RESEARCH STATEMENT
TIM SUSSE
My principal area of research is Geometric Group Theory, a field that studies the relationships between the intrinsic geometry of groups and their algebraic structure. Study of large
scale geometry, like quasi-isometry invariants and asymptotic cones, has produced notable
results relating Algebra to Geometry.
Geometric Group Theory also has strong ties to low dimensional topology, as coarse methods are increasingly common in the study of the mapping class groups of surfaces and three
manifold groups, as exhibited in [MM99, Bow06, BM08, BN08]. In particular, these references show a strong tie between geometry, topology and algebra. In my own work I focus
on stable commutator length in various classes of groups which connects algebra to the
topology of surfaces inside of a topological space. Understanding surfaces in three-manifolds
has been the goal of several recent results: specifically the resolution of the virtual Haken
conjecture in [Ago]. I will summarize questions which interest me about stable commutator
length in right-angled Artin groups, three-manifold groups, surface groups and connections
to topological questions.
1. Stable Commutator Length
Let G be a group and take g ∈ [G, G], the commutator subgroup. Consider [G, G] as
generated by the infinite set of all commutators. The commutator length of g, denoted cl(g)
is the word length of g in this subgroup. Culler showed in [Cul81] that in the free group
cl([a, b]3 ) = 2; i.e., that commutator length is not stable under taking powers. We define the
stable commutator length then by
cl(g n )
.
n→∞
n
The most important property of this quantity is its relationship to topology.
scl(g) = lim
Theorem 1.1 ([Cal09a]). Let X be a topological space with π1 (X) = G and let g ∈ [G, G].
Then
−χ− (S)
scl(g) = inf
,
n(S)
where the infimum is taken over all surfaces S with one boundary component and all continuous maps f : S → X with f∗ ([∂S]) = g n(S) .
Thus, understanding stable commutator length is the same as understanding maps of
surfaces with boundary to topological spaces. These definitions can also be extended to
1-boundaries of the group G without much extra work, making scl a pseudonorm on the
space of real one-boundaries of the
. group, denoted B1 (G). We can further consider scl as a
H
pseudo norm on B1 (G) = B1 (G) hg n − ng, hgh−1 − gi .
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Using this definition, Danny Calegari proved in [Cal09b, Cal11] that scl is a piecewise
rational linear norm on B1H (G) when G is a free group or free product of Abelian groups.
He also provided an algorithm to compute scl on any finite dimensional rational subspace.
We call a group with this property PQL; such a group has the property that the scl of any
element of the commutator subgroup is rational. Alden Walker recently provided an efficient
algorithm in [Wal] to compute scl in free products of cylic groups.
In my own work, I show that a larger class of groups is PQL.
Theorem 1.2 (Susse [Sus]). Let G = A ∗Zk B, where A and B are free abelian groups or
rank at least k. Then G is PQL and there exists an algorithm to compute scl on any finite
dimensional subspace of B1H (G).
The class of groups in Theorem 1.2 includes many interesting groups as specific examples.
These examples will be discussed further in the following two sections.
The results in the above theorem can be extended beyond just two free Abelian groups,
by considering a family of free Abelian groups amalgamated over a single subgroup.
Theorem 1.3 (Susse [Sus]). Let G = ∗Zk Ai , where {Ai } is a family of free abelian groups of
rank at least k amalgamated over a single shared subgroup. Then G is PQL and there exists
an algorithm to compute scl on any finite dimensional subspace of B1H (G).
The methods I developed can also be applied to study other families of groups; for instance,
Clay-Forrester-Louwsma recently employed similar methods to show that certain elements
of Baumslag-Solitar groups have rational scl [CFL].
2. Three Manifolds
In general, almost nothing is known about the scl spectrum in a group. Based on the
examples above where scl is known, Calegari conjectured the following.
Conjecture 2.1 (Calegari). Let M be a three-manifold. π1 (M ) is PQL.
While the conjecture is known to hold in the cases listed in the previous section, and some
other trivial examples (specifically Nil and Sol manifolds), many cases still remain open.
There are several questions motivated by this conjecture that should be answered before it
is evaluated.
