**Stable commutator length in graphs of groups- Lvzhou (Joe) Chen (University of Chicago)**

Abstract: The stable commutator length (scl) is a relative version of the Gromov-Thurston norm. For a given null-homologous loop L in a space X, its scl is the infimal complexity of surfaces bounding L, measured in terms of Euler characteristics. Surfaces realizing the minimal complexity are called extremal. They are pi_1-injective, and can only exist when scl is rational. We show that scl takes rational values for all loops in a space X if pi_1(X) is certain graphs of groups, inclusing Baumslag-Solitar groups. Moreover, there is a linear programming algorithm to compute scl.

**Flows, Thurston norm, and surfaces: homology to isotopy- Michael Landry (Washington University)**

Abstract: Let M be a closed hyperbolic 3-manifold. We begin by reviewing a classical picture due to Thurston, Fried, and Mosher in which a single pseudo-Anosov flow organizes the data of a fibered face F of the Thurston norm ball of M as well as certain nice surface representatives of homology classes lying in the cone over F. We then announce a new theorem which strengthens the above by collating all isotopy classes of incompressible surfaces representing classes in the cone over F. We explain that the result follows from a more general theorem, involving veering triangulations, which also applies to other faces of the Thurston norm ball (possibly nonfibered, possibly lower-dimensional).

**Dehn filling and knot complements that irregularly cover- William Worden (Rice University)**

Abstract: It is a longstanding conjecture of Neumann and Reid that exactly three knot complements can irregularly cover a hyperbolic orbifold---the figure-8 knot and the two Aitchison--Rubinstein dodecahedral knots. This conjecture, when combined with work of Boileau--Boyer--Walsh, implies a more recent conjecture of Reid and Walsh, which states that there are at most 3 knot complements in the commensurability class of any hyperbolic knot. We give a Dehn filling criterion that is useful for producing large families of knot complements that satisfy both conjectures. The work we will discuss is partially joint with Hoffman and Millichap, and partially joint with Chesebro, Deblois, Hoffman, Millichap, and Mondal.

## 10 replies on “Hyperbolic geometry and manifolds: Triangulations and complexity”

[…] Triangulations and Complexity: […]

Hi everyone. I welcome all comments and questions! Also please note my zoom office hour takes place 11a eastern on Monday 6/8.

Michael Landry

Hey Michael; thanks for the nice talk and the shout-out at the end.

Did I understand correctly? The question of if, for an embedded surface minimizing its Thurston norm almost transverse to a pA flow, that surface is unique in its isotopy class is still open (for non-fibers)?

So, e.g. to integer lattice points in the boundary of a face, you can construct a taut embedded surface by taking an integer number of sheets of cells in tau^(2) and gluing them together in the right way–the filling information tells you how to glue up this surface near the missing curves? But then we’re worried that there is another taut embedded surface that intersects the flow lines of the pA with the same numerics (i.e. represents the same homology class) but is not isotopic to the one you built?

You can isotope your mysterious taut rep. to be near tau^(2) by your theorem. It has to have the same numbers of sheets as the one you built, right? So the issue is understanding exactly what’s happening in the tori M – N?

Have you tried constructing two non-isotopic surfaces with the same homological information–just changing how sheets glue up in the tori, if there was some leeway?

Or maybe I misunderstood, and you know that taut embedded surfaces are isotopically unique in their homology class when they are almost transverse to a pA flow.

Thanks!

Hi James, thanks for the question. A surface which is almost transverse to a pseudo-Anosov flow, while taut, is not always unique in its isotopy class. Likewise, two surfaces representing the same homology class which are carried by $\tau^{(2)}$ are not necessarily isotopic. The type of surgery you suggest (changing how the surface is glued up in the tori) is what I have in mind here. However, I think you’re also asking a question about how unique the position of the carried surface with respect to $\tau^{(2)}$ is *outside* the tori. As far as I know, if S is carried by $\tau^{(2)}$ it may be possible to crumple S up and then flatten it out again in a different way, if that makes sense. So my answer to “It has to have the same numbers of sheets as the one you built, right?” is also “not necessarily.”

