Stable Commutator Length in Amalgamated Free
Products
Tim Susse
CUNY Graduate Center
GST Seminar
University of Nebraska – Lincoln
April 15, 2014
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Definitions
Let G be a group, and g ∈ [G, G].
cl(g) is the minimal number of commutators whose product is g.
In other words, cl(g) is the word length of g in
[G, G] = h[a, b] : a, b ∈ Gi.
Let G = F2 = ha, bi, then
cl([a, b]) = 1, cl([a, b]2 ) = 2, but cl([a, b]3 ) = 2
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Definitions
Let G be a group, and g ∈ [G, G].
cl(g) is the minimal number of commutators whose product is g.
In other words, cl(g) is the word length of g in
[G, G] = h[a, b] : a, b ∈ Gi.
Let G = F2 = ha, bi, then
cl([a, b]) = 1, cl([a, b]2 ) = 2, but cl([a, b]3 ) = 2
(Culler, 1981) [a, b]3 = [aba−1 , b−1 aba−2 ][b−1 ab, b2 ]. So, cl is not
homogeneous with respect to taking powers.
Definition.
cl(g n )
. Since n 7→ cl(g n ) is subadditive, this limit always
n→∞
n
scl(g) = lim
exists.
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Basic Facts
1
If cl(g) is uniformly bounded on G, then scl is identically zero.
2
If φ : G → H is a homomorphism then sclH (φ(g)) ≤ sclG (g).
3
If r : G → H is a retraction homomorphism, then r preserves scl (is
an isometry).
4
sclG×H ((g, h)) = max{sclG (g), sclH (h)}.
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Reformulation
Definition.
Let π1 (X ) = G. A map f : S → X of a surface S is called admissible for
h ∈ [G, G] if for some representative γ : S 1 → X of h, we have:
i
∂S −−−−→


y∂f
S


yf
γ
S 1 −−−−→ X
such that (∂f )∗ [∂S] = n(S)[S 1 ] and the diagram commutes up to
homotopy.
If S has one boundary component and genus g, this implies that hn(S)
is a product of g commutators. This was first used by Culler in 1981 to
study solutions to quadratic equations over free groups.
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Reformulation
If S has one boundary component and genus g, this implies that hn(S)
is a product of g commutators. This was first used by Culler in 1981 to
study solutions to quadratic equations over free groups.
Theorem.
Let X be a topological space with π1 (X ) = G. Then for any g ∈ G
−χ− (S)
scl(g) = inf
,
2n(S)
where the infimum is taken over all admissible surfaces for g with one
boundary component and χ− (S) the sum of the Euler characteristics
of all components of S with non-positive Euler characteristic.
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Homology and One-Boundaries
The bar complex of a group is a chain complex used to compute the
homology of a group. Here, Cn (G, R) = R[Gn ], of some ring R with
unit (usually R, Q or Z) and:
∂
∂
∂
∂
∂
··· −
→ Cn (G, R) −
→ Cn−1 (G, R) −
→ ··· −
→ C2 (G, R) −
→ C1 (G, R) → 1
with
∂(g1 , g2 , . . . , gn+1 ) =
+
(g2 , g3 , . . . , gn+1 )
n
X
(−1)i (g1 , . . . , gi−2 , gi−1 gi , gi+1 , . . . , gn+1 )
i=1
(−1)n+1 (g1 , g2 , . . . , gn ).
+
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Homology and One-Boundaries
The bar complex of a group is a chain complex used to compute the
homology of a group. Here, Cn (G, R) = R[Gn ], of some ring R with
unit (usually R, Q or Z) and:
∂(g1 , g2 , . . . , gn+1 ) =
+
(g2 , g3 , . . . , gn+1 )
n
X
(−1)i (g1 , . . . , gi−2 , gi−1 gi , gi+1 , . . . , gn+1 )
i=1
(−1)n+1 (g1 , g2 , . . . , gn ).
+
For example: ∂ : C2 (G) → C1 (G) is given by
∂(g, h) = h − gh + g = g + h − gh.
.
G
Further H1 (G, Z) =
, the abelianization.
[G, G]
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Reformulation Part II
Theorem.
Let X be a topological space with π1 (X ) = G. Then for any g ∈ G
−χ− (S)
,
scl(g) = inf
2n(S)
where the infimum is taken over all admissible surfaces for g.
