Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Ruth Charney (charney@brandeis.edu), Kiyoshi Igusa (igusa@brandeis.edu), Danny Ruberman (ruberman@brandeis.edu)

September 6: Organizational meeting

September 13:  Assaf Bar-Natan (Brandeis)

Title:  Big Surfaces, Big Mapping Class Groups, and Grand Arcs

Abstract: A surface S is called infinite-type if it has an infinite pair-of-pants decomposition (and a really funny name). The mapping class group, or, the group of homeomorphisms of a surface up to homotopy is a mysterious object for finite-type surfaces, and even more mysterious for infinite-type. In the past few years, an explosion of interest has yielded many new research directions and areas related to so-called big mapping class groups. One way to study this group is to find a good graph upon which it acts. We will do exactly that in this talk by introducing the grand arc graph. This is based on joint work with Y. Verberne. This talk is an introductory talk to the subject of infinite-type surfaces, and is meant to expose the audience to problems, ideas, and research directions in the subject.

September 20: Abdul Zalloum (University of Toronto)

Title: Hyperbolic models for CAT(0) spaces

Abstract: Two of the most studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatorial-like structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.

September 27:  No seminar (Rosh Hashanah)

October 4: Beibei Liu (MIT)

Title: The critical exponent: old and new

Abstract: The critical exponent is an important numerical invariant of discrete groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete groups acting on hyperbolic spaces (i.e., Kleinian groups), which is a family of important examples of these three types of spaces. In particular, I will review the classical result relating the critical exponent to the Hausdorff dimension using the Patterson-Sullivan theory and introduce new results about Kleinian groups with small or large critical exponents. 

October 11: Hannah Hoganson (University of Maryland)

Title: Big Out(F_n) and its Coarse Geometry

Abstract: Recently, Algom-Kfir and Bestvina introduced mapping class groups of locally finite graphs as a proposed analog of Out(F_n) in the infinite-type setting. In this talk we will introduce the classification of infinite-type graphs, their mapping class groups, and some important types of elements in these groups.  Using a framework established by Rosendal, we will then discuss the coarse geometry of the pure mapping class groups and related properties, including results on asymptotic dimension. This is joint work with George Domat and Sanghoon Kwak.

October 18: No seminar (Monday schedule)

October 25: Lorenzo Ruffoni (Tufts)

Title: BBGs not isomorphic to RAAGs

Abstract: Given a graph, the right-angled Artin group (RAAG) on it is the group generated by vertices, in which two generators commute if and only if the corresponding vertices are adjacent. Vice versa, when one is given a group in which all relators are commutators, it is natural to ask if it is secretly isomorphic to the RAAG on some graph. Bridson has shown that this problem is in general undecidable. In joint work with Y.-C. Chang we consider this problem for the class of finitely presented Bestvina-Brady groups (BBGs), and propose a complete solution in dimension 2. In this talk I will describe the obstruction that we use to certify when a BBG is not a RAAG. This obstruction works in any dimension and comes from the BNS-invariant of the BBG. No previous knowledge of BBGs or BNS-invariants is needed.

November 1: Yvon Verberne (University of Toronto)

Title: Automorphisms of the fine curve graph

Abstract: The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.

November 8:  Eduardo Martinez-Pedroza (Memorial University of Newfoundland)

Title: Quasi-isometries of group pairs

Abstract: A recent trend in geometric group theory is to understand the large scale geometry of a group with respect to a collection of subgroups.  The objects of study are pairs consisting of a finitely generated group and a finite collection of subgroups, and there is a notion of quasi-isometry of pairs inducing an equivalence.  This relation captures classical phenomena in the study of quasi-isometric rigidity and brings up natural questions.  The talk will introduce some of these topics and discuss recent results obtained in joint work with Sam Hughes and Luis Sánchez Saldaña.

November 15:  Corey Bregman (University of Southern Maine)

Title:  Hyperbolic groups with 3-sphere boundary

Abstract:   Bartels, Lück, and Weinberger showed that a torsion-free hyperbolic group G whose boundary is the n-sphere (n≥5) is the fundamental group of a closed aspherical manifold. When n=1, this follows from Eckmann's classification of PD(2) groups, and when n=2, this is the Cannon conjecture. This leaves open the cases when n=3 and 4.  In this talk, we investigate the n=3 case by studying the structure of quasiconvex subgroups. We show that if G contains a quasiconvex, codimension-1 subgroup, then G contains a quasiconvex subgroup whose boundary is either the 2-sphere, or a space with prescribed Cech cohomology, which we conjecture is the Pontryagin sphere. This is joint work with M. Incerti-Medici.

November 22: CANCELLED

November 29: Gage Martin (MIT)

Title:  Annular links, double branched covers, and annular Khovanov homology

Abstract: Given a link in the thickened annulus, you can construct an associated link in a closed 3-manifold through a double branched coverconstruction. In this talk we will see that perspective on annular links can be applied to show annular Khovanov homology detects certain braid closures. Unfortunately, this perspective does not capture all information about annular links. We will see a shortcoming of this perspective inspired by the wrapping conjecture of Hoste-Przytycki. This is partially joint work with Fraser Binns.

December 6:  Jason Manning (Cornell University)

Title:  Stability of the action of a hyperbolic group on its boundary

Abstract:  A hyperbolic group can be compactified in a nice way by its Gromov boundary, and the left action of the group on itself extends to an action by homeomorphisms on this space.  We show that this action is dynamically stable, in the sense that any perturbation of the action is related to the standard action by a semi-conjugacy (an equivariant surjection).  Our proof additionally gives information about what form the semi-conjugacy can take.  This is joint work with Katie Mann and Teddy Weisman.

THURSDAY, December 8, at 2 pm: Steve Gindi (Brandeis) **Note the unusual day/time**

Title:  Long Time Limits of Generalized Ricci Flow

Abstract: We derive rigidity results for generalized Ricci flow blowdown limits on classes of nilpotent principal bundles. We accomplish this by constructing new functionals over the base manifold that are monotone along the flow.  This overcomes a major hurdle in the nonabelian theory where the expected Perleman-type functionals were not monotone and did not yield results. Our functionals were inspired and built from subsolutions of the heat equation, which we discovered using the nilpotency of the structure group and the flow equations. We also use these and other new subsolutions to prove that, given initial data, the flow exists on the principal bundle for all positive time and satisfies type III decay bounds. As a next step, we will apply these results to study the collapsing of generalized Ricci flow solutions and to classify type III pluriclosed flows on complex surfaces.