The many friends of
Thompson’s group F
The participants of AIM’s conference,
photo by Matt Brin.
John Meier, Lafayette College
Atelier autour des groupes de Thompson
Tatihou, France, 9/2004
1
F ’s friend T
Thompson’s group T consists of PL homeomorphisms of S 1 = [0, 1]/0 ∼ 1 such that
• The maps take dyadic rationals to dyadic rationals; and
• The maps are differentiable at all but finitely
many singularities, which occur at dyadic rationals; and
• Away from the break points the derivatives
are all powers of 2.
A typical element of T might look like:
2
Remark. Every element of F satisfies these
conditions, and in fact F ,→ T .
Theorem. Thompson’s group T is generated
by x0, x1 (the generators of F ) and the element
c illustrated below.
3
One can make tree diagrams for elements of
T . A standard way to do this is to use a small
circle on a leaf in the range tree to denote the
target of the leftmost leaf of the domain tree.
For example, the diagram for c is:
and we can also represent this element on a
cylinder, where the left and right edges are
identified:
4
Surprisingly ...
Theorem.
T is conjugate to a group of
diffeomorphisms of S 1.
[Ghys & Sergiescu, Comm. Math. Helv., 1987]
They also determine the rational cohomology:
H ∗(T, Q) = Q[α, χ]/{α · χ = 0, |α| = |χ| = 2}
The group T and the next group V were the
first known examples of the following kind:
Theorem. The groups T and V are finitely presented, infinite simple groups.
[Thompson, widely circulated but unpublished notes;
details written down in Cannon-Floyd-Parry]
5
F ’s friend V
Here are two ways one can think about V .
A. V is just like T except that you only assume
continuity from the right. (So F ,→ T ,→ V .)
This can also be represented by pairs of trees
with a matching of their leaves.
1
2
3
2
1
3
or
6
B. V is the group of dyadic homeomorphisms
of the Cantor set C.
One can think of C as the end result of the
infinite recursive process
[0, 1] → [0,1/3]∪[2/3,1]
→ [0,1/9]∪[2/9,3/9]∪[6/9,7/9]∪[8/9,1] → · · ·
C is also {L, R} × {L, R} × {L, R} × · · · .
Rooted binary trees describe partitions of C
into copies of itself. The tree
L
RL
RR
corresponds to
C = CL ∪ CRL ∪ CRR
where for example CRL can be thought of either as all strings of Rs and Ls that begin
“RL · · · ” or as points in C that are contained
in [6/9, 7/9].
7
Given two such partitions of C into n pieces,
and a permutation π ∈ Σn, one can form a
homeomorphism of C. Taking both decompositions to be
C = CL ∪ CRL ∪ CRR
and setting π = (12) ∈ Σ3, one gets:
which is the same element of V as
or
1
2
3
2
1
3
or
Restricted to any of the sub-Cantor sets, such
a homeomorphism is an affine transformation.
8
Remark. Thompson’s group V contains a copy
of every finite group.
Why? Let T be the binary tree with n leaves
formed by continually adding carets to the right
edge. Then the elements of V described by
tree diagrams with both trees ' T with any
permutation of the leaves, are all non-trivial.
Thus Σn ,→ V .
1
2
3
4
1
3
4
2
(The permutation (1342) ∈ V )
9
Why is V simple?
Here’s the outline of a short argument that I
learned from Matt Brin (who says it came from
Mati Rubin). Let T be a rooted, binary tree
with at least three leaves. Call the element
of V formed by transposing two leaves of T , a
proper transposition.
1
4
2
3
4
3
2
1
10
Lemma: The proper transpositions generate
V.
Lemma: The proper transpositions are all conjugate.
Lemma: Given any v ∈ V , the normal closure
of hvi contains a proper transposition.
To see why this last lemma is true ...
Every element v ∈ V can be represented with
the domain and range sufficiently refined that
some subCantor set is moved off of itself by v.
Call the initial green subCantor set the domain and the terminal green subCantor set the
range.
11
Let g be the element of V that transposes the
domain. Then h = [g, v] transposes the domain
and range
The transposition g
The commutator h = [g, v]
Let k transpose the left half of the domain and
the left half of the range. Then the element
[k, h] = [k, [g, v]] is a proper transposition and
is in the normal closure of v.
The transposition k
The commutator [k, h]
[This argument is written up in Brin’s Higher dimensional Thompson groups paper. There is a general result of Epstein (Comp. Math, 1970) that essentially says
one should expect V and “groups like V ” to be simple.]
12
Higman’s groups
Higman: Why not look at versions of V that
are based on n-ary trees, not just binary trees?
And why not start with a planted forest of r
trees, not just one tree?
