Geometry and Topology Online

Tuesday April 28, 16:00 (UK time).

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Genevieve Walsh (Tufts)

Incoherence of free-by-free and surface-by-free groups

Abstract: A group is said to be coherent if every finitely generated subgroup is finitely presented. This property is enjoyed by free groups, and the fundamental groups of surfaces and 3-manifolds. A group that is not coherent is incoherent; it is very interesting to try and understand which groups have which property. We will discuss some of the geometric and topological aspects of this question, particularly quasi-convexity and algebraic fibers. We show that free-by-free and surface-by-free groups are incoherent, when the rank and genus are at least two.

This is joint work with Robert Kropholler and Stefano Vidussi.

Tuesday April 28, 16:30 (UK time).

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Ian Agol (Berkeley)

Virtually algebraically fibered congruence subgroups

Abstract: Addressing a question of Baker and Reid, we give a criterion to show that an arithmetic group has a congruence subgroup that is algebraically fibered. Some examples to which the criterion applies include: a hyperbolic 4-manifold group containing infinitely many Bianchi groups and a complex hyperbolic surface group.

This is joint work with Matthew Stover.

Tuesday May 5, 16:00 (UK time)

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Nathan Dunfield (UIUC)

Counting incompressible surfaces in three-manifolds

Abstract: Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic three-manifold, with the key difference that now the surfaces themselves have a more intrinsic topology. As there are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic three-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples.

This is joint work with Stavros Garoufalidis and Hyam Rubinstein.

Tuesday May 5, 16:30 (UK time)

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Priyam Patel (Utah)

Isometry groups of infinite-genus hyperbolic surfaces

Abstract: Allcock, building on the work of Greenburg, proved that for any countable group \(G\), there is a a complete hyperbolic surface whose isometry group is exactly \(G\). When \(G\) is finite, Allcock’s construction yields a closed surface. Otherwise, the construction gives an infinite-genus surface.

In this talk, we discuss a related question. We fix an infinite-genus surface \(S\) and characterise all groups that can arise as the isometry group for a complete hyperbolic structure on \(S\). In the process, we give a classification type theorem for infinite-genus surfaces and, if time allows, two applications of the main result.

This talk is based on joint work with T. Aougab and N. Vlamis.

Tuesday May 12, 16:00 (UK time)

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Neil Hoffman (OSU)

High crossing knot complements with few tetrahedra

Abstract: It is well known that given a diagram of a knot \(K\) with \(n\) crossings, one can construct a triangulation of \(S^3 - K\) with at most \(4n\) tetrahedra. A natural question is then: given a triangulation of a knot complement with \(t\) tetrahedra, is the minimum crossing number (for a diagram) of \(K\) bounded by a linear or polynomial function in \(t\)? We will answer the question in the negative by constructing a family of hyperbolic knot complements where for each knot \(K_n\) in \(S^3\) whose minimum crossing number is exponential in \(n\) but whose minimal tetrahedron number (of the knot complement) is only linear in \(n\). Similar constructions exist for torus and satellite knot complements.

This is joint work with Robert Haraway.

Tuesday May 12, 16:30 (UK time)

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Martin Scharlemann (Santa Barbara)

A strong Haken's theorem

Abstract: Suppose that \(T\) is a Heegaard splitting surface for a compact orientable three-manifold \(M\); suppose that \(S\) is a reducing sphere for \(M\). In 1968 Haken showed that there is then also a reducing sphere \(S^*\) for the Heegaard splitting. That is, \(S^*\) is a reducing sphere for \(M\) and the surfaces \(T\) and \(S^*\) intersect in a single circle. In 1987 Casson and Gordon extended the result to boundary-reducing disks in \(M\) and noted that in both cases \(S^*\) is obtained from \(S\) by a sequence of operations called one-surgeries. Here we show that in fact one may take \(S^* = S\), at least in the case where \(M\) contains no \(S^1 \cross S^2\) summands.

Tuesday May 19, 16:00 (UK time)

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Henry Segerman (OSU)

From veering triangulations to Cannon-Thurston maps

Abstract:Agol introduced veering triangulations of mapping tori, whose combinatorics are canonically associated to the pseudo-Anosov monodromy. In previous work, Hodgson, Rubinstein, Tillmann and I found examples of veering triangulations that are not layered and therefore do not come from Agol's construction.

