## Virtual Topology & Group Theory Seminar Vanderbilt University Spring 2021

Links to seminar schedule for previous years:
2010/11, 2011/12, 2012/13, 2014/15, Spring 2016, Fall 2016, Spring 2017, 2017/18, 2018/19 2019/20 Fall 2020

Organizers: Spencer Dowdall & Denis Osin

Time: Wednesdays, 4:10–5:00pm US Central Time

Format: Due to COVID the seminar will meet Online via Zoom Meeting.
Please email one of the organizers for the Zoom Meeting invitation and join link.

Wednesday, February 17, 2021

Spencer Dowdall (Vanderbilt)

Title: Geometric finiteness and Veech group extensions of surfaces groups

Abstract: The notion of "convex cocompactness" and its generalization "geometric finiteness" play an important role the classical theory of Kleinian groups, that is, discrete subgroups of isometries of hyperbolic space. By means of analogy, in 2002 Farb and Mosher defined a subgroup of the mapping class group of a closed surface to be convex cocompact if it acts cocompactly on a quasi-convex subset of Teichmuller space. These subgroups have received much attention and it is known that they are precisely the subgroups whose corresponding surface group extensions are Gromov hyperbolic. However, it remains unclear precisely how geometric finiteness should manifest in mapping class groups nor how it should relate to the geometry of surface group extensions. This talk will look at perhaps the most compelling candidates for geometric finiteness in mapping class groups, namely Veech subgroups. I will explain the structure of Veech groups and show that their corresponding surface group extensions exhibit strong aspects of negative curvature and in fact are hierarchically hyperbolic. Joint work with Matt Durham, Chris Leininger, and Alex Sisto.

The Zoom recording of this talk can be viewed here (until March 3): https://vanderbilt.zoom.us/rec/share/7uTkNOOAqGG9QPg3kJRq1Qb-S6fjrX2fj6jpgIvFIK6GVd7LKxx0ZMSxGenar5T1.dvrvTqWhKDrtD8WB.

Wednesday, February 24, 2021
Special time: 5:10-6:00pm US Central Time

Hoang Thanh Nguyen (Peking University)

Title: Subgroup distortion of 3-manifold groups

Abstract: A finitely generated group G can be considered as a geometric object by equipping the group with a word length metric. A finitely generated subgroup of G itself admits a word length metric, but it also inherits an induced metric from the group G. The distortion of the subgroup in G compares these metrics on the subgroup. For an arbitrary subgroup of a finitely generated 3-manifold group, we show that the subgroup distortion can be only linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3-manifold group is closely related to the separability of this subgroup. This is a joint work with Hongbin Sun.

The Zoom recording of this talk can be viewed here (until March 10): https://vanderbilt.zoom.us/rec/share/Fi1j3wGbSBlBdBX2YBItK0ocUQiVRf5wXDgQ3puwZ3YPO6dOw4tK8F-VlSPsTZ0.RECHit5Vu1h5GL4g.

Wednesday, March 3, 2021

Ian Runnels (University of Virginia)

Title: RAAGs in MCGs

Abstract: Inspired by Ivanov's proof of the Tits alternative for mapping class groups via ping-pong on the space of projective measured laminations, Koberda showed that right-angled Artin subgroups of mapping class groups abound. We will outline an alternate proof of this fact using the hierarchy of curve graphs, which lends itself to effective computations and stronger geometric conclusions. Time permitting, we will also discuss some applications to the study of convex cocompact subgroups of mapping class groups.

The Zoom recording of this talk can be viewed here (until March 17): https://vanderbilt.zoom.us/rec/share/4-H-kT03zBUYZxwquSiTpH5WF_G-AVHX0sURIRH6dfNQ6o6-ur19l2CLVQufWWfg.nULBptEzZdpyRMyU .

