexity of Sets of 2-Colorings 147
8.1. Smallness in measure and category 147
8.2. Σ0-hardness and Π0-completeness 148
2 3
8.3. Flecc groups 151
8.4. Nonflecc groups 156
Chapter 9. The Complexity of the TopologicalConjugacy Relation 169
9.1. Introduction to countable Borel equivalence relations 169
9.2. Basic properties of topological conjugacy 170
9.3. Topological conjugacy of minimal free subflows 176
9.4. Topological conjugacy of free subflows 185
Chapter 10. Extending Partial Functions to 2-Colorings 205
10.1. A sufficient condition for extendability 205
10.2. A characterizationfor extendability 207
10.3. Almost equality and cofinite domains 218
10.4. Automatic extendability 229
Chapter 11. Further Questions 233
11.1. Group structures 233
11.2. 2-colorings 234
11.3. Generalizations 235
11.4. Descriptive complexity 236
Bibliography 237
Index 239
Abstract
InthispaperwestudythedynamicsofBernoulliflowsandtheirsubflowsover
generalcountable groups. One of the main themes of this paper is to establish the
correspondence between the topological and the symbolic perspectives. From the
topologicalperspective, we are particularly interested in free subflows (subflows in
whicheverypointhastrivialstabilizer),minimalsubflows,disjointnessofsubflows,
andthe problemofclassifyingsubflowsuptotopologicalconjugacy. Ourmaintool
to study free subflows will be the notion ofhyper aperiodic points; a point is hyper
aperiodic if the closure of its orbit is a free subflow. We show that the notion of
hyper aperiodicity corresponds to a notion of k-coloring on the countable group, a
key notion we study throughout the paper. In fact, for all important topological
notionswestudy,correspondingnotionsingroupcombinatoricswillbeestablished.
Conversely, many variations of the notions in group combinatorics are proved to
be equivalent to some topological notions. In particular, we obtain results about
the differences in dynamical properties between pairs of points which disagree on
finitely many coordinates.
Another main theme of the paper is to study the properties of free subflows
and minimal subflows. Again this is done through studying the properties of the
hyper aperiodic points and minimal points. We prove that the set of all (minimal)
hyper aperiodic points is always dense but meager and null. By employing notions
and ideas from descriptive set theory, we study the complexity of the sets of hyper
aperiodicpoints andofminimalpoints, andcompletely determine their descriptive
complexity. In doing this we introduce a new notion of countable flecc groups and
study their properties. We also obtain the following results for the classification
problem of free subflows up to topological conjugacy. For locally finite groups the
topologicalconjugacy relationfor all (free) subflows is hyperfinite and nonsmooth.
For nonlocally finite groups the relation is Borel bireducible with the universal
countable Borel equivalence relation.
The third, but not the least important, theme of the paper is to develop con-
structive methods for the notions studied. To construct k-colorings on countable
groups, a fundamental method of construction of multi-layer marker structures is
ReceivedbytheeditorDecember28,2011.
2010 Mathematics Subject Classification. Primary37B10, 20F99; Secondary 03E15, 37B05,
20F65.
Key words and phrases. colorings, hyper aperiodic points, orthogonal colorings, Bernoulli
flows, Bernoulli shifts, Bernoulli subflows, free subflows, marker structures, tilings, topological
conjugacy.
SuGao’sresearchwassupportedbytheU.S.NSFGrantsDMS-0501039andDMS-0901853.
SteveJackson’s researchwassupportedbytheU.S.NSFGrantDMS-0901853.
BrandonSeward’sresearchwassupportedbytwoREUsupplementstotheU.S.NSFGrant
DMS-0501039andanNSFGraduateResearchFellowship.
v
vi ABSTRACT
developed with great generality. This allows one to construct an abundance of k-
colorings with specific properties. Variations of the fundamental method are used
in many proofs in the paper, and we expect them to be useful more broadly in
geometric group theory. As a special case of such marker structures, we study
the notion of ccc groupsand prove the ccc-ness for countable nilpotent, polycyclic,
residually finite, locally finite groups and for free products.
