Research Statement
Collin Bleak
1
Introduction
My research primarily uses close analysis of the dynamics of a group or semigroup action on a space
to discern properties of the group or semigroup. In particular, I am primarily interested in groups
of automorphisms of a space and in semigroups of endomorphisms of a space. Supporting this, I have a
secondary focus on topological and symbolic dynamics, which itself has lead me to an interest in automata
and formal language theory.
Most of my past and current work centers around studies of large groups of homeomorphisms. Typical
groups of interest are PL(I n ), Homeo(S 1 ), and Homeo(C), the groups of piecewise-linear homeomorphisms
of the n-cube I n = [0, 1]n , the group of homeomorphisms of the unit circle S 1 , and the group of homeomorphisms of the Cantor Set C. A consequence of this is that many of my results have direct bearing on
the theory of the family of R. Thompson groups F ≤ T ≤ V and their relatives.
In the discussion which follows, I will primarily describe the impact of my work on the family of
“Thompson-like” groups. Along the way I will also mention other areas impacted by my research.
2
Some theory for the R. Thompson’s groups F ≤ T ≤ V
Richard Thompson defined the groups F , T , and V in 1965. In [45], he and McKenzie use V to construct
finitely presented groups with unsolvable word problems. In his unpublished notes, Thompson shows that
T and V are finitely-presented, infinite simple groups (these were the first such examples).
For each of the groups F , T , and V , we will briefly discuss some of their known properties, and
then point to the (admittedly incomplete) bibliography for the interested reader. The reference [28] is a
commonly studied introductory survey to the theory of the Thompson groups.
The group F has been the most thoroughly studied of these groups. It can be thought of as a subgroup
of the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval (denoted here
by P Lo (I)). The group F has a simple first derived group, is finitely presented, and is torsion free. F
satisfies no group laws, contains no non-abelian free subgroups [22], and is not elementary amenable [28].
F has a universal conjugacy idempotent, and is an infinitely iterated HNN extension [32, 25]. Brown and
Geoghegan also show in [25] that F has property F P∞ , and so give the first example of a torsion-free
infinite-dimensional F P∞ group. Also, F has integral homology groups free-abelian of rank two in every
positive dimension [24, 53]. Every non-abelian subgroup of F contains a copy of the standard restricted
wreath product Z ≀ Z [36, 12, 11]. Guba in [35] shows that F has quadratic Dehn function, while Burillo
in [26] shows that it has unknown but exponential growth rate [26]. The papers [30, 29, 5] show that F
has a Cayley graph (under the standard choice of generators) which is not combable by geodesics and
which is maximally nonconvex. Also, F has a solvable simultaneous conjugacy problem [38], and a very
fast solution to its word problem [36]. The group F has been studied in connection with cryptography
[51, 43]. The structures of the solvable subgroups of F are described by my work in [12, 11], while [14]
describes a non-solvable subgroup W of F which embeds in every non-solvable subgroup of F .
The group T has been less extensively studied than F . The group T can be represented as a subgroup
of the orientation-preserving piecewise-linear homeomorphisms of the circle (denoted P Lo (S 1 )), but T
can also be faithfully embedded in Dif f+∞(S 1 ), [33]. From this last perspective, there is quite a bit
known about T due to the efforts of mathematicians studying low-dimensional topology and dynamics.
For instance, it is known that every element in T has rational rotation number [33, 27, 41], hence, every
element of T admits a power with a fixed point. T does not satisfy the Tit’s alternative, but one can
apply Margulis’ theorem (see [34, 42]) to see that if H is a subgroup of T without free subgroups, then
H supports an invariant probability measure. The conjugacy and simultaneous conjugacy problems for T
are solved in [6, 15]. The automorphism groups of both T and F are classified in [20]. In [16] we classify
the structure of the solvable subgroups of T . This last paper also contains a Tit’s Alternative type of
classification of various subgroups of T and separately of Homeo(S 1 ) in the solvable and non-solvable
cases.
The group V can be thought of as a subgroup of the self-homeomorphism group of the Cantor set C.
