Brandeis Topology Seminar, Spring 2023

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

January 17: First day of classes, no seminar

January 24:  Hokuto Konno (Univ. of Tokyo, MIT)

Title:  Homological instability for moduli spaces of 4-manifolds

Abstract: We prove that homological stability with respect to connected sums of S^2×S^2 fails for moduli spaces BDiff(X) of simply-connected closed 4-manifolds X. This makes a striking contrast with other dimensions: in all even dimensions except for 4, analogous homological stability for moduli spaces has been established by work of Harer and of Galatius and Randal-Williams. The proof of the above result is based on a characteristic class constructed using the Seiberg-Witten equations. This is joint work with Jianfeng Lin.

January 31: Carolyn Abbott (Brandeis University)

Title: Morse boundaries of CAT(0) cube complexes

Abstract: The visual boundary of a hyperbolic space is a quasi-isometry invariant that has proven to be a very useful tool in geometric group theory.  In particular, there is a well-defined notion of the visual boundary of a hyperbolic group.  When one considers CAT(0) spaces, however, the situation is more complicated, because the visual boundary is not a quasi-isometry invariant.  Instead, one can consider a certain subspace of the visual boundary, called the (sublinearly) Morse boundary.  In this talk, I will describe a new topology on this boundary and use it to show that  the Morse boundary with the restriction of the visual topology  is a quasi-isometry invariant in the case of (nice) CAT(0) cube complexes.  This is in contrast to Cashen's result that the Morse boundary with the visual topology is not a quasi-isometry invariant of CAT(0) spaces in general.  This is joint work with Merlin Incerti-Medici.

February 7: Rose Morris-Wright (Middlebury College)

Title:  Rewrite systems in 3-free Artin groups

Abstract:  Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. In particular, it is unknown whether the word problem is solvable for all Artin groups. I will discuss a new algorithm for solving the word problem in 3-free Artin groups. This is based on work by Holt and Rees for large type and sufficiently large type groups (2012 and 2013). Our work significantly broadens the class of Artin groups with solvable word problem. This algorithm gives an explicit way to reduce a word to a geodesic form without ever increasing the length of the word.  (Joint work with Ruben Blasco-Garcia and Maria Cumplido.)

February 14:  Rocky Klein (Brandeis)

Title:  Elementary equivalence of Baumslag-Solitar groups and related results

Abstract: Every field of math has a notion of equivalence of objects. In algebra there is isomorphism, in geometric group theory there is quasi-isometry, and in model theory there is elementary equivalence. Two models are said to be elementarily equivalent if they have the same first order theory, i.e. if they satisfy the same sentences in first order logic. Groups are models for the group axioms in the language of groups and therefore one can ask whether two groups are elementarily equivalent. These are questions of the form, "from the perspective of first order logic, when are two groups the same?" In this expository talk I will be discussing two recent (2020) results from Casals-Ruis and Kazachkov on elementary equivalence of Baumslag-Solitar groups while mentioning related results. No logic background is required as I will review basic definitions and work up to the notion of elementary equivalence. 

February 21: February break, no seminar

February 28:  Double Header!

Talk 1: Rylee Lyman (Rutgers, Newark), 2:15pm

Title:  What's So Cool About Deformation Spaces of Tree Actions?

Abstract:  Many groups interesting to geometric group theorists act on trees: for example, free groups, surface groups, Baumslag–Solitar groups, fundamental groups of many 3-manifolds, groups with more than one end or with a nontrivial JSJ decomposition, and more. When a group acts without global fixed point on a tree, it typically acts on many trees in many ways. Work of Culler–Morgan, Culler–Vogtmann, Forester, Clay and Guirardel–Levitt organizes these different actions into deformation spaces. The goal of this talk is to introduce deformation spaces of tree actions (and their spines) and think about ways to begin to work with them. I will attempt to mention at least one open problem and at least one theorem of mine.

Talk 2: Michael Hull (University of North Carolina, Greensboro), 3:30pm

Title: Random walks and convex cocompacntess in acylindrically hyperbolic groups

Abstract: For a group G acting acylindrically on a hyperbolic metric space X, we say a subgroup  H is convex cocompact if the orbit map quasi-isometrically embedds H into X. We study how the elements produced by random walks on G interact with these convex cocompact subgroups. In particular, we show that the subgroup generated by H and a random element is a convex cocompact free product and that the action of G on the space of all infinite index convex cocompact subgroups is topologically transitive. This is joint work with C. Abbott, A. Minasyan, and D. Osin.