CHAPTER 1
Introduction
In this paper we study Bernoulli flows over arbitrary countable groups (these
arealso knownas Bernoullishifts, Bernoullisystems, and Bernoulli schemes). The
overall focus of this paper is on the development and application of constructive
methods, with a particular emphasis on questions surrounding free subflows. The
topics, methods, and results presented here should be of interest to at least re-
searchers in descriptive set theory, symbolic dynamics, and topological dynamics,
and may be of interest to researchers in C -algebras, ergodic theory, geometric
∗
grouptheory, and percolation theory. In Section 1.1 we remind the reader the def-
initions of Bernoulli flow and subflow and also discuss the importance of Bernoulli
flowstovariousareasofmathematics. InSection1.2weintroducesomebasicnota-
tionandterminologywhichisneededforthischapter. InSection1.3wediscussthe
questionoftheexistenceoffreesubflows. Thisquestionhasbeenrecentlyanswered
and is of importance to this paper. In Sections 1.4, 1.5, 1.6, and 1.7 we discuss
the main results of this paper and at the same time discuss relevance to and mo-
tivation from various areas of mathematics, namely descriptive set theory, ergodic
theory, geometric group theory, symbolic dynamics, and topological dynamics. A
significant aspect of this paper is the invention of some versatile tools which add
structure to arbitrary countable groups and offer significant aid in constructing
points in Bernoulli flows. These tools are developed in great generality and likely
haveapplications beyondtheir use here. These constructivemethods andtheir po-
tentialutility to variousareasofmathematics arediscussedinSection1.8. Finally,
inSection1.9wegiveabriefoutlinetothepaperanddiscusschapterdependencies.
We encourage the readerto make use of the detailed index found at the end of the
paper which includes both terminology and notation.
1.1. Bernoulli flows and subflows
Let us first begin by presenting the most general definition of a Bernoulli flow
(also known as Bernoulli shift, Bernoulli system, and Bernoulli scheme). If G is a
countable group and K is a set with the discrete topology and with a probability
measure ν, then the Bernoulli flow over G with alphabet K is defined to be
KG = x : G K = K
{ → }
g G
Y∈
together with the product topology, the product measure νG, and the following
action of G: for x KG and g G, g x KG is defined by (g x)(h)=x(g 1h).
−
∈ ∈ · ∈ ·
ThesetK isalwaysassumedtohaveatleasttwoelementsasotherwiseKG consists
of a single point.
TheactionofGonKG isquiteintuitive. Forexample,ifG=ZandK = 0,1
thenKG = 0,1 Z canbe viewedasthe spaceofallbi-infinite sequencesof0’{s and}
{ }
1
2 1. INTRODUCTION
1’s. Z then acts by shifting these sequences left and right (the action of 5 Z
shifts these sequences 5 units to the right). Similarly, 0,1 Z2 can be visualized∈as
{ }
the spaceof 0,1 -labelingsofthe twodimensionallattice Z2 R2 with the action
{ } ⊆
of Z2 moving the labels in the obvious fashion. Comprehension of these examples
should lead to an intuitive understanding of the action of G on KG.
Under the producttopology,the basicopensetsofKG arethe setsofthe form
x KG : 1 i n x(h )=k
i i
{ ∈ ∀ ≤ ≤ }
where h ,h ,...,h G, k ,k ,...,k K, and n 1. Thus the action of G on
1 2 n 1 2 n
∈ ∈ ≥
KG is continuous. It is not difficult to see that the basic open sets of KG are both
open and closed (i.e. clopen). Since every point is the intersection of a decreasing
sequence of basic open sets, it follows that KG is totally disconnected (meaning
that the only connected sets are the one point sets). KG is also seen to be perfect
(meaning that there are no isolated points). Furthermore, KG is compact if and
onlyifK isfinite. Thusbyawellknowntheoremoftopology,KG ishomeomorphic
tothe CantorsetwheneverK is finite. Onbasic opensets the measureνG is given
by
νG( x KG : 1 i n x(h )=k )= ν(k ).
i i i
{ ∈ ∀ ≤ ≤ }
1 i n
≤Y≤
Therefore the action of G on KG is measure preserving. It may not be so clear,
but this action is in fact ergodic.