As a special case of Theorem 1.2, fundamental groups of all torus knot complements are
PQL: since their fundamental groups are isomorphic to two copies of Z amalgamated over
Z. The methods I used in the proof can also be used to study other topological properties
of these manifolds. In particular, in [Sus] I provide a new and simple proof of Waldhausen’s
classical theorem that any embedded incompressible surface in a torus knot complement is
a boundary parallel torus.
Question 2.2. Are fundamental groups of hyperbolic link complements PQL? Do there exist
surfaces realizing the infimum in Theorem 1.1?
The approach I take in [Sus] allows for a complete understanding of surfaces with boundary
inside torus knot complements, giving an understanding of the relative Gromov-Thurston
norm. A reasonable approach to this larger question is suggested by the topological resolution
of this question I developed in that special case: to study the topology of surfaces with
2
boundary mapping to the knot or link complement, or equivalently, its Dehn complex. Giving
a method for all knots and links is likely a long term project: in the near term I plan to look at
examples with low crossing numbers. In particular I would like to produce an understanding
of scl for the figure-8 knot complement.
I would like to develop and use a normal surface theory for surfaces with boundary to
attack this question. Recently work has been done by Tillman-Cooper [CT09] and JacoRubenstein-Tillman [JRT13] on relating normal surfaces to the Thurston norm on the second
homology of a three-manifold (with torus boundary). In particular, they show the the unit
ball in the Thurston norm is the projection of a polyhedron in the normal surface space.
Since Thurston norm is a natural analog for stable commutator length (an absolute version),
this approach seems promising.
Since it is known that every three-manifold is obtainable by Dehn surgery on a link complement [Lic62, Wal60], the next step in a systematic approach to Calegari’s conjecture is
the following question:
Question 2.3. How does Dehn surgery affect stable commutator length? Are the effects
quantifiable?
A positive solution of the conjecture requires that Dehn surgery preserves the property
PQL. Previous results, found in [Cal09a] give bounds for scl of some elements based on the
geometry of the cusp in a 3-manifold and the geometry of a hyperbolic Dehn filling, but no
precise measurements are known.
Another interesting approach to the conjecture is to expand the work in [Cal11, Sus] to
study connected sums of Nil-manifolds; i.e., G is a free products of lattices in the real Heisenberg group. I have already thought about how to study these groups and have found some
evidence that there may be elements with irrational scl. Since very little is known about
elements and group with irrational scl, a precise calculation here would be interesting. Further, if the element has algebraic irrational scl, it would be the first such element discovered
in a finitely generated group.
3. Right-Angled Artin Groups
Given a graph Γ with vertex set V (Γ) and edges E(Γ) we form the right-angled Artin
group A (Γ) given by the presentation:
A (Γ) = hV (Γ) | [v, w] = 1 if and only if (v, w) ∈ E(Γ)i .
These groups have recently played an important role in theory of cubulated groups [HW08,
Wis11] and Ian Agol’s proof of the virtual Haken conjecture [Ago]. In particular, the fundamental group of every closed hyperbolic three-manifold virtually embeds in a right-angled
Artin group. Connected to Calegari’s conjecture in the previous subsection is the following
question:
Question 3.1. Are right-angled Artin groups PQL? If not, under what conditions are they?
My work in [Sus] provides some insight into a more specific question: whether rightangled Artin groups whose generating graphs are trees are PQL. These groups can naturally
be decomposed as a tree of groups with free abelian vertex groups, a situation that is similar
to the one in Theorem 1.2. I plan to explore this connection and expand my methods and
algorithm to cover this case, with the Croke-Kleiner group (i.e., Γ is a path of length 3)
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as a test case. Those groups are exactly the right-angled Artin groups which are also the
fundamental groups of three-manifolds [HM99] and studied by Behrstock and Neumann in
[BN08]. This puts those groups in the intersection of Question 3.1 and Conjetcure 2.1 and
an excellent test case for both.
Since it is known that virtually special groups (such as hyperbolic three-manifold groups)
are virtual retracts of right-angled Artin groups, they are virtually isometrically embedded
with respect to scl. This naturally leads to the following question, and answer to which would
shed light on Conjecture 2.1 and be very interesting to topologists and group theorists.
Question 3.2. Which right-angled Artin groups can contain three-manifold groups? What
about surface groups?
