We give examples of finitely presented groups containing elements with
irrational (in fact, transcendental) stable commutator length, thus answering
in the negative a question of M. Gromov. Our examples come from 1-dimensional
dynamics, and are related to the generalized Thompson groups studied by M.
Stein, I. Liousse and others.Comment: 8 pages, 1 figur

Zhuang, Dongping

English

arXiv

We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse and others

Caltech Authors - Main

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IRRATIONAL STABLE COMMUTATOR LENGTH IN FINITELY PRESENTED
GROUPS
DONGPING ZHUANG
ABSTRACT. We give examples of finitely presented groups containing elements with ir-
rational (in fact, transcendental) stable commutator length, thus answering in the negative
a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are
related to the generalized Thompson groups studied by M. Stein, I. Liousse and others.
1. INTRODUCTION
LetG be a group, and let a be an element of the commutator subgroup, which we always
denote by [G,G]. The commutator length of a, which we denote cl(a), is defined to be the
minimum number of commutators whose product is equal to a. That is,
cl(a) = min{n|a = [b1, c1] · · · [bn, cn]}
cl(a) is a subadditive function, so the limit of cl(a
n)
n , as n→∞, exists.
Definition 1.1. G, a ∈ [G,G], the stable commutator length of a, denoted by scl(a), is
defined to be
scl(a) = lim
n→∞
cl(an)
n
Set scl(a) =∞, if no power of a is in [G,G].
Commutator length and stable commutator length in groups have long been studied,
often under the name of the genus problem. If G = π1(V ) for some aspherical space V ,
and γ is a loop in V representing the conjugacy class of an element a, then the commutator
length of a is the minimal genus of a surface S for which there is a map f : S → V taking
∂S to γ. More generally, given a class H ∈ H2(V, γ), one can ask for the least genus
immersed surface in the relative class H . Stabilizing, one obtains a norm on H2(V, γ). If
G is finitely presented, H2(V, γ) is finitely dimensional, and one can try to minimize this
stable norm on the subspace which is the preimage of the class of [γ] under the boundary
map H2(V, γ)→ H1(γ). This infimum is the stable commutator length of a.
M. Gromov (in [10] 6.C2) asked the question of whether such a stable norm onH2(V, γ),
or in our content, the stable commutator length in a finitely presented group, is always
rational, or more generally, algebraic. The purpose of this note is to give simple (and
even natural) examples which show that the stable commutator length in finitely presented
groups can be transcendental.
We now state the contents of this note. In §2 we state the fundamental Duality Theorem
of C. Bavard [2], which gives a precise relationship between stable commutator length and
homogeneous quasimorphisms on groups. In §3 we give our examples and demonstrate
that they have the desired properties, which are based on the work of M. Stein [14], D.
Calegari [4] and I. Liousse [13]. The examples are central extensions of certain (finitely
generated) groups of piecewise linear homeomorphisms of the circle. C. Bavard’s duality
1
2 DONGPING ZHUANG
theorem connects dynamics (rotation numbers, as studied by I. Liousse [13]) with stable
commutator length. We include an appendix presenting basic properties of rotation num-
bers. They are used in §2 and §3.
1.1. Acknowledgements. The fact that these examples have irrational stable commutator
length was first conjectured by D. Calegari, who also suggested the problem of trying to
calculate stable commutator length exactly in certain finitely presented groups. I am very
grateful to Danny Calegari for giving generous support and advice during the preparation
of this work.
2. STABLE COMMUTATOR LENGTH AND QUASIMORPHISMS
Definition 2.1. Let G be a group. A quasimorphism on G is a function
φ : G→ R
for which there is a constant D(φ) ≥ 0 such that for any a, b ∈ G, we have an inequality
|φ(a) + φ(b) − φ(ab)| ≤ D(φ)
In other words, a quasimorphism is like a homomorphism up to a bounded error. The least
constant D(φ) with this property is called the defect of φ.