This reformulation of stable commutator length allows S to have
multiple boundary components, and extends the scl to 1-boundaries of
G. We consider B1 (G; R) the set of all real one-boundaries of G.
Proposition.
.
scl is a pseudonorm on B1H (G; R) = B1 (G; R) g n − ng, hgh−1 − g .
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Bavard Duality Theorem
There is also an alternate way to compute stable commutator length
from quasimoriphisms.
Definition.
A quasimorphism is a function φ : G → R so that there is a least
non-negative number D(φ), called the defect, so that for every
g, h ∈ G:
|φ(gh) − φ(g) − φ(h)| ≤ D(φ).
Further, a quasimorphism is called homogenous is φ(g n ) = nφ(g) for
every integer n.
For example, any homomorphism or bounded function to R is naturally
a quasimorphism.
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Bavard Duality Theorem
Definition.
A quasimorphism is a function φ : G → R so that there is a least
non-negative number D(φ), called the defect, so that for every
g, h ∈ G:
|φ(gh) − φ(g) − φ(h)| ≤ D(φ).
Further, a quasimorphism is called homogenous is φ(g n ) = nφ(g) for
every integer n.
For example, any homomorphism or bounded function to R is naturally
a quasimorphism.
The space of homogenous quasimorphisms
of a group is denoted
.
HQH(G)
is the kernel of the
HQH(G). The vector space
H 1 (G; R)
comparison map Hb2 (G; R) → H 2 (G; R) between second real bounded
cohomology and ordinary second real cohomology.
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Examples
Example (Brooks)
If G = Fr is the free group, and w ∈ Fr is some reduce word let cw (g)
be the maximal number of disjoint copies of w appears as a subword
of g.
Let hw (g) = cw (g) − cw −1 (g). This is a quasimorphism with defect at
most 2, called a Brooks Counting quasimorphism.
Brooks originally did not have the disjoint condition. Removing it leads
to a defect of at most 3(|w| − 1).
Rotation
Let G y S 1 . Lift this action to an action on R. Then
g n (0)
rot(g) = lim
. This is the (lifted) rotation number. This a
n→∞
n
quasimorphism on the induced central extension of G. Its defect is 1
(this is due to Poincare).
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Theorem. (Bavard Duality Theorem)
Given a chain η ∈ B1H (G), the following holds:
scl(η) =
sup
φ∈HQH/H 1
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φ(η)
.
2D(φ)
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Theorem. (Bavard Duality Theorem)
Given a chain η ∈ B1H (G), the following holds:
scl(η) =
sup
φ∈HQH/H 1
φ(η)
.
2D(φ)
Why?
Let φ ∈ HQH. Then firstly, φ(g) = φ(hgh−1 ) for all h ∈ G. So, given a
commutator [a, b], we have:
|φ([a, b])| = |φ(abAB)| ≤
D(φ) + |φ(a) + φ(bAB)|
=
D(φ) + |φ(a) + φ(A)|
=
D(φ)
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Theorem. (Bavard Duality Theorem)
Given a chain η ∈ B1H (G), the following holds:
scl(η) =
sup
φ∈HQH/H 1
φ(η)
.
2D(φ)
Why?
Let φ ∈ HQH. Then firstly, φ(g) = φ(hgh−1 ) for all h ∈ G. So, given a
commutator [a, b], we have:
|φ([a, b])| = |φ(abAB)| ≤
D(φ) + |φ(a) + φ(bAB)|
=
D(φ) + |φ(a) + φ(A)|
=
D(φ)
So, if cl(w) = k, then |φ(w)| ≤ (2k − 1)D(φ).
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Computing SCL
In general, scl is very hard to compute. It is known to be identically
zero in some cases:
Abelian groups
Amenable groups
Lattices in higher rank Lie groups
Subgroups of PLHomeo+ (I), where I ⊂ R is an interval (e.g.,
Thompson’s Group F ).
Precise computations are only known in the following classes of finitely
generated groups:
1
Free Groups [Calegari, 2009 & Walker, 2013 with an efficient
algorithm]
2
Free products of Abelian groups [Calegari, 2011]
3
Stein-Thompson Groups [Zhuang, ’08] (only known irrational
examples)
4
Some elements of Baumslag-Solitar groups
[Clay-Forester-Louwsma, ’13]
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PQL Groups
In all but the fourth example above, the scl is rational on the group and
is a piecewise rational linear (semi-)norm on B1H (G). Such a group is
called PQL.