Even more specifically, it is known that many right-angled Artin groups contain surface
subgroups. The first such instance of this was discovered by Servatius-Droms-Servatius in
[SDS89] inside the right-angled Artin group on the pentagon. In fact, this is true for any
graph containing a cycle of length at least 5. There are, however, right-angled Artin groups
whose generating graphs do not have long cycles which do contain surface subgroups, as
shown by Crisp-Sageev-Sapir in [CSS08] and Kim in [Kim10]. Kim also obtained negative
results, showing that right-angled Artin groups on chordal graphs cannot contain surface
subgroups.
4. Surface Groups
Following the computation of scl in free groups, maps of surfaces with boundary into
non-compact surfaces have been extensively studied. In particular, it is shown in [Cal09a]
that if S is a surface with non-empty boundary and a chain in π1 (S) virtually bounds
an immersed surface, then that surface realizes the scl of the chain. In particular, the
rotation quasimorphism of the immersed surface is extremal for the chain. This, however,
is not necessarily true when S is closed. Further, he shows that every chain, after adding
sufficiently many copies of ∂S, virtually bounds an immersed surface.
Question 4.1. Let S be a closed surface. Is π1 (S) PQL?
In the case where S is a genus 2 surface we can decompose π1 (S) = F2 ∗Z F2 , a situation
similar to the one I studied previously in [Sus]. Critically different, though, is that the
vertex groups of the amalgamation are free and not free Abelian and thus themselves have
non-trivial stable commutator length. It is possible, though, to use the same analysis of the
amalgamation as in the free Abelian case.
Question 4.2. Are there surfaces realizing the infimum in Theorem 1.1 for surface groups?
In the free group case, it is possible to exploit the structure of surfaces realizing the
infimum in Theorem 1.1. In particular, Calegari-Walker used the combinatorics of fat graphs
in [CWb, CWa, CW13] to show that random groups contain surface subgroups and that
there are precise asymptotics for scl of generic elements in free groups. A positive answer to
question 4.2 will open up new avenues for exploring applications listed in the next section
and give a better understanding of the spectrum of scl in these groups.
I am currently collaborating with Alden Walker using a new and promising approach to
question 4.1. The approach combines Calegari-Walker’s train tracks over the free groups
developed in [CWb] and the amalgamation machinery I developed in [Sus].
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5. Future Applications
One of the most important outstanding questions in three-manifold topology is the Simple
Loop conjecture of Gabai.
Conjecture 5.1 (Gabai [Gab85]). Let S be a surface, M a 3-manifold and f : S → M
a continuous map. If the f∗ : π1 (S) → π1 (M ) is not injective, then the kernel contains a
conjugacy class represented by a simple loop.
Cooper-Manning and Louder showed in [MC, Lou] that there are representations of surface
groups into SL(2, C) which do not kill any simple loops, but they do not show that the
image of the homomorphism lies in any hyperbolic three-manifold group. Calegari showed
in [Cal13] that the Gromov-Thurston norm is a certificate to show that certain surfaces in
a 3-manifold are geometrically incompressible, which implies that the map f∗ contains no
simple loop in its kernel. Further, he links scl and the Gromov-Thurston norm in graphs
of spaces amagamated over an S 1 . This situation is very similar to the one in the previous
section and the spaces I studied in [Sus].
Along the same lines, the following question is directly relevant to the simple loop conjecture, distortion of scl and Gromov-Thurston norms of essential surfaces in 3-manifolds.
Question 5.2. If M is a 3-manifold and S ⊂ M is an essential surface, how does the stable
commutator length of a nullhomotopic curve in π1 (S) relate to the Gromov-Thurston norm
of [S] ∈ H2 (M ; Z)?
The eventual goal of this is to produce examples to model counter-examples to the conjecture, which is still open in the hyperbolic manifold case.
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References
[Ago]
[BM08]
[BN08]
[Bow06]
[Cal09a]
[Cal09b]
[Cal11]
[Cal13]
[CFL]
[CSS08]
[CT09]
[Cul81]
[CWa]
[CWb]
[CW13]
[Gab85]
[HM99]
[HW08]
[JRT13]
[Kim10]
[Lic62]
[Lou]
[MC]
[MM99]
[SDS89]
[Sus]
[Wal]
Ian Agol, The virtual haken conjeture, Preprint: arXiv:1204.2810. http://arxiv.org/pdf/1204.
2810.pdf. Appendix by Ian Agol, Daniel Groves and Jason Manning.