Hi Michael, thanks for the nice talk! I have two questions about your main theorem:

1. Can you explain a bit more what you mean by a surface in $M$ being carried by $\tau^{(2)}$? If I understand correctly, there are veering triangulations whose 2-skeleton do not carry any surfaces (listed as ‘non-measurable’ in the veering triangulation census), so I presume the surfaces in your theorem escape $\tau^{(2)}$ near $\partial N$ in some way?

2. Suppose the veering triangulation $\tau$ comes from a pseudo-Anosov flow $\phi$. $\mathcal{C}_\tau$ contains the cone spanned by periodic orbits of $\phi$ (and even its homology directions), to what degree does the reverse containment hold?

Hi Chi Cheuk, glad you liked the talk and thanks for the question.

1. The definition I swept under the digital rug in my talk is the following: S is carried by $\tau^{(2)}$ if $S\cap N$ is carried by $\tau^{(2)}$ (as an honest branched surface in $N$, the manifold with boundary) and $S\cap (M-N)$ is a union of disks and annuli which $\pi_1$-inject into their respective components of $M-N$. You ask about nonmeasured triangulations. It’s possible that $\tau^{(2)}$ could carry nothing at all according to the definition I just gave. But this will happen exactly when $\mathcal{C}_\tau=H_1(M)$. In this case the codimension statement I gave should be interpreted as saying that $\tau$ determines the “empty face” of $B_x$.

2. The two cones are equal (and also equal to the cone of the homology directions of the flow, but that’s just because it is pseudo-Anosov). That’s not too hard to see, and I’d be happy to discuss more if you’re curious about it.

Hi Michael,

Thanks for answering my questions. I have one more question, although this might be slightly out of scope: Do you expect all the faces of the Thurston unit ball to arise in the situation of your theorem? This should be equivalent to the problem of: given a surface in $M$, does there exist a pseudo-Anosov flow (without perfect fits) transverse to it, which seems difficult in general. But perhaps something more can be said using the tool of veering triangulations?

Thanks!

Hi Chi Cheuk, that’s a question I’m very curious about. I wouldn’t feel comfortable making a guess at this point. There’s (unpublished) machinery of Gabai and Mosher that, given a taut surface S, produces a pseudo-Anosov flow almost transverse to S. However, I don’t think it is understood how to get a handle on whether that flow has perfect fits or not. It would be interesting if one could get around this by somehow directly constructing a veering triangulation carrying S.

Will,

Thanks for your nice talk: no pressure to respond to this on the weekend.

Is Theorem 4 something that you were looking for, or did you realize you could get it after proving the Lemma about (\epsilon, d_N)-twisted fillings? The cores of the filling tori have to be definitely shorter (d_N-smaller) than the injectivity radius of the manifold that you start with–is this just some quantitative way to say that the length of the curve that you’re filling is long enough to guarantee that the geometry of N is close enough to the geometry of M? What is d_N?

Thanks!

Hi James,

Thanks for the question! Theorem 4 actually doesn’t use the Lemma at all, or even Dehn filling. It’s actually pretty simple, and follows pretty immediately from work of Adams and Neumman-Reid, plus past work of Hoffman. Specifically, from Adams and Neumann-Reid you can get lower bounds for volumes of non-arithmetic rigid cusped orbifolds, and from Hoffman you get lower bounds on degrees of manifold covers of rigid cusped orbifolds (e.g., for an orbifold with a (3,3,3) cusp the volume is at least v_0/2 and a cover by a manifold has degree at least 12). These volume bounds of Adams and Neumann–Reid were needed for other results in our work, and so the 6v_0 bound just fell out of noticing that the other ingredient needed was already proved by Hoffman.

For the second question: Yes, that’s the idea. The intuition is, if the cusp c_0 that’s left unfilled doesn’t cover a rigid cusp, then for long enough Dehn fillings the corresponding cusp in the filling should be geometrically similar to c_0, and so also won’t cover a rigid cusp. The (\epsilon,d_N) definition is meant to nail down at what point geometric convergence kicks in, for the purpose of questions involving finite covers. It also frames the question in a way that allows for effectivization using work of Futer–Purcell–Schleimer.

d_N is the maximal degree cover from N to a non-arithmetic orbifold, and in particular one can show that d_N is bounded by 4vol(N)/v_0, so if you like you can just take d_N=4vol(N)/v_0.

Thanks!