Definition 2.2. A quasimorphism is homogeneous if it satisfies the additional property
φ(an) = nφ(a)
for all a ∈ G and n ∈ Z.
A homogeneous quasimorphism is a class function by its definition. Denote the vector
space of all homogeneous quasimorphisms on G by Q(G).
Example 2.3. Let ˜Homeo+(S1) = {f ∈ Homeo+(R)|f(x + 1) = f(x) + 1}. It’s the set
consisting of all possible lifts of elements in Homeo+(S1) under the covering projection
π : R→ S1. We have the central extension:
0→ Z→ ˜Homeo+(S1)→ Homeo+(S1)→ 1
where Z is generated by the unit translation and p : ˜Homeo+(S1) → Homeo+(S1) is the
natural projection from its definition. For g ∈ ˜Homeo+(S1), define
rot(g) = lim
n→∞
gn(0)
n
With this definition, rot is a homogeneous quasimorphism with defect 1. (See appendix
for a proof.)
We have the following fundamental theorems of C. Bavard, which state the duality
between scl and homogeneous quasimorphisms. See C. Bavard [2] or D. Calegari [5] for
a reference.
Theorem 2.4 (C. Bavard).
(1) Let G be a group. Then for any a ∈ [G,G], we have an equality
scl(a) =
1
2
sup
φ∈Q(G)/H1(G;R)
|φ(a)|
D(φ)
IRRATIONAL STABLE COMMUTATOR LENGTH IN FINITELY PRESENTED GROUPS 3
(2) There is an exact sequence
0→ H1(G;R)→ Q(G)→ H2b (G;R)→ H
2(G;R)
(3) Let G be a group. Then the canonical map from bounded cohomology to ordinary
cohomology H2b (G;R) → H2(G;R) is injective if and only if the stable commu-
tator length vanishes on [G,G].
Given a group G, if Q(G) is small enough, we can use (1) of Theorem 2.4 to determine
scl. For the previous example G = ˜Homeo+(S1), we have dimRQ(G) = 1.
Proposition 2.5 (J. Barge and ´E. Ghys [1]). rot : H˜omeo+(S1) → R is the unique
homogeneous quasimorphism which sends the unit translation to 1.
Proof: Suppose τ ∈ Q( ˜Homeo+(S1)) is another such map , then we consider
rot− τ : ˜Homeo+(S1)→ R
which is also a homogeneous quasimorphism, and since any homogeneous quasimorphism
on abelian groups, especially Z⊕ Z, must be a homomorphism, we have
(rot − τ)(f1) = (rot − τ)(f2)
if p(f1) = p(f2). It therefore induces a homogeneous quasimorphism on Homeo+(S1),
denote it still by rot− τ : Homeo+(S1)→ R. But Homeo+(S1) is uniformly perfect, i.e.
cl is bounded ([9]), so the induced map is bounded. Since it’s homogeneous, it must be a
zero map, i.e. rot = τ
By Theorem 2.4 and Proposition 2.5 together, we have scl(g) = 12rot(g), for any g
in ˜Homeo+(S1). Suppose G is a subgroup of Homeo+(S1) which is uniformly perfect.
Let G˜ be the preimage of G in ˜Homeo+(S1). Then by the same argument, scl(a) =
rot(a)/2D for any a in G˜, where D is the defect of rot restricted to G˜.
Our goal therefore is to find the subgroups of Homeo+(S1) which are finitely presented
and uniformly perfect, and contain elements with interesting rotation numbers. As a first
example, consider Thompson’s well-known group of dyadic piecewise linear homeomor-
phisms.
Let G˜ consist of piecewise linear homeomorphisms f of R with the following proper-
ties:
(1) For each point xi of discontinuity of the derivative of f (hereafter a “break point”),
both xi and f(xi) are dyadic rational numbers (i.e. of the form p2q, p, q ∈ Z);
(2) The derivatives of the restrictions of f to (xi, xi+1) are powers of 2 (i.e. of the
form 2m,m ∈ Z);
(3) f preserves dyadic rational numbers and f(x+ 1) = f(x) + 1.