Conjecture. (Calegari)
Let M be a 3-manifold. Then π1 (M) is PQL.
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PQL Groups
In all but the fourth example above, the scl is rational on the group and
is a piecewise rational linear (semi-)norm on B1H (G). Such a group is
called PQL.
Conjecture. (Calegari)
Let M be a 3-manifold. Then π1 (M) is PQL.
Why the Conjecture?
Let M be a (closed) 3-manifold. Consider the Thurston norm on
H2 (M; R) (or relative to the boundary). For some class α ∈ H2 (M; R) :
kαkT = inf −χ− (S), taken over all S representing α.
S
So, scl is a natural relativization of the Thurston norm! Since the
Thurston norm is PQL, perhaps the scl norm on B1H (π1 (M)) is as well.
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Results
Theorem. (Susse)
Let A and B be two free Abelian groups of rank at least k , then A ∗Zk B
is PQL. Further, there is an algorithm to compute scl on rational chains.
Theorem. (Susse)
Let {Ai } be a collection for free Abelian groups of rank at least k . Then
∗Zk Ai , their free product amalgamated over a single shared subgroup,
is PQL. Further, there is an algorithm to compute scl on rational chains.
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Results
Theorem. (Susse)
Let A and B be two free Abelian groups of rank at least k , then A ∗Zk B
is PQL. Further, there is an algorithm to compute scl on rational chains.
Theorem. (Susse)
Let {Ai } be a collection for free Abelian groups of rank at least k . Then
∗Zk Ai , their free product amalgamated over a single shared subgroup,
is PQL. Further, there is an algorithm to compute scl on rational chains.
Note that, up to isomorphism,
G = a1 , . . . an , b1 , . . . , bn | [ai , aj ] = [bi , bj ] = 1, a1r1 = b1s1 , . . . , akrk = bksk .
If A = B = Z, then
A ∗Z B = a, b : ap = bq .
If (p, q) = 1, then G = π1 (S 3 \ K ), where K is a torus knot.
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The Set-up
Let G = A ∗Zk B and let TA and TB be tori whose fundamental
groups are identified with A and B respectively. Let X be the
space formed by taking a cylinder T k × [0, 1] and gluing T k × {0}
to TA and T k × {1} to TB so that π1 (X ) = G.
Let f : S → X be a map of a surface with (possible empty)
boundary, and let C = T k × { 12 }.
WLOG we can assume f −1 (C) is a disjoint collection of arcs with
end points on ∂S and simple closed curves.
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Parameterizing Surfaces
WLOG, components of S \ f −1 (C) are planar surfaces with
boundary, whose boundaries are made of arcs alternating
between:
Arcs from ∂S (τ -edges);
Arcs from f −1 (C) (σ-edges)
Some boundary components may be loops from f −1 (C) (σ-loops)
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Parameterizing Surfaces
WLOG, components of S \ f −1 (C) are planar surfaces with
boundary, whose boundaries are made of arcs alternating
between:
Arcs from ∂S (τ -edges);
Arcs from f −1 (C) (σ-edges)
Some boundary components may be loops from f −1 (C) (σ-loops)
Using the σ- and τ -edges we can parameterize all of the surfaces
S → X.
Problem: We need to control contributions from σ-loops.
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Outline of the proof
Let η ∈ B1H (G; Z)
1
Control σ-loops by investigating gluing equations and winding
numbers.
2
Given a vector which can parameterize a surface (in one of the
two tori), estimate the Euler characteristic in a piecewise rational
linear way.
3
Given vectors parameterizing surfaces which are compatible,
ensure that there is a way to glue them, after considering winding
numbers – do this by adding loops.
4
Show that the Euler characteristic estimate from the second step
is still good.
5
Maximize the Euler characteristic over a rational polyhedron
parameterizing all surfaces with boundary (a multiple of) η using
linear programming.
Since we are looking at a piecewise rational linear function over a
rational polyhedron, the maximum is rational.
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Step 1: Winding Numbers and Gluing Equations
Take a component of S \ f −1 (C). Attached to each σ-edge is a k-tuple
of integers, giving the class of the edge in H1 (C) ∼
= Zk .
4
a
−1
−1
a4
a4
0
Figure: A disc in the (6, n) torus knot complement
For every component, the sum of the τ -edges and labels of the
σ-edges must be 0 in homology. Here, the τ -edges contribute 12a and
each −1 on the σ-edges contributes −6a.