InadditiontoBernoulliflows,wearealsoveryinterestedintheirsubflows (also
known as subshifts or subsystems). A subflow of a Bernoulli flow KG is simply a
closed subset of KG which is stable under the action of G. Bernoulli flows and
their subflows show up in many areas of mathematics. One reason is that they
have a rich diversity of dynamical properties which allow them to model many
phenomenon. This “modeling” shows up in many contexts, such as in ergodic
theory, descriptive set theory, percolation theory, topological dynamics, and sym-
bolic dynamics. In ergodic theory and descriptive set theory, the orbit structures
of Bernoulli flows are used to model the orbit structures of measurable group ac-
tions on other measure spaces. More generally, they are used to model countable
Borel equivalence relations as a well known result of Feldman-Moore states that
every countable Borel equivalence relation on a standard Borel space is induced
by a Borel action of a countable group ([FM]). In the site percolation model of
percolation theory, Bernoulli flows of the form 0,1 G are used to model the flow
{ }
of liquids through porous materials. In topological dynamics, it is known that if
a group G acts continuously on a compact topological space X and the action is
expansive, then there is a Bernoulli flow KG over G, a subflow S KG, and a
⊆
continuous surjection φ : S X which commutes with the action of G (meaning
→
φ(g s)=g φ(s)forallg Gands S). Furthermore,ifX istotallydisconnected
· · ∈ ∈
then φ can be chosen to be a homeomorphism. Similarly, if X can be partitioned
by a collectionofclopen sets, then there is a subflow S of a Bernoulliflow KG and
a continuous surjection φ : X S which commutes with the action of G. These
→
typesoffactscanbeusedtostudyBernoulliflowsviatopologicaldynamics(forex-
ample,asin[GU]),butmorefrequentlytopologicaldynamicalsystemsarestudied
viaBernoulliflows. Thislatterapproachledtothe inventionofsymbolicdynamics
([MH])anditssubsequentgrowthoverthepastseventyyears. Aclassicalexample
of the use of symbolic dynamics is the modeling of geodesics flows on manifolds by
1.2. BASIC NOTIONS 3
(the suspensionof)subflowsofBernoulliflowsoverZ. TraditionallyonlyBernoulli
flowsoverZandZn arestudiedin symbolicdynamics,but morerecentlyBernoulli
flows over hyperbolic groups have been used to model the dynamics of hyperbolic
groups acting on their boundary ([CP]).
AkeyaspectoftheimportanceofBernoulliflowsistheirmodelingcapabilities,
but there are several other reasons to study them as well. Indeed, Bernoulli flows
may be considered interesting in and of themselves. This viewpoint can be seen in
at least descriptive set theory, ergodic theory, and symbolic dynamics. Bernoulli
flows serve as very naturalexamples of orbit equivalence relations, of measure pre-
serving ergodic group actions, and of continuous groupactions on compact spaces.
At the same time, Bernoulli flows have very simple definitions yet their dynamical
propertiesareverydifficulttofullyunderstand. Aparticularlyniceandmanytimes
useful aspect of Bernoulli flows is that they are susceptible to combinatorial argu-
ments, something which is typically not seen in other dynamical systems. Indeed,
combinatorial approaches are a predominant feature both in symbolic dynamics
and in this paper. Another source of motivation for studying Bernoulli flows is to
understand the relationship between the algebraic properties of the acting group
and the dynamical properties of the Bernoulli flow (a research program suggested
by Gottschalk in [Go]). There are some known results of this type. For example,
with complete knowledge of the dynamical properties of a Bernoulli flow KG, one
candetermineifGisamenable([CFW]),ifGhasKazhdan’sproperty(T)([GW]),
and the rank of G if G is a nonabelian free group ([Ga]), to name a few. This is
another aspect of Bernoulliflows which appears on severaloccasions in this paper.
Finally, in topological dynamics Bernoulli flows are also studied in order to reveal
properties of the greatest ambit of G, since it is known that the greatest ambit of
G is the enveloping semigroup of the Bernoulli flow 0,1 G (see [GU]).
{ }
In this paper we study the dynamics of Bernoulli flows from the symbolic and
topologicalviewpointsandemployideasfromdescriptivesettheorytogainfurther
understanding. Although we do not study Bernoulli flows from the ergodic theory
perspective, there is a topic we study (tileability properties of groups)which could
be of interest to researchersin ergodic theory and geometric group theory.