The group V is infinite, simple, finitely presented, and contains copies of every finite group. The theory of
V interacts with the theory of interval exchange maps from dynamics [3], algorithmic group theory [40],
circuit complexity[9, 8, 7], and logic and combinatorial group theory [45] in the large. For each positive
integer index n, V = 1V has a natural generalization to a higher-dimensional analogue group nV , as well
as other forms of generalization springing from its other main description as the group of automorphisms
of a particular algebra [37]. The conjugacy problem has been solved in V in three separate, but related
ways [37, 49, 6], and many of the homological properties of V and its generalizations (excluding the groups
nV ) have been investigated [23]. Recently it has been shown that V has a context free co-word problem
[40]. The structure of element centralizers for arbitrary elements of V was classified by my research group
in the Cornell 2007 summer REU project [1]. Finally, while V contains many copies of non-abelian free
groups, and other free products of groups such as PSL(2, Z) ∼
= Z2 ∗ Z3 , V has some perhaps surprising
restrictions on the free products it contains; it contains many copies of Z × Z, but it does not contain a
copy of Z ∗ (Z 2 ) [19].
2.1
A general discussion of my research program
My research primarily flows along two related streams. One of these uses dynamics to analyze groups of
homeomorphisms, as alluded above, and is inspired by a theorem of M. Rubin. The other project relates
to building new geometric invariants for monoids in the spirit of the constructions in the main body of
geometric group theory.
In the later sections of this statement, I will focus primarily on the first stream, as the discussion
of the second stream below seems sufficient to describe its nature and to suggest its potential long-term
impact.
2.1.1
Ideas which flow from Rubin’s theorem
The first stream involves the study of groups that can be realized as large groups of homeomorphisms.
(H(X) below will denote a group of homeomorphisms of a space X.) This research is primarily motivated
by a beautiful theorem of M. Rubin (proved using model theory in [48]).
Rubin’s theorem roughly states that if φ is an isomorphism between two groups acting “nicely” on
two “friendly” spaces, then φ is induced by a topological homeomorphism of the spaces. This theorem
essentially implies that any algebraic property of a group which acts nicely on a friendly space can be
discovered through a careful analysis of the dynamics of the action.
We thus have the following general outline of research.
1. Use a careful study of the dynamics of H(X)’s action on X to understand how elements interact
(do they commute, are they conjugate, etc.)
2. Attempt to classify the groups of germs of the action of H(X) on X.
3. Explore the subgroup structure of H(X) by considering what subgroups satisfy specific dynamical
criteria.
4. Test the effectiveness of the above three projects by using their results to support work on hard
open questions for the group in question or by finding new proofs of known hard theorems. Typical
test projects are as follows.
(a) Classify the structure of the standard subgroups of H(X) (eg., element centralizers).
(b) Answer standard problems from algorithmic group theory for H(X) (eg., the conjugacy or
membership problems).
2
(c) Answer embedding and isomorphism questions related to H(X).
(d) Understand the automorphism group of H(X).
2.1.2
Ideas which flow from Squier’s FDT construction
The second stream is closer to pure geometric group theory, although in my case I have been applying it
to the study of finiteness properties of monoids. This stream of work has developed over the last two years
during a collaboration with Susan Hermiller. We have been working on extending Squier’s definition of
Finite Derivation Type (FDT) for monoids.
In the original definition, Squier associated a specific 2-complex Σ2 with a presentation. The group or
monoid then has FDT if, modulo the action of the free group/monoid ring, the one-dimension homotopy
of Σ2 is finitely generated.
FDT is a finiteness property for monoids, which has some correspondence to FP3 for groups. (In fact,
a group has FDT if and only if it has FP3 [31].) There has been work to extend the concept of FDT
(FDT3 in our notation) to higher indices, producing properties FDTn for monoids. To date, there has
been no full extension of the concept of FDT3 beyond FDT4 (this work in McGlashan’s dissertation is
reported in the paper [44]). Weaker properties such as “homological finite derivation type” have been
defined for all n in [46, 47, 31, 2].
The Squire complex Σ2 has a second use as a foundation for the theory of diagram groups [39, 36].
Diagram groups are an important class of groups which are computed as fundamental groups of Squier
complexes.
With Susan Hermiller, I have been working to extend the research she and John Meier began, where
they provide an independent definition of FDT4 and significant work towards a definition of FDT5 . We
have been working to build an inductive definition of FDTn for all natural n > 1 and have made significant
progress in this work to date.
