# Geometry and Topology Online

## Term III, 2019-2020

Please contact Saul Schleimer if you would like to speak or to suggest a speaker. Please contact Gillian Kerr if you have questions about using Zoom. The ICMS page for the seminar has a time zone calculator, a link to the slack channel, links to the videos, and other invariant information.

The seminar will run weekly, with two 25 minute talks separated by a five minute break. The first talk will start on the hour, and the second on the half hour. We'll open the Zoom session 15 minutes before we start; we'll close the Zoom session about 30 minutes after we finish. Note that no password is required; links to the zoom session for each talk are below.

 Tuesday April 28, 16:00 (UK time). 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). 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) Eric Samperson (UIUC) How helpful is hyperbolic geometry? Abstract: Hyperbolic geometry serves dual roles at the intersection of group theory and 3-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) Corey Bregman (Brandeis) TBA Abstract: TBA

 Tuesday June 16, 16:30 (UK time) TBA (TBA) TBA Abstract: TBA

 Tuesday June 23, 16:00 (UK time) TBA (TBA) TBA Abstract: TBA

 Tuesday June 23, 16:30 (UK time) TBA (TBA) TBA Abstract: TBA

Information on past talks. This line was last edited 2020-04-17.

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