Wednesday, March 10, 2021

Michael Mihalik (Vanderbilt)

Title: Relatively hyperbolic groups with semistable fundamental group at $$\infty$$

Abstract: We are interested in two long standing questions concerning the asymptotic behavior of finitely presented groups. The first, attributed to H. Hopf (probably in the 1940's) asks: Is $$H^2(G;\mathbb ZG)$$ free abelian for all finitely presented groups $$G$$? The second arose in the late 1970's and asks: Do all finitely presented groups have semistable fundamental group at $$\infty$$? We consider these questions for relatively hyperbolic groups. In 2017, we answered both questions in the affirmative when the boundary of the relatively hyperbolic group did not have a cut point. In the presence of cut points, one outstanding problem remained. We solved the cohomology version that problem in 2018 and recently we have shown: If $$G$$ is a finitely presented 1-ended group that is hyperbolic relative to a finite collection of finitely presented groups $$P_i$$ and each $$P_i$$ has semistable fundamental group at $$\infty$$, then $$G$$ has semistable fundamental group at $$\infty$$. A key idea is that of nearly geodesic proper homotopies in a cusped space for $$G$$. (Joint with Matt Haulmark)

The Zoom recording of this talk can be viewed here (until March 24): https://vanderbilt.zoom.us/rec/share/iJMO4xUP3GsMGVR16s3FciQRjoGutWr6T-hiR29N12ZJObMHu3wVWbiIW6SjHT0.NQ6rO-Z4zXkXdVR6 .

Wednesday, March 17, 2021

Anna Marie Bohmann (Vanderbilt)

Title: Algebraic K-theory for Lawvere theories: assembly and Morita invariance

Abstract: Lawvere theories are a way of encoding algebraic structures, such as those of modules over a ring or sets with a group action. They are simultaneously very flexible and very rigid and are useful tools in universal algebra as well as algebraic topology and other fields. In this talk, we discuss an extension of the construction of algebraic K-theory, usually an invariant of rings, to the context of Lawvere theories. Our construction of algebraic K-theory builds in information about automorphism groups of these structures. We'll discuss both particular examples and general constructions in the K-theory of Lawvere theories, including examples showing the limits of Morita invariance and the construction of assembly-style maps. This is joint work with Markus Szymik.

The Zoom recording of this talk can be viewed here (until March 31): https://vanderbilt.zoom.us/rec/share/saqquvWly04i-Tq7PXZUGuT77uq8ZxDbRkhxuj9WM_0hmVhu2qK6a0OMRK_U_8GR.d3PbQxYfYw_cZX4r .

Wednesday, March 24, 2021

Emily Stark (Wesleyan)

Title: Action Rigidity for Graphs of Manifold Groups

Abstract: The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we prove certain graphs of manifold groups are action rigid. Consequently, we obtain examples of quasi-isometric groups that do not virtually have a common model geometry. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

The Zoom recording of this talk can be viewed here (until April 7): https://vanderbilt.zoom.us/rec/share/1PiSWqhFzHhsLF60-2GS0fQnvbq7cKodj0T64Mh-t_B_qQjN4GezCdUqt3etNAhJ.fXCsRw6yl0hSSMDC .

Wednesday, March 31, 2021

Michael Hull (University of North Carolina, Greensboro)

Title: Maps between 3-manifolds

Abstract: We study the maps between compact 3-manifolds by considering the induced homomorphisms between fundamental groups, and we prove that these sets of homomorphisms satisfy a uniform group-theoretic analogue of the Hilbert Basis Theorem. We apply this theorem to give a descending chain condition on $$\pi_1$$-surjective maps between compact 3-manifolds. This is joint work with D. Groves and H. Liang.

The Zoom recording of this talk can be viewed here (until April 14): https://vanderbilt.zoom.us/rec/share/qTNdrkV3qVU4eYtI3XWOMvhtd1Q857WQ4q_fi5bKj-Y4dgT_DOdvtOUBRCGPukhO.CBT9_h0cJWAam1el .

Wednesday, April 7, 2021

Srivatsav Kunnawalkam Elayavalli (Vanderbilt)

Title: A primer to proper proximality

Abstract: I will define the notion of proper proximality introduced by Boutonnet-Ioana-Peterson and provide a friendly introduction to the kind of techniques involved in showing this property for groups. There is an interest at the moment to identify new examples of such groups, both at the von Neumann algebraic and at the group theoretic level. I will make a brief attempt at explaining the von Neumann algebraic side of things, since this seminar is more geared to group theorists. However, the main purpose of this talk is to prepare the audience for the next talk in our seminar wherein the speaker and co authors prove proper proximality for a vast family of groups including mapping class groups.