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Winding Numbers and Gluing Equations
For each component the sum of the winding numbers σ-edges and
σ-loops must make up the “homology deficit” of the τ -edges in the
component.
For each torus we get a system of gluing equations where each
σ-edge or loop appears with a coefficient ±1 in exactly one equation.
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Winding Numbers and Gluing Equations
For each component the sum of the winding numbers σ-edges and
σ-loops must make up the “homology deficit” of the τ -edges in the
component.
For each torus we get a system of gluing equations where each
σ-edge or loop appears with a coefficient ±1 in exactly one equation.
Analyzing these gluing equations we get the following:
Proposition.
Let f : S → X be a continuous map of a surface. Then, up to replacing
S with a surface of higher Euler characteristic, we can assume that all
loop components of f −1 (C) are separating.
The key ingredient in the proof is the following Lemma.
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Winding Numbers and Gluing Equations
Lemma.
Consider a linear system of equations as follows




N1
MA
v = 
,

MB
N2
such that all entries of MA and MB are either 0 or 1; each column in MA
and each column in MB has a 1 in exactly one place; N1 and N2 are
integral vectors and the system is consistent. Letting Aj be a row from
MA and Bi a row from MB , it is possible to row reduce the matrix so
that each fully reduced row is of the form
X
X
i Bi −
δj Aj ,
where i , δj ∈ {0, 1}. Consequently, the system has an integral
solution.
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Proof of the Proposition
Look for the equation lC = n in the reduced system, where lC is the
variable corresponding to a given σ-loop. If not it’s not there, it is
possible to set lC = 0, compress the loop, and still get a solution to the
system.
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Proof of the Proposition
Look for the equation lC = n in the reduced system, where lC is the
variable corresponding to a given σ-loop. If not it’s not there, it is
possible to set lC = 0, compress the loop, and still get a solution to the
system.
Rows represent a components of S \ f −1 (C). From the Lemma,
reduction corresponds to taking a union of components and gluing
along their common σ-edges. The reduced row then describes the
remaining free σ-edges.
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Proof of the Proposition
l4
l1
l3
l2
l2
−l1
l4
−l3
(a) Unglued
(b) Glued Together
Components
Figure: The rows in (a): (1, 1, 1, 1) and (−1, 0, −1, 0) are glued together in
(b). The rows are added and result is (0, 1, 0, 1).
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Proof of the Proposition
Thus, the equation lC = n occurs only if there is some union of
components mapping to TB and TA that results in a surface whose only
remaining σ-edge is our loop. Thus, the loop is separating.
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Parameterizing
Let T (A) be a set of all τ -edges from TA . C(A) be the span of
T (A).
Let T2 (A) be the set of all σ-edges connecting elements of T (A).
T2 (A) is the set of ordered pairs of elements of T (A) — they are
the initial and terminal τ -edges.
C2 (A) = span{T2 (A), a1r1 , . . . , akrk }.
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Parameterizing
Let T (A) be a set of all τ -edges from TA . C(A) be the span of
T (A).
Let T2 (A) be the set of all σ-edges connecting elements of T (A).
T2 (A) is the set of ordered pairs of elements of T (A) — they are
the initial and terminal τ -edges.
C2 (A) = span{T2 (A), a1r1 , . . . , akrk }.
Given a surface S, we get a vector v (S) ∈ C2 (A), which is the
formal sum of all σ-edges in S and the sum of their winding
numbers.
There is a system of (rational) linear equations that determines
what combinations of σ-edges gives rises to a surface mapping to
TA .
The equations come in two types: closing equations and
homology equations. The set of solutions with all non-negative
T2 (A) components is called VA .
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Computing χ
The map that takes surfaces to vectors in VA is many-to-one — so
we try to find the one with largest Euler characteristic
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Computing χ
The map that takes surfaces to vectors in VA is many-to-one — so
we try to find the one with largest Euler characteristic
To compute χ, we need to count the (maximal) number of disc
components in a surface parameterized by v .
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The Klein Function
The set of disc vectors DA is the set of all integral vectors in v ∈ VA so
that the surface parameterized by v has one boundary component.
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The Klein Function
The set of disc vectors DA is the set of all integral vectors in v ∈ VA so
that the surface parameterized by v has one boundary component.
Definition.
Let κ(v ), called the Klein function be the function which counts discs.