The elements of G˜ induce piecewise linear homeomorphisms of S1 ≃ R/Z. The collec-
tion of these homeomorphisms is the Thompson group G ([15], [6]). Thompson group is
simple, FP∞ and uniformly perfect([6], [7] and [8]), so we have (The defect is still 1. See
appendix.)
scl(a) =
1
2
rot(a), a ∈ G˜
About the rotation numbers of elements in the Thompson group, we have
Theorem 2.6 ( ´E. Ghys and V. Sergiescu [8]). rot(a) is rational for any a ∈ G˜.
4 DONGPING ZHUANG
So scl takes only rational values on the group G˜.
3. GENERALIZED THOMPSON GROUPS
Our definition of generalized Thompson groups is from [14] by M. Stein. Let P be
a multiplicative subgroup of the positive real numbers and let A be a ZP -submodule of
the reals with PA = A. Choose a number l ∈ A, l > 0. Let F (l, A, P ) be the group of
piecewise linear homeomorphisms of [0, l] with finitely many break points, all in A, having
slopes only in P . Similarly define T (l, A, P ) to be the group of piecewise linear homeo-
morphisms of [0, l]/{0, l} (the circle formed by identifying endpoints of the closed interval
[0, l]) with finitely many break points in A and slopes in P , with the additional requirement
that the homeomorphisms send A∩ [0, l] to itself. In these notations, T (1,Z[ 12 ], 〈2〉) is the
Thompson group. In our study of generalized Thompson groups, we always assume that P
is generated by the set of positive integers {n1, n2, . . . , nk} and A = Z[ 1n1 ,
1
n2
, . . . , 1nk ],
here {n1, n2, . . . , nk} forms a basis for P . An important theorem in studying generalized
Thompson groups is the following Bieri-Strebel criterion (See [14] appendix for a proof.).
Theorem 3.1 (R. Bieri and R. Strebel [3]). Let a, c, a′, c′ be elements of A with a < c and
a′ < c′. Then there exists f , a piecewise linear homeomorphism of R, with slopes in P
and finitely many break points, all in A, mapping [a, c] onto [a′, c′]⇔ c′ − a′ is congruent
to c− a modulo IP ∗A.
Here IP ∗A is the submodule of A generated by elements of the form (1− p)a, where
a ∈ A and p ∈ P . Let d = gcd(n1 − 1, . . . , nk − 1) and from now on, we assume that
d = 1. In this case, IP ∗A = P (dZ) = PZ = A, so the Bieri-Strebel criterion from Thm
3.1 is vacuously satisfied.
Take an arbitrary f ∈ T = T (l, A, P ) with the assumptions above. Choose points
a < b, c < d ∈ [0, l)∩A, f([a, b]) = [a1, b1] such that [c, d]∩([a, b]∪[a1, b1]) = ∅(This can
be achieved by taking [a, b] small enough.). We can find piecewise linear homeomorphisms
g1 and g2 with slopes in P and break points in A, sending [b, c] to [b1, c] and [d, a] to [d, a1]
respectively. Construct g ∈ T as follows.
g =


f if x ∈ [a, b]
g1 if x ∈ [b, c]
id if x ∈ [c, d]
g2 if x ∈ [d, a]
Now f = (fg−1)g, both f ◦ g−1 and g fix some nonempty open arcs.
Write f = g1g2 where gi’s fix some open arcs for i = 1, 2. For each such gi, there
exists a rotation through θi, Rθi ∈ T such that supp(Rθi ◦ gi ◦ R−θi) ⊆ (0, l). (Rθi ∈ T
if and only if θi ∈ A which is dense in [0, l], so we only need to choose θi in the open arc
fixed by gi.). Write hi = Rθi ◦ gi ◦R−θi , so hi ∈ F = F (l, A, P ) and hi lies furthermore
in the kernel of the following homomorphism
ρ : F → P × P
ϕ 7→ (ϕ′(0+), ϕ′(l−)), taking derivatives at endpoints. Let B = kerρ, so hi ∈ B.