The Zoom recording of this talk can be viewed here (until April 23): https://vanderbilt.zoom.us/rec/share/KzDWv3f5k21zR7ePyOZBBFUELGS-XCFAyA3RUJ6oOYKzw9jpkfVy-fxzDwX2ZaBy.V6EUcUvIQ1la7Zzf .

Wednesday, April 14, 2021

Jingyin Huang (The Ohio State University)

Title: Proper proximality for groups with features of non-positively curvature

Abstract: Proper proximality of a countable group is a notion introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In this talk, we prove proper proximality for a few classes of groups with features of non-positive curvature, including CAT(0) groups with rank one elements, groups acting properly and minimally on locally finite affine buildings, and mapping class groups (or more generally many hierarchically hyperbolic groups). As a consequence, we deduce a rigidity result for weakly compact actions of mapping class groups from proper proximality and an orbit equivalence rigidity result of Kida. Our proof relies on a dynamical criterion for proper proximality, which is a variation of the north-south dynamics. I will explain why this criterion holds in each of the above-mentioned case. This is joint work with Camille Horbez and Jean Lecureux.

The Zoom recording of this talk can be viewed here (until April 30): https://vanderbilt.zoom.us/rec/share/JpjsD3-PZJVFEbGbcgAfSrXPDTbZipAgf-Lvs43HQBqhgHBCQknszUPhfIeH6Q6q.OOft4C0h5FtCld5g .

Wednesday, April 21, 2021

Michael Landry (Washington University in St. Louis)

Title: 3-manifolds, surfaces, and the veering polynomial

Abstract: I will describe work concerning the Thurston norm on the second homology of a 3-manifold and its interaction with foliations and flows. This norm is a 3-manifold invariant with connections to many areas: geometric group theory, foliation theory, Floer theory, and more. There are some beautiful clues due to Thurston, Fried, Mosher, Gabai, McMullen, and others that indicate there should be a dictionary between the combinatorics of the norm's polyhedral unit ball and the geometric/topological structures existing in the underlying manifold. The picture is incomplete, and mostly limited to the case when the manifold fibers as a surface bundle over the circle. I will explain some new results which hold not just in the fibered case but also the more general setting of manifolds admitting veering triangulations (introduced by Agol). The main focus will be on joint work with Yair Minsky and Samuel Taylor, in which we use a veering triangulation to define a polynomial invariant which generalizes McMullen's Teichmuller polynomial.

The Zoom recording of this talk can be viewed here (until May 7): https://vanderbilt.zoom.us/rec/share/YEMN-TKlT9mkZIGcb_i1nQzQvhSCq5oBurZ4HvezIuJEPV3zWlrrkzkYI36aGwrR.NYG2l5uDIZBbKIuw .

Wednesday, April 28, 2021
Special Time: 10:10-11:00am US Central Time

Yash Lodha (Korea Institute for Advanced Study)

Title: Some new constructions and directions in the theory of left orderable groups.

Abstract: I will define two new constructions of finitely generated simple left orderable groups (in recent joint work with Hyde and Rivas). Among these examples are the first examples of finitely generated simple left orderable groups that admit a minimal action by homeomorphisms on the Torus, and the first family that admits such an action on the circle. I shall also present examples of finitely generated simple left orderable groups that are uniformly simple (these were constructed by me with Hyde in 2019). And present new examples that, somewhat surprisingly, have infinite commutator width. In another part of the talk I shall discuss ongoing work with Elayavalli and Goldbring, where we initiate a study of the Polish space of enumerated groups that satisfy one among various notions of orderability.

The Zoom recording of this talk can be viewed here (until May 14): https://vanderbilt.zoom.us/rec/share/iMExZ7QsrF99OPFBvuchuo6ZlwitL7sSQAMfx4HPgblGz4N8CYZFULBXldyxxxI.F4zeHOTUqqPmDNx9 .