1.2. Basic notions
We study Bernoulli flows from the symbolic and topological perspectives. We
therefore only want to consider Bernoulli flows over finite alphabets (these are
precisely the compact Bernoulli flows, as mentioned in the previous section). So
throughout the paper the term “Bernoulli flow” will always mean “Bernoulli flow
over a finite alphabet.” We will also not make use of any measures (aside from
a single lemma). So we will never specify measures on the alphabets or on the
Bernoulli flows. Since the alphabet K is always finite and the particular elements
of K are unimportant, we will always use K = 0,1,...,k 1 for some positive
{ − }
integer k >1. As is common in logic and descriptive set theory, we let the positive
integerkdenotetheset 0,1,...,k 1 . WethereforewritekG = 0,1,...,k 1 G.
{ − } { − }
Let G be a countable groupand let X be a compact Hausdorffspace on which
G acts continuously (such as the Bernoulli flow kG). A closed subset of X which
is stable under the group action is called a subflow of X. We denote the closure of
sets A X by A. If x X, then the orbit of x is denoted
⊆ ∈
[x]= g x : g G .
{ · ∈ }
4 1. INTRODUCTION
Notice that [x] is the smallest subflow of X containing x X. If g G 1 ,
G
∈ ∈ −{ }
x X, and g x = x then we call g a period of x. We call x X periodic
∈ · ∈
if it has a period and otherwise we call x aperiodic (notice that here “periodic”
and“aperiodic”differ fromconventionaluse sincemostcommonly these twoterms
relate to whether or not the orbit of x is finite). A subflow of X is called free
if it consists entirely of aperiodic points, and x X is called hyper aperiodic if
∈
[x] is free (in [DS] such points are called limit aperiodic). In the specific case
where X is the Bernoulli flow kG, we use k-coloring interchangeably with “hyper
aperiodic.” Notice that x X is hyper aperiodic if and only if x is contained in
∈
some free subflow, and furthermore the collection of all hyper aperiodic points is
precisely the union of the collection of free subflows. A subflow S X is minimal
⊆
if [s] = S for all s S. Similarly, a point x X is minimal if [x] is minimal (this
∈ ∈
again differs from conventional terminology, since such points x are usually called
“almost periodic”). Two points x,y X are called orthogonal if [x] and [y] are
∈
disjoint. Finally, two subflows S ,S X are topologically conjugate if there is
1 2
⊆
a homeomorphism φ : S S which commutes with the action of G (meaning
1 2
→
φ(g s) = g φ(s) for all g G and s S ). From the viewpoint of symbolic and
1
· · ∈ ∈
topological dynamics, topologically conjugate subflows are essentially identical.
Asmentionedintheprevioussection,ausefulpropertyofBernoulliflowsisthat
many topological and dynamical properties are found to have equivalent combina-
torialcharacterizations. In fact, it is knownthat hyper aperiodicity, orthogonality,
minimality,andtopologicalconjugacycanallbeexpressedinacombinatorialfash-
ion. We heavily rely on the combinatorial characterizations of these properties
within the paper, and as a convenience to the reader we include proofs of these
characterizations. Our heavy use of the combinatorial characterization of hyper
aperiodicity led us to frequently use the term “k-coloring” in place of “hyper ape-
riodic.” The term emphasizes the combinatorial condition and is also reminiscent
of the term “coloring” in graph theory as both roughly mean “nearby things look
different.” We use the term “hyper aperiodic” within this chapter in order to em-
phasize the dynamical property as well as to avoid the possibility of the reader
confusing “k-colorings”with arbitrary elements of kG.
Nowhavinggonethroughthebasicdefinitions,letusrepeatthesecondsentence
of this introduction. The overall focus of this paper is on the development and
applicationof constructivemethods for Bernoulliflows,with a particularemphasis
on questions surrounding free subflows.
1.3. Existence of free subflows
The most basic, natural, and fundamental question one can ask about free
subflows is:
Does every Bernoulli flow contain a free subflow? Equivalently,
does every Bernoulli flow contain a hyper aperiodic point?
This question is an important source of motivation for this paper, so we discuss it
here at some length. One may at first hope that this question is answered by an
existential measure theory or Baire category argument. Indeed, a promising well
known fact is that the collection of aperiodic points in a Bernoulli flow always has
full measure and is comeager (i.e. second category, the countable intersection of
dense open sets). However, it is not clear if a comeager set of full measure must
contain a subflow, and furthermore a simple argument (included here in Section