If it turns out that we can succeed in building these two-complexes and in checking if a monoid has
property FDTn , then we will be able to define “higher isoperimetric functions” for monoids, and new
flavors of “higher diagram groups”. In any case, the process under scrutiny produces the same results for
monoids in terms of FDT3 and FDT4 as the original definitions.
3
Results and projects related to the dynamics approach
That part of my research which is motivated in part by Rubin’s theorem also splits into two primary
areas. The first area has direct bearing on the Thompson-like family of groups, while the second area has
bearing on properties of actions of groups on high dimensional manifolds.
3.1
Results and projects involving the Thompson family of groups
For the Thompson family of groups, one can see aspects of all four tests of the dynamical approach in
my results. I will discuss briefly the dynamical side of a family of related results, and then I will make a
short list of those results.
3.1.1
Dynamics on the interval
For groups of homeomorphisms of R or the interval I (such as PLo (I) and Thmpson’s group F ), one can
discover many algebraic facts by looking at the dynamics associated with how the supports of elements
overlap. Almost all of the results in this section are derived using techniques related to studies along
these lines.
In order to state some of these results, we need to define various groups.
Let G0 be the trivial group, and define
M
Gn =
(Gn−1 ≀ Z)
(here, n is not the sum index)
Z
3
where “≀” denotes the standard resticted wreath product of groups. Similarly, define W0 to be the trivial
group, and
Wn = Wn−1 ≀ Z
Now define W = ⊕n∈N Wn .
A close analysis of groups in PLo (I) with all element supports being nested with respect to other
elements produces the following result. (Note that all results below still apply if we globally replace
“PLo (I)” with “F ”.)
Theorem 3.1 ([11]) Suppose H is a solvable group with derived length n. H embeds in PLo (I) if and
only if H embeds in Gn .
A consequence of this is the result that Z ≀(Z 2 ) does not embed in PLo (I), which answers a question of
Sapir. I provided a dynamical proof of this second result in [12] while Ol’shanski independently produced
the same answer to Sapir’s question using an algebraic proof based on the fact that Z ≀ (Z 2 ) does not
embed in any Gn .
One can produce the following result (with some effort) by allowing more complex support overlaps.
Theorem 3.2 (Bleak [14]) If H ≤ PLo (I) where H is non-solvable, then W embeds in H.
A neat fact which arises from the previous result is that, as all of the groups Gn embed in W , every
solvable subgroup of F embeds in every non-solvable subgroup of F !
The previous result can be stengthened if one passes to finitely generated subgroups of PLo (I). The
group B mentioned below is used in [21] as a model for generating elementary amenable non-solvable
subgroups in PLo (I) (Brin calls this group G1 ).
Theorem 3.3 (Bleak [10]) Suppose H is finitely generated and H embeds in PLo (I), then B embeds in
H.
3.1.2
Dynamics on the circle
For orientation preserving homemorphism groups of the circle, one can take advantage of various dynamical properties; there is a dichotomy of behavior. If there is no non-abelian free subgroup, one can use
a careful study of dynamics to deduce that the rotation number map Rot: Homeo+ (S 1 )→ Q/Z restricts
over the group in hand to a homomorphism. In this case, the elements of the group act with simpler
dynamics, and one obtains the following Tit’s Alternative type of theorem. The following theorem represents some of my joint work with Martin Kassabov and Francesco Matucci. Many of these results (in
factor form) are known by work of Beklaryan [4] and Solodov [52], although our method is different.
Theorem 3.4 ([16]) Let G ∈ Homeoo (S 1 ), with no non-abelian free subgroups. Then there are subgroups
H0 and Q of Homeoo (S 1 ), such that
G ֒→ H0 ≀ Q
that is, G embeds in the unrestricted wreath product of H0 and G. Further, we have the following five
properties:
1. Q ∼
= G/ (ker (Rot) ∩ G) is isomorphic to a subgroup of R/Z, which is at most countable if ker (Rot) ∩
G is non-trivial, and
Q
2. H0 embeds into N Homeoo (I), where N is a countable (possibly finite) index set
3. H0 has no non-abelian free subgroups,
4. H0 is trivial if ker (Rot) ∩ G is trivial, and
4
5. the subgroups H0 , Q ≤ Homeoo (S 1 ) generate a subgroup isomorphic to the restricted wreath product
H0 ≀ Q. This subgroup can be “extended” to an embedding of the unrestricted wreath product into
Homeoo (S 1 ) where the embedded extension contains G.