In particular:
nX
o
X
0
0
κ(v ) = max
ti : v =
ti vi + v , ti > 0, vi ∈ DA , v ∈ VA
χo (v ) = κ(v ) −
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|v |
, where |v | is the l1 -norm of the vector v .
2
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The Klein Function
Definition.
Let κ(v ), called the Klein function be the function which counts discs.
In particular:
nX
o
X
κ(v ) = max
ti : v =
ti vi + v 0 , ti > 0, vi ∈ DA , v 0 ∈ VA
χo (v ) = κ(v ) −
|v |
, where |v | is the l1 -norm of the vector v .
2
Lemma.
Given a vector v ∈ VA and > 0 there exists and integer n and a
surface S parameterized by nv so that nχo (v ) > χo (S) and
χo (v ) − χo (S) < .
n Further, χo (S) is a piecewise rational linear function on VA .
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Computing SCL
Theorem.
Let A and B be two free Abelian groups of rank at least k , then A ∗Zk B
is PQL. Further, there is an algorithm to compute scl on rational chains.
Proof.
From the previous Lemma χo (vA ) + χo (vB ) is a sum of two piecewise
rational linear functions which estimate the Euler characteristic of a
surface (we skipped some steps here). Thus, using linear
programming, we can minimize it over the rational polyhedron
Y ⊆ VA × VB of compatible vectors with n(S) = 1, where vA and vB
produce surfaces to give us the boundary η that we desire. (Note that if
the vector is not an integer vector, we clear the denominators to obtain
an integer vector admissable for η). Since we are minimizing a
piecewise rational linear function over a rational polyhedron, the result
is rational.
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Examples
a
b
−1 1
0
a
A
−1 1
B
B
0 0
A
0
b
A mirror in the trefoil knot complement
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Examples
a2
b3
A2
B3
A2
b3
B3
b
3
a2
a2
B3
A2
An extremal surface for [a2 , b3 ] in the (7, 9) torus knot
complement. cl7,9 ([a2 , b3 ]3 ) = 1.
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A Formula for SCL
Proposition.
In the group G = ha, b | ap = bq i:
m
1
n
1
m n
sclG ([a , b ]) = max min
−
, −
,0
2 lcm(m, p) 2 lcm(n, q)
.
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A Formula for SCL
Proposition.
In the group G = ha, b | ap = bq i:
m
1
n
1
m n
sclG ([a , b ]) = max min
−
, −
,0
2 lcm(m, p) 2 lcm(n, q)
.
For fixed p and q this depends only on the modulus of m (mod p)
and n (mod q).
For fixed m and n this formula is the quotient of two linear
quasipolynomials (polynomials of degree 1 whose coefficients
depend only on the modulus of p and q with respect to some
modulus π – here, lcm(m, n).
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Graphs of SCL
On the left, a graph of sclp,q ([a2 , b3 ]). On the right, a graph of
sclp,q (aba−4 b2 a2 b−1 a2 b−3 a−1 b).
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Quasirationality
The phenomenon seen in the previous slides is not a coincidence.
Theorem.
Let w ∈ F2 = ha, bi and let wp,q be the projection to the group
G(p, q) = ha, b | ap = bq i. Let sclp,q (w) = sclG(p,q) (wp,q ). Then
sclp,q (w) varies as the quotient of two quasipolynomials for p and q
sufficiently large.
To prove this Theorem, we need to investigate the way that the
polyhedron VA (p) and the function κp (v ) depend on p. Thus, we need
to investigate the geometry of conv(DA ). A Theorem of
Calegari-Walker implies that the vertices of this polyhedron are
quasilinear functions in p.
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Open Questions and Conjectures
Conjecture.
Given a finite tree Γ, is the right-angled Artin group with generating
graph Γ PQL?
Such a group is a graph of groups whose underlying graph is a tree
with cyclic edge groups and vertex groups Z2 . The methods above
should extend to cover these groups, but it is not clear how to count
discs. These groups are also precisely the right-angled Artin groups
that are 3-manifold groups.
Conjecture.
sclp,q (w) is actually quasilinear in p1 and q1 [Walker]. Further,
quasilinearity (or more weakly: quasirationality) occurs for
p ≥ lcm(set combinations of a-exponents) and
q ≥ lcm(set combinations of b-exponents).
Tim Susse CUNY Graduate Center
SCL in AFP
April 11, 2014
31 / 32
Thank You!
Tim Susse CUNY Graduate Center
SCL in AFP
April 11, 2014
32 / 32