However, non-layered veering triangulations retain many of the good properties enjoyed by mapping tori. For example, Schleimer and I constructed a canonical circular ordering of the cusps of the universal cover of a veering triangulation. Its order completion gives the veering circle; collapsing a pair of canonically defined laminations gives a surjection onto the veering sphere.

In work in progress, Manning, Schleimer, and I prove that the veering sphere is the Bowditch boundary of the manifold's fundamental group (with respect to its cusp groups). As an application we produce Cannon-Thurston maps for all veering triangulations. This gives the first examples of Cannon-Thurston maps that do not come, even virtually, from surface subgroups.

Tuesday May 19, 16:30 (UK time)

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Baris Coskunuzer (UT Dallas)

Minimal surfaces in hyperbolic three-manifolds

Abstract: The existence of minimal surfaces in three-manifolds is a classical problem in both geometric analysis and geometric topology. In the past years, this question has been settled for closed, and also for finite volume, riemannian three-manifolds. In this talk, we will prove the existence of smoothly embedded, closed, minimal surfaces in any infinite volume hyperbolic three-manifold, barring a few special cases. For further details, please see the paper.

Tuesday May 26, 16:00 (UK time)

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Daniel Allcock (UT Austin)

Big mapping class groups fail the Tits alternative

Abstract: Let \(S\) be a surface with infinitely many punctures, or infinitely many handles, or containing a disk minus Cantor set. (This accounts for almost all infinite-type surfaces.) Then the mapping class group of S fails to satisfy the Tits alternative. Namely, we construct a finitely generated subgroup which is not virtually solvable and contains no free group of rank greater than one. The Grigorchuk group is a key element in the construction.

Tuesday May 26, 16:30 (UK time)

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Talia Fernós (UNC)

Boundaries and \(\CAT(0)\) cube complexes

Abstract: The universe of \(\CAT(0)\) cube complexes is rich and diverse thanks to the ease by which they can be constructed and the many of natural metrics they admit. As a consequence, there are several associated boundaries, such as the visual boundary and the Roller boundary. In this talk we will discuss some relationships between these boundaries, together with the Furstenberg-Poisson boundary of a "nicely" acting group.

Tuesday June 2, 16:00 (UK time)

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Daniel Woodhouse (Oxford)

Quasi-isometric rigidity of graphs of free groups with cyclic edge groups

Abstract: Let \(F\) be a finitely generated free group. Let \(w_1\) and \(w_2\) be suitably random/generic elements in \(F\). Consider the HNN extension \( G = \group{F, t}{t w_1 t^{-1} = w_2}\). It is already known that \(G\) will be one-ended and hyperbolic. What we have shown is that \(G\) is quasi-isometrically rigid. That is, if a finitely generated group \(H\) is quasi-isometric to \(G\), then \(G\) and \(H\) are virtually isomorphic. The main argument involves applying a new proof of Leighton's graph covering theorem.

Our full result is for finite graphs of groups with virtually free vertex groups and and two-ended edge groups. However the statement here is more technical; in particular, not all such groups are quasi-isometrically rigid.

This is joint work with Sam Shepherd.

Tuesday June 2, 16:30 (UK time)

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Rylee Lyman (Tufts)

Outer automorphisms of free Coxeter groups

Abstract: A famous theorem of Birman and Hilden provides a close link between the mapping class group of a punctured sphere and the centraliser, in the mapping class group of a closed surface, of a hyperelliptic involution. There is a group theory analogue of this in \(\Out(F_n)\), the outer automorphism group of a free group. Namely, the outer automorphism of a free Coxeter group is linked to the centraliser, in \(\Out(F_n)\), of a hyperelliptic involution. In this talk we will meet the outer automorphism group of a free Coxeter group, try to understand the analogy with mapping class groups, and survey some recent results and interesting questions.

Tuesday June 9, 16:00 (UK time)

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Kathryn Mann (Cornell)

Large-scale geometry of big mapping class groups

Abstract: Mapping class groups of infinite type surfaces are not finitely generated; they are not even locally compact. Nonetheless, in many cases it is still meaningful to discuss their large scale geometry. We will explore which mapping class groups have nontrivial coarse geometry.