Many other interesting results flow easily from this result. One such is Margulis’ theorem that a
subgroup H of Homeo+ (S 1 ) which admits no free subgroups must act in such a way as to support an
invariant probability measure on the circle (see [42]). A second result is an extension of the classification
of the solvable subgroups of PLo (I) (and of F ) to a classification of the solvable subgroups of PLo (S 1 )
(and of T ); they are subgroups of wreath products of the groups Gn mentioned before with the group
Q/Z.
In separate work with Kassabov and Matucci (see [15]), we classify roots and element centralizers
for certain groups of homeomorphisms of the circle (including T ). This is an important step towards
providing an algorithm solving the simultaneous conjugacy problem for T .
3.1.3
Dynamics on the Cantor Set
As mentioned before, the groups V and nV (for natural n) are groups of homoeomorphisms of the Cantor
Set C. My results to date come from analysis of the dynamics of element actions on C. I am unaware of
any purely algebraic method for discovering any of the following results.
The first result is obtained by calculating the local groups of germs of the action of nV on C. Dan
Lanoue and I jointly thus found a short answer to a question of Brin.
Theorem 3.5 ([18]) Given two indices m 6= n, the groups mV and nV are not isomorphic.
The second result comes by building a geometric object (the flow graph) modeling element dynamics
(flow graphs label attracting and repelling periodic orbits, and describe flows from sources to sinks) and
studying what actions commute with actions along the flow graph. This is joint with Hannah AbbotSmith, Alison Gordon, Garret Graham, Jacob Hughes, Francesco Matucci, and Jenya Sapir.
Theorem 3.6 ([1])
Let n > 1 be a positive integer and suppose α ∈ Vn . Let CVn (α) = {β ∈ Vn | αβ = βα} denote the
centralizer in Vn of α. Then, there are non-negative integers s, m1 , m2 , . . . , ms , and t so that
t
s
Y
Y
Ej ) ⋊ Pt ).
⋊
V
)
×
((
M
CV (α) ∼
(
=
mi
j=1
i=1
The above needs some explanation.
Each mi represents a specific finite cycle size, and Ci represents a fundamental domain of the action
of α restricted to the subset of C where α acts as a torsion element with order mi . For each mi , we
have Mmi = M aps(C, Cmi ), where Cmi is the cyclic group on mi elements with the discrete topology.
Finally, the number t is the number of components of the flow graph, and for each such component, Ej
is an extension of Z by that component’s finite group of flow graph automorphisms, while Pt ≤ Σt is a
permutation group swapping isomorphic flow graph components.
The final results arise as consequences of building techniques to modify the repeller-to-attractor flow
structure of elements in V in a controlled fashion, through the use of various algebraic processes.
The results are stated in counterpoint to each other, as they draw a nice dichotomy of behavior; on
one hand, V is writhe with free products, but on the other, there are some startling restrictions on which
free products actually occur.
Let A be the smallest class of groups which contains all finite groups, contains the integers, and
contains Q/Z and which is closed under isomorphism, passing to subgroup, and taking direct products of
any finite member by any member.
Olga Salazar-Dı́az and I show the following.
5
Theorem 3.7 ([19])
If K1 , K2 ∈ A, then K1 ∗ K2 embeds in V .
On the other hand
Theorem 3.8 ([19])
The group Z 2 ∗ Z does not embed in V .
This last work answered a question of Sapir.
3.2
Further work
There are quite a few questions about F , T , and V which are interesting, and where I think my developing
techniques have bearing. However, there are three particular questions which refuse to let me go. I will
mention these questions, and briefly describe my work to date on them.
Question 1 Must every subgroup of F which is not elementary amenable contain a copy of F ?
It is a conjecture of Brin in [21] that the answer to this question is “Yes.” Sapir also finds this question
of interest, as he posted it in his list of interesting group theory questions in [50].
In any case, the dynamical tools developed in [14] have allowed me to reduce the truth of the conjecture
down to a careful study of a small specific family S of two-generator subgroups of F . There are now many
restrictions on the generators. If I can show that a copy of F must occur in every group in S, then the
conjecture is true. If there is an example of a group in S without a copy of F , then it is likely that the
conjecture is false, and the resulting group will represent a strange and interesting subgroup in F .