Answer: You certainly can. Denote the resulting group Vn,r .
For example, the group of PL homeomorphisms
of [0, 2] that are right continuous, preserve
Z[1/3] ∩ [0, 2], have finitely many break points
at triadic rationals, and have slopes powers of
3, is denoted V3,2. An element of V3,2 is indicated in the forest diagram below.
1
2 3
4
4
2
6
5
6
3
1
5
13
Theorem. The groups Vn,r are simple if n is
even, and contain a simple subgroup of index
2 if n is odd.
[Higman, 1974]
Thompson’s original group V is V2,1, hence this
generalizes Thompson’s result.
Of course one can also add the parameters n
and r to Thompson’s groups F and T . Thus
for fixed n and r one has Fn,r ,→ Tn,r ,→ Vn,r .
Ken Brown established various simplicity results along the lines above, as well as providing
a single proof-scheme that establishes:
Theorem. The groups Fn,r , Tn,r and Vn,r all
admit K(π, 1)s with finitely many cells in each
dimension.
[Brown, JPAA, 1987]
14
Higman studied the isomorphism types of the
Vn,r . It is easy to see that if r = s modulo
(n − 1) then Vn,r ' Vn,s. [Because one can split
one generator into n generators.] But there is
more going on than this.
In order to quote his results, we need one bit
of terminology. Let Zn−1 be the ring of integers modulo (n − 1). Let Un−1 be its group of
units, and let Dn−1 be the subgroup of Un−1
generated by divisors of n.
There are three main results:
• Vn,r 6' Vm,s if n 6= m.
• If r = s modulo Un−1 then Vn,r ' Vn,s.
• If Vn,r ' Vn,s then r = s modulo Dn−1.
[My source for this is Brin & Guzmán, J. Algebra, 1998]
Question: What’s the complete classification
up to isomorphism? Quasi-isometry?
15
Stein’s groups
Let P = hn1, . . . , nk i be the multiplicative subgroup of (0, ∞) generated by distinct positive
integers {n1, . . . , nk }. Let l ∈ A = Z[1/n1 · · · nk ].
F (l, hn1, . . . , nk i) is the group of PL homeomorphisms of [0, l], taking elements of A to
elements of A, with finitely many singularities
(all in A) and slopes in P .
T (l, hn1, . . . , nk i) is the group of PL homeomorphisms of S 1 = [0, l]/0 ∼ l with finitely many
singularities (all in A), with slopes in P .
V (l, hn1, . . . , nk i) is the group of right continuous bijections of S 1 = [0, l]/0 ∼ l that are PL
except at finitely many break points (all in A),
with slopes in P .
Theorem. All of these groups admit K(π, 1)s
with finitely many cells in each dimension.
[Stein, TAMS, 1992]
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Stein’s complexes are sufficiently concrete that
she can do a number of interesting cohomology
calculations.
Pick two distinct positive integers n1 and n2
and let d = gcd(n1 − 1, n2 − 1). Abbreviate the
name of F (1, hn1, n2i) to Fn1,n2 .
Theorem. The homology groups Hi(Fn1,n2 )
are free abelian. If hi is the rank of Hi(Fn1,n2 )
then:
h0 = 1, h1 = 2(d + 1), h2 = (1 + 4d)(d + 1)
and
hj = dhj−2 + 2dhj−1 for j > 2 .
For example, H0(F5,7) = Z, H1(F5,7) = Z6,
H2(F5,7) = Z27, H3(F5,7) = Z66, usw.
Cleary has shown that you don’t have to restrict to
rational slopes.
17
BNS Invariants
Bieri, Neumann and Strebel introduced a geometric invariant for finitely generated groups.
The analysis of what these invariants look like
for Thompson’s group F and the related groups
that Stein studied provided important examples.
Theorem. Let φ : F → Z. Then the kernel of
φ is finitely generated except if φ(x1) = 0 or
φ(x0) = φ(x1). The commutator subgroup is
not finitely generated.
Contrast this with:
Theorem. The kernel of any map φ : F2,3 → Z
is finitely generated. However, the commutator subgroup of F2,3 is not finitely generated.
[See the Bieri-Neumann-Strebel reference and the Brown
reference, both in Invent. math., 1987]
18
Braided V
The picture of V in terms of permuting subCantor sets of C makes one think about what
happens if one puts braids in the middle instead
of permutations.
This group has shown up (independently) in a
number of different places.
19
Matt Brin discovered braided-V , denoted BV ,
while thinking about coherence of braided categories. Patrick Dehornoy recently ran into the
same group from similar motivations, namely
studying the structure group of the associative
law combined with a twisted version of commutativity.
