Brandeis Topology Seminar, Spring 2025

Organizers: Carolyn Abbott (carolynabbott@brandeis.edu), Dani Álvarez-Gavela (dgavela@brandeis.edu) Kiyoshi Igusa (igusa@brandeis.edu), Thomas Ng (thomasng@brandeis.edu), Danny Ruberman (ruberman@brandeis.edu)

January 16: No seminar

January 23: Kasia Jankiewicz (UCSC / IAS).

Title: Graph braid groups and their topological complexity

Abstract: Graph braid groups are the fundamental groups of configuration spaces of particles in a graph. They can be expressed as the fundamental groups of special cube complexes. In the introductory part, I will discuss those groups, some of their properties, and their associated cube complexes. In the second part, I will talk about my joint work with Kevin Schreve, where we study a question of whether certain sets of elements in a graph braid group generate a right-angled Artin group, and use it to compute the topological complexity of graph braid groups with sufficiently many particles.

January 30: No seminar (colloquium)

February 6: Chenyang Wu (Brandeis)

Title: Bounded Geodesics on Locally Symmetric Spaces

Abstract: For a general noncompact complete Riemannian manifold, it is of particular interest to know whether there exists a bounded geodesics on it or not. In 1980s, S. G. Dani proved that for a Riemannian manifold M of constant curvature and finite Riemannian volume, the set of bounded geodesics on M has the same Hausdorff dimension as the unit tangent bundle of M. In a recent paper we generalize Dani's result to any locally symmetric space with finite volume. Moreover, for a special locally symmetric space SO_3(Z)\SL_3(R)/SL_3(Z), we can prove a winning property (stronger than full Hausdorff dimension) of the aforementioned set. This is a joint work with Lifan Guan.

February 13: Nima Hoda (Tufts)

Title: Strong shortcuts and generating sets

Abstract: A group is strongly shortcut if it has a Cayley graph in which circles cannot embed at arbitrarily large scales with arbitrarily good bilipschitz constants. This can be shown to be a special case of the Gromov mesh condition implying simply connected asymptotic cones and polynomial Dehn function. Most classes of nonpositively curved groups are strongly shortcut, including CAT(0) groups, Helly groups, systolic groups and hierarchically hyperbolic groups. I will discuss various results on strongly shortcut groups, including recent joint work with Timothy Riley in which we showed that the strong shortcut property is not invariant under change of generating sets.

February 20: No seminar

February 27: Khánh Lê (Rice)

Title: Order-preserving outer automorphisms of free and surface groups

Abstract: A group is bi-orderable if it admits a total ordering that is left and right invariant. Orderable groups have received recent attention due to their connection with dynamical group theory and with 3-manifold groups via the L-space conjecture. Given a bi-orderable group G, it is natural to ask which outer automorphisms of G preserve a bi-ordering on G since these correspond precisely to cyclic extension of G that is bi-orderable. Motivated by the connection with 3-manifolds, we focus on the case when G is a free group or a bi-orderable surface group. In this talk, I will also describe a criterion for an outer automorphism of a free group induced by a braid action to be order-preserving using the reduced Burau representation. This is a work joint with Jonathan Johnson.

March 6: Ian Biringer (Boston College)

Title: Ranks of hyperbolic manifolds and their covers

Abstract: The rank of a manifold is the minimal number of elements needed to generate its fundamental group. We’ll show that infinite sequences of bounded rank, finite sheeted covers of a fixed closed hyperbolic 3-manifold can only come from taking cyclic covers of a mapping torus. This uses an earlier theorem with Souto that gives a geometric decomposition of closed hyperbolic 3-manifolds with bounded rank and injectivity radius bounded below.

March 13: Jonathan Zung (MIT)

March 20: Joshua Perlmutter (Brandeis)

Title: The Morse Local-to-Global Property for Graph Products

Abstract: Morse local-to-global (MLTG) groups are a generalization of hyperbolic groups that includes CAT(0) groups, hierarchically hyperbolic groups (HHGs), and injective groups. MLTG groups have many nice properties including a regular language for Morse geodesics and a growth rate gap for stable subgroups. It is natural to ask how large the class of MLTG groups is. In this talk, I will expand the class of known MLTG groups by using HHG techniques to show that graph products of MLTG groups are MLTG.

March 27: Lea Kenigsberg (UC Davis)

April 3: Tam Cheetham-West (Yale)

Background talk title: Finite quotients of groups

Abstract: The finite index subgroups of a finitely generated group generate a topology on the group. We will discuss, using examples, how to organize a group's finite quotients. We’ll also give concrete examples and non-examples of groups determined by their finite quotients. 

Seminar talk title: Splittings and finite quotients of 3-manifold groups

Abstract: Embedded essential surfaces in a 3-manifold correspond to non-trivial splittings of its fundamental group. We give some conditions on the fundamental group of a Haken hyperbolic 3-manifold which guarantee that any other 3-manifold group with the same finite quotients must have a non-trivial splitting. Using one of these conditions, we obtain restrictions on the possible first Betti numbers of regular covers of aspherical integer homology spheres. This is joint work with Khánh Lê.

April 10: Rachel Skipper (Utah)

Intro Talk Title: Finiteness Properties via an Example

Abstract: A group has type F_n if it admits a classifying space with a compact n-skeleton. The class of "Thompson-like groups" are a good source of groups with interesting finiteness properties. We'll go through a proof that Thompson's group F has type F_\infty following a paper of Zaremsky.

Main Talk Title: Finiteness properties of simple groups

Abstract: Relatively little is known about the world of finitely presented infinite simple groups. In this talk, we see how to build a simple group of type F_{n-1} but not F_n for each n. This is based on joint work with Zaremsky and Witzel.

April 17: No seminar

April 24: Brandis Whitfield (Temple)
Title: Short curves of end-periodic mapping tori

Abstract: One creates a fibered 3-manifold by thickening a surface by the interval and gluing its ends by a surface homeomorphism. In the finite-type setting, much is known about how the topological data of the gluing homeomorphism determine geometric information about the hyperbolic 3-manifold. Currently, there is a lot of research activity surrounding end-periodic homeomorphisms of infinite-type surfaces. 
As an "infinite type" analogue to work of Minsky in the finite-type setting, we show that given a subsurface Y of S, the subsurface projections between the "positive" and "negative" Handel-Miller laminations provide bounds for the geodesic length of the boundary of Y as it resides in the end-periodic mapping torus. 
In these talks, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic 3-manifolds, show these techniques may be used in the infinite-type setting, and how our main theorems return results to the closed, fibered setting.

May 1: Michelle Chu (U Minnesota)

Title: Salem numbers and lengths of geodesics in arithmetic hyperbolic orbifolds 

Abstract: Salem numbers are algebraic integers all of whose conjugates have absolute value at most one. There is a direct relationship between Salem numbers and the lengths of geodesics in arithmetic hyperbolic orbifolds. In this talk, I will introduce this relationship and describe several natural questions relating the number theory to the geometry. I will also discuss recent progress on counting and realizing Salem numbers.