Theorem 3.2 (M. Stein [14]). B′ is simple and B′ = F ′
Theorem 3.3 (K. S. Brown). H∗(F ) ∼= H∗(B)⊗H∗(P × P ).
(For a proof, see M. Stein [14].)
IRRATIONAL STABLE COMMUTATOR LENGTH IN FINITELY PRESENTED GROUPS 5
Let’s further assume that the slope group P has rank 2, i.e. P = 〈p, q〉, A = Z[ 1p ,
1
q ],
p, q ∈ Z+ and d = gcd(p − 1, q − 1) = 1. Fp,q, Tp,q are the corresponding groups. M.
Stein, in [14], computed the homology groups of Fp,q by using its action on a complex.
Theorem 3.4 (M. Stein). H1(Fp,q) is a free abelian group with rank 2(d + 1), where
d = gcd(p− 1, q − 1).
By assumption d = 1, rkZ(H1(Fp,q)) = 4. By Theorem 3.3, H1(Fp,q) ∼= H1(B) ⊕
H1(P × P ) and here P × P ∼= Z × Z × Z × Z, so rkZ(P × P ) = 4 = rkZ(H1(Fp,q)),
which implies that H1(B) = B/B′ is trivial. So B = B′ = F ′p,q .
So hi ∈ B = F ′p,q can be written as a product of commutators of Fp,q . So is gi, which
is conjugate to hi in Tp,q. So overall we proved that Tp,q is perfect, i.e. Tp,q = T ′p,q.
Let’s compute the set Q(Tp,q). For this purpose, we need the theorem of D. Calegari
[4] about the scl of elements in subgroups of PL+(I).
Theorem 3.5 (D. Calegari). Let G be a subgroup of PL+(I). Then the stable commutator
length of every element of [G,G] is zero.
Lemma 3.6. Let Tp,q = T (1,Z[ 1p ,
1
q ], 〈p, q〉) and d = gcd(p − 1, q − 1) = 1, then
Q(Tp,q) = {0}.
Proof: By the argument above, f = g1g2 and gi = R−θi ◦ hi ◦ Rθi , i = 1, 2. Since
every homogeneous quasimorphism φ is a class function, it follows that φ(gi) = φ(hi).
Since d = 1, hi ∈ F ′p,q ⊆ PL+(I) and Theorem 3.5 and Theorem 2.4 imply φ(hi) = 0, so
φ(gi) = 0. For any n, suppose fn = g1ng2n, then φ(g1n) = φ(g2n) = 0
|nφ(f)| = |φ(fn)| = |φ(g1ng2n)| = |φ(g1ng2n)− φ(g1n)− φ(g2n)| ≤ D(φ)
so
|φ(f)| ≤
D(φ)
n
for any n ∈ Z+. Let n→ +∞, we get φ(f) = 0. By Theorem 2.4,Q(Tp,q)/H1(Tp,q;R) =
{0}, but we have Tp,q = T ′p,q, which implies that H1(Tp,q;R) is trivial, thus Q(Tp,q) =
{0}.
Theorem 3.7. Suppose gcd(p − 1, q − 1) = 1. Let T˜p,q be the preimage of Tp,q in
H˜omeo+(S1). Then rot is the unique homogeneous quasimorphism which sends the unit
translation to 1.
Proof: By the same argument in Proposition 2.5 with Lemma 3.6 replacing the uni-
formly perfectness.
By Bavard’s Theorem 2.4, if gcd(p − 1, q − 1) = 1, then for all a in T˜p,q we have an
equality scl(a) = rot(a)/2D. In the appendix we show that D = D(rot) = 1 on the
groups T˜p,q, and therefore this simplifies to scl(a) = 12rot(a). So it remains to determine
the rotation numbers of elements in generalized Thompson groups.