Question 2 (The Membership Problem) Is there an algorithm which, when given a list {g1 , g2 , . . . , gn+1 }
of n + 1 elements in F , determines if the element gn+1 is in the group H = hg1 , g2 , . . . , gn i?
I have done significant work on this question. Firstly, there is an algorithm which detects if H is solvable,
and which can also determine the derived length of H.
If H is solvable, then the answer to the question will depend on whether or not there exists an
algorithm to detect if (. . . ((Z ≀ Z) ≀ Z) . . .) ≀ Z has an algorithm to solve its membership problem.
If H is not solvable, it becomes easier to find an algorithm; if H contains a copy of F , then it is easier
to build gn+1 as a product of elements in H by the nature of F . If H does not have a copy of F , then
the answer quickly reduces to the existence of an algorithm very similar to the one which would solve
the question for H solvable, if such an algortihm exists. In these last two cases, the real difficulty lies
in a tricky intersection problem; one potential algorithm solves the question over the components of the
support of H individually. After that, one needs to detect if these problems can be solved simultaneously.
Question 3 What is the group Out(V )?
In the Summer 2008 REU, my student Daniel Lanuoe and I spent four weeks understanding properties
of this automorphism group. By Rubin’s theorem, any such automorphism is carried by a conjugation of
V by a homeomorphism of C. We classified the local groups of germs of V (which are preserved by such
automorphisms), and understood why Brin’s classifications of the automorphism groups of F and T (see
[20]) could not be pushed forward in the case of V .
Over the Summer of 2009, while directing my REU students in other projects, I gave this project
further thought. I was able to show that any such automorphism could not be described by a transducer
(except the trivial automorphism and the automorphism of V generated by a conjugation using the
x → 1 − x homeomorphism of C). (A transducer can be thought of as a finite state automata that will
read an infinite string from some alphabet, and make regular string substitutions in order from a finite list
of substitutions.) This last result result greatly reduces the possibilities of viable automorphisms, given
the constraints that all such conjugating homeomorphisms must map V to V . In any case, I consider this
one of my most interesting and difficult projects.
6
3.3
Higher dimensional actions
My research in [12] showing Z ≀ (Z 2 ) does not embed in F (recall this answered a question of Mark Sapir
which came from the growing theory of diagram groups) lead to further study [13] where I show that
Z ≀ (Z 2 ) also does not embed in T (this occurred in response to a “continuing” question put to me by
Ken Brown). This overall work raised many other questions for me, and I became interested in higherdimensional analogues of these questions for general piecewise-linear manifolds. The discussion which
follows details some of that work.
For each integral dimension n > 1, fix polyhedrons Ipn and Spn which are homeomorphic to I n and S n ,
respectively, and where the polyhedron Spn has a natural decomposition as two polyhedra (intersecting
each other on their boundaries), each of which is piecewise-linearly homeomorphic with Ipn . Let ZWn be
the group Z ≀ (Z n ). The following is an easy theorem.
Theorem 3.9 For each positive integer n, ZWn embeds faithfully in the group of piecewise-linear selfhomeomorphisms of Ipn and of Spn .
On the other hand, the nature of the proofs of the existence of these embeddings, and the proofs of the
non-existence of the embeddings of ZW2 into PLo (I) and PLo (S 1 ), lead me to ask the following question
in [13].
Question 4 Is it true that there is no integer n > 1 so that ZWn embeds in either of the groups of
piecewise-linear self-homeomorphisms of the spaces Ipn−1 and Spn−1 ?
In the Summer 2009 Cornell REU, I continued investigating this question with my student Ben Krause.
If Fix(I n ) ≤ PL(I n ) is the subgroup of elements which act as the identity in a collar on the boundary
of I n , then we have shown that if Z n+1 is embedded in Fix(I n ) and x ∈ I n , then the orbit of x in I n
under the action of Z n+1 is induced from the action of a subgroup Z j ≤ Z n+1 for some j ≤ n ([17],
in preparation). This local obstruction in acting dimension is the key ingredient needed in the previous
non-embedding results, and so I am lead to make the following conjecture.
Conjecture 5 Let M be a closed, n-dimensional piecewise-linear manifold, for some natural number n.
There is no faithful piecewise-linear action of ZWn+1 on M .
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