Jason Behrstock and Yair Minsky, Dimension and rank for mapping class groups, Annals of Mathematics. Second Series 167 (2008), no. 3, 1055–1077.
Jason Behrstock and Walter Neumann, Quasi-isometric Classification of Graph Manifold Groups,
Duke Mathematical Journal 141 (2008), no. 2, 217–240.
Brian Bowditch, Intersection numbers and the hyperbolicity of the curve complex, Journal für die
reine und angewandte Mathematik 598 (2006), 105–129.
Danny Calegari, scl, MSJ Memoirs, Mathematical Society of Japan, 2009.
, Stable Commutator Length is Rational in Free Groups, Journal of the American Mathematical Society 22 (2009), no. 4, 941–961.
, SCL, Sails and Surgery, Journal of Topology 4 (2011), no. 2, 305–326.
, Certifying incompressibility of non-injective surfaces with scl, 2013, pp. 257–262.
Matt Clay, Max Forrester, and Joel Louwsma, Stable Commutator Length in Baumslag-Solitar
Groups and Quasimorphisms for Tree Actions, Preprint: arXiv:1310.3861. http://arxiv.org/
pdf/1310.3861.pdf.
John Crisp, Michah Sageev, and Mark Sapir, Surface Subgroups of Right-angled Artin Groups,
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Daryl Cooper and Stephan Tillmann, Thurston Norm via Normal Surfaces, Pacidic Journal of
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Mark Culler, Using Surfaces to Solve Equations in Free Groups, Topology 20 (1981), 133–145.
Danny Calegari and Alden Walker, Random Groups Contain Surface Subgroups, Preprint:
arXiv:1304.2188. http://arxiv.org/pdf/1304.2188.pdf.
, Surface Subgroups from Linear Programming, Preprint: arXiv:1212.2618. http://arxiv.
org/pdf/1212.2618.pdf.
, Random Rrigidity in the Free Group, Geometry and Topology 17 (2013), no. 3, 1707–1744.
David Gabai, The simple loop conjeture, Journal of Differential Topology 21 (1985), no. 1, 143–149.
Susan Hermiller and John Meier, Artin groups, rewriting systems and three-manifolds, Journal of
Pure and Applied Alebra 136 (1999), 141–156.
Frédéric Haglund and Daniel T. Wise, Special cube complexes, Geometric and Functional Analysis
17 (2008), no. 5, 1551–1620.
William Jaco, Hyam Rubenstein, and Stephan Tillmann, Z2-Thurston Norm and Complexity of
3-manfolds, Mathematische Annalen 366 (2013), no. 1, 1–22.
Sang-hyun Kim, On Right-angled Artin Groups without Surface Subgroups, Groups, Geometry and
Dynamics 4 (2010), no. 2, 275–307.
W. Lickorish, A Representation of Orientable Combinatorial 3-maifolds, Annals of Mathematics
76 (1962), no. 3, 531–540.
Larsen Louder, Simple Loop Conjecture for Limit Groups, Preprint:arXiv:1106.1350. http://
arxiv.org/pdf/1106.1350.
Jason Manning and Daryl Cooper, Non-faithful Representations of Surface Groups into sl(2,c)
which Kill No Simple Closed Curve, Preprint: arXiv:1104.4492. http://arxiv.org/pdf/1104.
4492.
Howard Masur and Yair Minky, Geometry of the complex of curves I: Hyperbolicity, Inventiones
Mathematicae 138 (1999), 103–149.
Herman Servatius, Carl Droms, and Brigitte Servatius, Surface Subgroups of Graph Groups, Proceedings of the American Mathematical Society 106 (1989), no. 3, 573–578.
Tim Susse, Stable Commutator Length in Amalgamated Free Products, Preprint: arXiv:1310.2254.
http://arxiv.org/pdf/1310.2254.pdf.
Alden Walker, Stable Commutator Length in Free Products of Cyclic Groups, Preprint:
arXiv:1304.6312. http://arxiv.org/pdf/1304.6312.pdf.
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[Wal60] Andrew Wallace, Modifications and cobounding manifolds, Canadian Journal of Mathematics 12
(1960), 503–528.
[Wis11] Daniel T. Wise, From riches to raags: 3-manifolds, right-angled artin groups, and cubical geometry,
Lecutre Notes from the Cubical Geometry and Right-Angled Artin Groups, 2011.
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