Isabelle Liousse (in [13]) has studied the rotation numbers of elements in generalized
Thompson groups, and proved the following theorem:
Theorem 3.8 (I. Liousse). Let l ∈ A = Z[ 1p , 1q ], p, q ∈ Z+ independent and d = gcd(p−
1, q − 1). Suppose d is prime or 1, then Tl,(p,q) = T (l,Z[ 1p ,
1
q ], 〈p, q〉) contains elements
with irrational rotation numbers. In particular, if d = 1, Tl,(p,q) ∼= T1,(p,q), for any l ∈ Z.
6 DONGPING ZHUANG
Example 3.9 (from [13]). Let p = 2, q = 3, T˜2,3 satisfies the assumption. We have the
element f ∈ T2,3:
f =
{
2
3x+
2
3 if x ∈ [0,
1
2 ]
4
3x−
2
3 if x ∈ [
1
2 , 1]
0
1
1
2
2
3
1
1
2
2
3
FIGURE 1.
The transformation h(x) := 2− 12x−1 , x ∈ [0, 1] conjugates f to the rotation by ρ where
ρ = log 3log 2 − 1 ≈ 0.58496250072 · · · . Thus any lift of f has the rotation number
log 3
log 2 + n,
n ∈ Z. By the celebrated theorem of Gel’fand-Schneider [12], the rotation number ρ is
transcendental.
f lies in the class of the homeomorphisms in PL+(S1), satisfying the property D
[13]. I. Liousse showed that for any f ∈ PL+(S1) with the property D, there exists
h ∈ Homeo+(S1) such that h−1fh is a rotation, i.e. in SO(2). Thus we have a measure µ
on S1 that is invariant under f and furthermore∫
log(Df)dµ = 0
here Df is the function of the derivative of f , which is a piecewise constant function,
so to write down the left side of the equality, we only need to get the measures of the
intervals between break points. This can be obtained by the assumed property D, without
any knowledge of the conjugate function h. This equality gives a linear equation of the
rotation number of f , so in this way, we can get the rotation number.
4. APPENDIX
In this appendix, we will justify the claims in previous sections on defects of rotation
numbers as homogeneous quasimorphisms. We will use a lemma by C. Bavard on defect
estimation. See C. Bavard [2] or D. Calegari [5] for a proof.
Lemma 4.1 (C. Bavard). Let φ be a homogeneous quasimorphism on G. Then there is an
equality
sup
a,b∈G
|φ([a, b])| = D(φ)
Proposition 4.2. rot : H˜omeo+(S1) → R is a homogeneous quasimorphism with defect
1.
IRRATIONAL STABLE COMMUTATOR LENGTH IN FINITELY PRESENTED GROUPS 7
Proof: Refer to [11] for basic properties of rotation numbers.
(1) Let f, g ∈ ˜Homeo+(S1). Without loss of generality, we can assume that 0 ≤
f(0), g(0) < 1. So 0 ≤ f ◦ g(0) < 2. And 0 ≤ rot(f) ≤ 1, 0 ≤ rot(g) ≤ 1, and
0 ≤ rot(f ◦ g) ≤ 2. Thus we have |rot(f ◦ g)− rot(f)− rot(g)| ≤ 2, rot is a quasimor-
phism. That rot is homogeneous is clear from its definition.
(2) We show that D(rot) = 1 by using Lemma 4.1.
Take any f, g ∈ ˜Homeo+(S1) and we are going to compute rot([f, g]). We can still
assume that 0 ≤ f(0), g(0) < 1. Suppose 0 ≤ g(0) ≤ f(0) < 1, then we have by using
that f, g are increasing functions:
g(f(0)) < g(1) = g(0) + 1 ≤ f(0) + 1 ≤ f(g(0)) + 1
so
f(g(0))− g(f(0)) > −1
Then we have two cases:
(i) If we also have f(g(0))− g(f(0)) ≤ 1, then
−1 ≤ f(g(0))− g(f(0)) ≤ 1
g(f(−1)) = g(f(0))− 1 ≤ f(g(0)) ≤ g(f(0)) + 1 = g(f(1))
which implies
−1 ≤ f−1g−1fg(0) ≤ 1
so
|rot([f, g])| ≤ 1
(ii) If instead we have f(g(0)) > g(f(0))+1, then g(f(0)) < f(g(0))−1 = f(g(0)−1) <
f(0). ConsiderH(x) = f−1g−1fg(x)−1−x, for x ∈ [0, 1]. H(0) = f−1g−1fg(0)−1 >
0 by assumption.
