one-hour lectures
Boundaries of hyperbolic and relatively hypebolic spaces
We give a brief survey of boundaries of metric spaces which
are hyperbolic or relatively hyperbolic in the sense of Gromov.
We give the basic definitions, describe some examples,
and outline some applications and questions.
Variations of the Morse property
The Morse property, in its most common form, is a quantified way to express that a set is almost convex with respect to quasigeodesics. I’ll talk about several reformulations of this property, why you might prefer different versions in different situations, and show that there are effective equivalences between all of them.
Heading for infinity
Boundaries encode different ways to “head for infinity”. The goal of this talk is to give an overview of boundaries and the role they play in geometric group theory. I will begin by reviewing the definition and key properties of boundaries of hyperbolic spaces and of CAT(0) spaces. Then I will discuss some newer notions of boundaries, namely Morse boundaries and sub-linearly Morse boundaries, that are defined for more general metric spaces and groups. The talk will be purely expository.
Martin boundaries, ratio limit boundaries and applications to C* algebras
The ratio limit boundary of a random walk on a group, recently defined by Adam Dor-On is a probabilistic compactification which relates asymptotic properties of the random walk on the group to algebraic properties of a certain associated C^* algebra, the so called Toeplitz algebra. In particular, minimality of the group action on the ratio limit boundary implies that the Toeplitz algebra has a minimal group invariant quotient. As such, it is interesting to compare ratio limit boundaries of random walks on groups with hyperbolic properties to more familiar (probabilistic) Martin boundaries and geometric boundaries such as Gromov, Bowditch and (sublinearly) Morse boundaries. Generalizing work of Woess for hyperbolic groups, Adam Dor-On and I show that for symmetric random walks on relatively hyperbolic groups, all conical of the Bowditch boundary embed into the ratio limit boundary. All new terms will be defined during the talk.
Connections between boundaries of Croke-Kleiner groups
In this talk we study and compare several types of boundaries on an important and famous example: the Croke-Kleiner group as appeared in [CK00]. We will examine the topology of its visual boundaries, Morse boundary and sublinear boundaries and study the connections between all three. If time permits we will generate the conclusions to all Croke-Kleiner admissible groups.
Visual and Tits Boundaries of CAT(0) spaces via Examples
We will introduce the visual and Tits boundaries of a CAT(0) space by examining several examples. We will also discuss how these two boundaries are related and how you can use them simultaneously to prove results about a group acting geometrically on your CAT(0) space.
Random walks in Outer space
We prove that for the harmonic measure associated to a random walk on Out(Fr) satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This is joint work with Ilya Kapovich, Joseph Maher, and Catherine Pfaff.
The sublinearly Morse boundary of the mapping class group via CAT(0) cube complexes
The theory of hierarchically hyperbolic spaces (HHSes) was motivated by the observation that many (conjecturally all) cocompact CAT(0) cube complexes (CCCs) enjoy a structure analogous to that of mapping class groups and Teichmüller spaces. In words, `` the powerful machinery of subsurface projections enjoyed by mapping class groups and Teichmüller spaces applies to all known CCCs". Such an observation led not only to the definition of an HHS, but also to asking whether the opposite of the statement above holds, which is: Can tools be exported from the world of CCCs to study mapping class groups, Teichmüller spaces, and more generally HHSes? Work of Beherstock, Hagen and Sisto addresses such a question where they show that convex hulls of finite sets of points in any HHS are approximated by CCCs. I will discuss a joint work with Durham where we show that finite sets of median rays in HHSes can also be approximated by CCCs, establish connections between the geometry of the curve graph and the combinatorics of hyperplanes in the approximating CCCs and use that to develop the geometric foundations of sublinear-Morseness in the mapping class group and Teichmuller space. For instance, we prove that their sublinearly-Morse boundaries admit continuous equivariant injections into the boundary of the curve graph. Finally, we completely characterize sublinear-Morseness in terms of the hierarchical structure on these spaces.