March 7: Ian Montague (Brandeis)

Title:  Equivariant Correction Terms in Seiberg-Witten Floer Homology

Abstract: In this talk I will discuss the computation of index-theoretic equivariant correction terms coming from the Seiberg-Witten Floer spectrum associated to spin rational homology three-spheres equipped with a cyclic group action defined in previous work, whose mod 2 reductions constitute equivariant analogues of the Rokhlin invariant. In the case of Seifert-fibered homology spheres, we can generalize Nicolaescu's computations of the eta invariants of the Dirac operator in order to provide explicit formulas for these equivariant correction terms. I will then give several conjectures about the behavior of these correction terms for certain families of Brieskorn spheres, and provide evidence for these conjectures using some computer-assisted computations.

March 14:  CANCELLED DUE TO WEATHER

March 21: Sally Collins (Georgia Tech)

Title: Homology cobordism & knot concordance

Abstract: We can associate to a knot K in the 3-sphere a 3-manifold, called the zero-surgery of K, via performing zero-framed Dehn surgery on the knot. It is a natural question to ask: if two knots have zero-surgeries which are homology cobordant via a cobordism that preserves the homology class of the two positively oriented knot meridians, does that imply that the knots must be smoothly concordant? We review the history and motivation of this question and questions like it and describe our own result, ending with a brief discussion of a proof technique of obstructing torsion in the smooth concordance group.

March 28:  Francesco Lin (Columbia)

Title:  Homology cobordism and the geometry of hyperbolic three-manifolds

Abstract:  A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, I will discuss how monopole Floer homology can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying some natural geometric constraints.

April 4: CANCELLED

April 11:  Passover break, no seminar

April 18:  Alexandre Martin (Heriot-Watt University)

Title:   Subgroups generated by large powers and the Tits Alternative for Artin groups

Abstract:   A well-known conjecture in geometric group theory asserts that non-positively curved groups satisfy a strong form of the Tits Alternative: their finitely generated subgroups are either virtually abelian or contain a non-abelian free subgroup. While this conjecture has been proved in many cases, it is still open in general, and in particular for the class of Artin groups. I will talk about a recent proof of this alternative for two-dimensional Artin groups. The approach is geometric in nature and relies on the action of these groups on a nice CAT(0) complex, their Deligne complex.  Along the way, we develop tools to construct free subgroups, and even prove in certain cases a rather surprising result: given any two elements of an Artin group of extra-large type, one can find suitable powers that either commute or generate a free subgroup. This answers in the positive a question of Wise for this class of Artin groups. 

April 25: Lam Pham (Brandeis)

Title:  Short closed geodesics in compact locally symmetric spaces

Abstract: Given a symmetric space X of non-compact type, we consider the closed geodesics of all possible locally symmetric spaces arising as quotients of X by a discrete (torsion-free) group of isometries. Margulis conjectured the existence of a uniform lower bound on the length of the shortest closed geodesic of any arithmetic locally symmetric space. In joint work with F. Thilmany, we proved that this conjecture is equivalent to a weak version of the Lehmer conjecture, a well-known problem from Diophantine geometry. In joint work with M. Fraczyk, we studied the bottom of the length spectrum when the group of isometries is a simple Lie group of any rank conditional on a uniform lower bound on Salem numbers, a much weaker -- but still open -- problem. In particular, our work establishes several unconditional results on the length spectrum including a precise obstruction to short lengths and a uniform lower bound for the combined lengths of pairs of pairs of closed geodesics. The proof of these results use tools from algebraic groups and Diophantine geometry.

THURSDAY April 27, 3:50-4:50pm, in Goldsmith 226: Jason Behrstock (CUNY) 

Title: Quasiflats and quasi-isometric rigidity in hierarchically hyperbolic spaces

Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. This discussion will center around work in which we classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. In particular, I will discuss how this study of quasiflats can be used to prove quasi-isometric rigidity for certain families of groups. This is joint work with Mark Hagen and Alessandro Sisto.

2021-2022 Topology Seminar Schedule