H(f(0)) = f−1g−1fg(f(0))− 1− f(0)
< f−1g−1f(f(0))− 1− f(0)
= f−1g−1f2(0)− 1− f(0)
We want to show that H(f(0)) < 0, which can be deduced from the inequality below
f−1g−1f2(0) < 1 + f(0)
which is equivalent to
f2(0) < g(f2(0)) + 1
This is always true since x < g(x) + 1, for any x ∈ R.
So we have H(0) > 0 and H(f(0)) < 0, here 0 < f(0) < 1. There must be a point
y ∈ (0, f(0)) such that
H(y) = f−1g−1fg(y)− 1− y = 0
that is
f−1g−1fg(y) = 1 + y
8 DONGPING ZHUANG
So
rot[f, g] = lim
n→∞
[f, g]n(y)− y
n
= 1
The proof for the case 0 ≤ f(0) ≤ g(0) < 1 is the same. Put all together and we get
D(rot) = 1 by C. Bavard’s Lemma 4.1.
Proposition 4.3. Let T = T (1, A, P ), whereA = Z[ 1n1 ,
1
n2
, . . . , 1nk ], P = 〈n1, n2, . . . , nk〉,
here ni’s are independent. Suppose d = gcd(n1 − 1, . . . , nk − 1) = 1, then T is dense in
Homeo+(S1) with C0 topology.
Proof: Take an arbitrary f in Homeo+(S1) and any ǫ > 0. f is uniformly continuous,
so we can choose 0 = x0 < x1 < . . . < xl < 1, xi’s are in A that is dense in S1 =
[0, 1]/{0 = 1}, such that |f(xi) − f(xi−1)| < ǫ3 . Since A is dense in [0, 1], we can find
y0 < y1 < . . . < yl < 1, yi’s are in A and |yi − f(xi)| < ǫ3 . By the Bieri-Strebel
criterion(Theorem 3.1), there exists g ∈ T such that g(xi) = yi. And from the choice of
xi’s and yi’s, it’s easy to see that ‖f − g‖C0 < ǫ.
Thus we also have that T˜ ⊆ ˜Homeo+(S1) is dense. On the other hand, the map rot,
thought of as a function from ˜Homeo+(S1) to R, is continuous in the C0 topology [11].
So we have that the defect of rotation number D(rot), restricted to T˜ , is also 1.
REFERENCES
[1] J. Barge and ´E. Ghys, Cocycles d’Euler et de Maslov, Math. Ann. 294 (1992), 235–265
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[3] R. Bieri and R. Strebel, On groups of PL homeomorphisms of the real line, Preprint (1985)
[4] D. Calegari, Stable commutator length in subgroups of PL+(I), preprint (2006)
[5] D. Calegari, scl, monograph, in preparation;
available from http://www.its.caltech.edu/∼dannyc/scl/scl.pdf.gz
[6] J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson’s group, L’Enseign.
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[9] ´E. Ghys, Groups acting on the circle, L’Enseign. Math. (2) 47 (2001), no. 3-4 329–407
[10] M. Gromov, Asymptotic invariants of infinite groups, LMS Lecture note series 182, (1993)
[11] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univer-
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[12] S. Lang, Algebra, revised third edition, Springer-Verlag, (2002)
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[15] R.J. Thompson, Handwritten notes
DEPARTMENT OF MATHEMATICS, CALTECH, PASADENA CA 91125
E-mail address: dongping@caltech.edu


Irrational Stable Commutator Length in Finitely Presented Groups

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Irrational Stable Commutator Length in Finitely Presented Groups

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