Theorem. BV admits a finite presentation.
(There is a presentation with 3 generators and
18 relations)
[Brin, preprint]
By its description it is not too surprising that
BV injects into the mapping class group of
a disk minus a Cantor set (Dehornoy). The
group BV also injects into the “universal mapping class group” constructed by Funar and
Kapoudjian.
There is another braided variant of V introduced by Greenberg and Sergiescu [CMH, 1991]
that is different from BV .
20
Higher dimensional V s
Matt Brin also introduced higher dimensional
versions of V . I’ll concentrate on 2V .
Just as C can be partitioned into sub-Cantor
sets, one can decompose C × C into (affine)
copies of itself.
We should be fairly specific about what decompositions are allowed.
21
A pattern is formed by taking the unit square,
dividing it in half either horizontally or vertically, and then repeating this process on the
subrectangles a finite number of times. Here
are two patterns:
(The one on the left indicates the process used to make
a pattern is not unique.)
You can form numbered patterns
1
3
3
2
4
5
5
1
4
2
22
Given two numbered patterns with the same
number of pieces, one can form the associated
element of 2V .
1
3
3
2
4
5
5
1
4
2
gives
Theorem. The group 2V is an infinite simple
group, it admits a finite presentation, and it is
not isomorphic to any Vn,r .
[Brin, preprints]
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How does Brin show 2V 6' V ?
Let β ∈ 2V be the “Baker’s map” shown below.
2
1
2
1
Lemma. There is no bound on the size of a
finite orbit under the action of the Baker’s map
on C × C.
Proposition. Given any v ∈ V there is a number
n(v) such that any orbit under v y C contains
at most n(v) points.
∼
If 2V → V , then there is an induced action
2V y C, and vice-versa. Brin then quotes a
theorem of Rubin to show that the “dynamical system” results above are then impossible,
as these dynamics are invariant in passing between “reasonable” actions.
24
Cinquième présentation
(F is a diagram group)
Consider the semigroup hx | x2 = xi. Using
the relation x2 = x one can build sequences of
words that start and end with x. These can
be encoded geometrically:
If you add “mirror image cancellation” to this
structure, for example:
you get a group (called a Diagram group). For
hx | x2 = xi, the group is Thompson’s F .
25
This presentation leads to a nice construction
of a cube complex C on which F acts freely.
The vertices are diagrams with top edge x and
bottom label any string of x’s. One forms
cubes when disjoint copies of x2 = x are attached at the bottom.
The group F acts “on the top” preserving the
cubical structure defined by adding cells on the
bottom.
x
=
26
Theorem. This cube complex C is a contractible,
locally finite, CAT(0) cube complex.
[The complex dates back to work of Brown and is presented in a more general context in Stein, TAMS, 1992.
That C is CAT(0) and that this construction works much
more generally is due to Farley, Topology, 2003]
The action F y C is not cocompact. However, if one defines Cn to be all cubes with at
most n x’s on the bottoms of the associated
diagrams, then F y Cn is cocompact. Further,
the connectivity of Cn increases as n increases.
The Baum-Connes conjucture claims that a
certain map between the algebraic K-theory
and the topological K-theory—for any countable group G—is an isomorphism. An immediate Corollary of Farley’s result is:
Corollary. Thompson’s group F satisfies the
Baum-Connes conjecture.
27
A few more references
Bieri, Neumann and Strebel: A geometric invariant of
discrete groups, Invent. math. 90 (1987) 451–477.
M. Brin: The algebra of strand splitting I and II, preprints.
M. Brin: Higher dimensional Thompson groups, preprint.
M. Brin and F. Guzmán: Automorphisms of generalized
Thompson groups, J. Algebra 203 (1998) 285–348.
K. Brown: Trees, valuations and the Bieri-NeumannStrebel invariant, Invent. math. 90 (1987) 479–504.
K. Brown: Finiteness properties of groups, JPAA 44
(1987) 45–75.
S. Cleary: Regular subdivision in
44 (2000) 453–464.
√
1+ 5
Z[ 2 ],
Illinois J. Math.
D. Epstein: The simplicity of certain groups of homeomorphisms, Compositio Math. 22 (1970) 165–173.
E. Ghys and V. Sergiescu: Sur un groupe remarquable
de difféomorphisms du cercle, Comm. Math. Helv. 62
(1987) 185–239.
V. Guba and M. Sapir: Diagram groups, MAMS 130
(1997) no. 620.
G. Higman: Finitely presented infinite simple groups,
Notes on Pure Mathematics 8, ANU, Canberra (1974).
M. Stein: Groups of piecewise linear homeomorphisms,
TAMS 332 (1992) 477–514.
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