This is joint work with Kasra Rafi.

Tuesday June 9, 16:30 (UK time)

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Eric Samperton (UIUC)

How helpful is hyperbolic geometry?

Abstract: Hyperbolic geometry serves dual roles at the intersection of group theory and three-manifold topology. It plays the hero of group theory — rescuing the field from a morass of uncomputability — but the anti-hero of low-dimensional topology—seemingly responsible for much of the complexity of three-manifolds. Where do these roles overlap?

I’ll give examples of group-theoretic invariants of three-manifolds (or knots) that are NP-hard to compute, even for three-manifolds (or knots) that are promised to be hyperbolic. The basic idea is to show that the right-angled Artin semigroups of reversible circuits (a kind of combinatorial abstraction of particularly simple computer programs) can be quasi-isometrically embedded inside mapping class groups. Recent uniformity results concerning the coarse geometry of curve complexes play a key role.

This is joint work with Chris Leininger that builds on previous work with Greg Kuperberg.

Tuesday June 16, 16:00 (UK time)

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Corey Bregman (Brandeis)

Isotopy and equivalence of knots in three-manifolds

Abstract: It is a well-known fact that the notions of (ambient) isotopy and equivalence coincide for knots in \(S^3\). This is because all orientation-preserving homeomorphisms of \(S^3\) are isotopic to the identity. In this talk, we compare the notions of equivalence and isotopy for knots in more general three-manifolds.

We show that the mapping class group of a three-manifold "sees" all the isotopy classes of knots; that is, if an orientation-preserving homeomorphism fixes every isotopy class, then it is isotopic to the identity. In the case of \(S^1 \cross S^2\) we give infinitely many examples of knots whose isotopy classes are changed by the Gluck twist. Along the way we prove that every three-manifold group satisfies Grossman's Property A.

This is joint work with Paolo Aceto, Christopher Davis, JungHwan Park, and Arunima Ray.

Tuesday June 16, 16:30 (UK time)

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Yulan Qing (Toronto)

The sub-linearly Morse boundary

Abstract: The Gromov boundary, of a hyperbolic metric space, plays a central role in many aspects of geometric group theory. In this talk, we introduce a generalization of the Gromov boundary that also applies to non-hyperbolic spaces. For a given proper geodesic metric space and a given sublinear function \(\kappa\), we define the \(\kappa\)-Morse boundary to be the space of all \(\kappa\)-sublinearly-Morse quasi-geodesics rays starting at a given base point.

We show that, equipped with a coarse version of the cone topology, the \(\kappa\)-boundary is metrizable and is a QI-invariant. For some groups, we show that their Poisson boundaries can be realized on the \(\kappa\)-boundary of their Cayley graphs. These groups include all \(\CAT(0)\) groups, mapping class groups, Teichmüller spaces, hierarchically hyperbolic groups, and relatively hyperbolic groups.

This talk is based on joint projects with Ilya Gekhtmann, Kasra Rafi, and Giulio Tiozzo.

Tuesday June 23, 16:00 (UK time)

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Caroline Series (Warwick)

Geometry in non-discrete groups of hyperbolic isometries: Primitive stability and the Bowditch Q-conditions are equivalent.

Abstract: There are geometrical conditions on a group of hyperbolic isometries which are of interest even when the group is not discrete. We explain two such conditions; these are stated in terms of the images of primitive elements of the free group \(F_2\) under a \(\SL(2,\CC)\) representation. One is Minsky’s condition of primitive stability; the other is the so-called BQ-conditions introduced by Bowditch and generalised by Tan, Wong, and Zhang.

These two conditions have been shown to be equivalent by Jaijeong Lee and Binbin Xu (Trans AMS 2020) and independently by the speaker (arxiv 2019 and 2020). We will explain the ideas using an combination of both methods. If time permits, we also explain another, closely related, condition which constrains the axes of palindromic primitive elements.

Tuesday June 23, 16:30 (UK time)

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William Worden (Rice)

Dehn filling and knot complements that do not irregularly cover

Abstract: It is a longstanding conjecture of Neumann and Reid that exactly three knot complements can irregularly cover a hyperbolic orbifold -- the figure-eight 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 three 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 also partially joint with Chesebro, Deblois, Hoffman, Millichap, and Mondal.