Topology Festival

The boundary of a hyperbolic group is the “space at infinity” of any hyperbolic space the group acts geometrically on. As a topological space that contains rich information of the hyperbolic group, the boundary is a powerful tool to understand the structure of the group and its subgroups. In this talk, we will introduce equivalent definitions of hyperbolic groups and boundaries. Through examples, we will see how a combinatorial splitting (JSJ decomposition) of a hyperbolic group can be detected through the topology of the boundary.

A marked hyperbolic metric on a closed surface of genus g > 1 is determined by the lengths of finitely many closed curves. However, a natural converse of this result can fail spectacularly: there are arbitrarily large finite collections of curves, all of whose lengths agree in every hyperbolic metric. Leininger gives an equivalence between curves whose lengths agree in every hyperbolic metric and elements of the fundamental group whose characters agree in any SL(2,C) representation. We first review various spectral rigidity (or lack-of-rigidity) results for linear representations of closed surface groups, and then focus on techniques relevant to applications to unmarked simple length spectral rigidity.

We start by recalling the definition for Geometric Structures on manifolds and some facts about Teichmüller space and Quasi-Fuchsian space, and then survey the classical theory of Pleated Surfaces defined by Thurston and further developed by Bonahon. In particular, we will discuss how to use pleated surfaces to provide a parametrization of Quasi-Fuchsian space via shearing and bending coordinates.

This talk will introduce symplectic embeddings starting with basic definitions in symplectic geometry. We will discuss the interplay of flexibility and rigidity that show up in symplectic embedding problems by looking at various examples such as Gromov's Nonsqueezing Theorem and 4-dimensional embeddings. There will be no symplectic background needed.

One of the simplest measurements you can make of the "size" of a compact symplectic manifold X is its Gromov width that measures the capacity of the largest ball that embeds into it. More generally, one can study the size of the largest ellipsoid of a given eccentricity that embeds into X. This function of the eccentricity has been (partially) calculated for certain 4-dimensional targets, such as the 4-ball or its one-point blowup, and turns out to have intricate arithmetic properties. This talk, which will be aimed at a nonspecialized audience, will describe some recent work with Nicki Magill and Morgan Weiler about the properties of this function when the target is a ball that has been blown up once with weight b.

In this talk I will describe some current efforts to understand the high-degree rational cohomology of SL_n(Z), or more generally the cohomology of SL_n(R) when R is a number ring. Although the groups SL_n(R) do not satisfy Poincare duality, they do satisfy a twisted form of duality, called (virtual) Bieri--Eckmann duality. Consequently, their high-degree rational cohomology groups are governed by an SL_n(R)-representation called the Steinberg module. The key to understanding these representations is through studying the topology of certain associated simplicial complexes. I will survey some results, conjectures, and ongoing work on the Steinberg modules, and the implications for the cohomology of the special linear groups. This talk includes work joint with Brück, Kupers, Miller, Patzt, Sroka, and Yasaki. The talk is geared for topologists and will not assume prior expertise on the cohomology of arithmetic groups!

We’ll describe some features of the global topology of the space Sub(G) of all closed subgroups of G=PSL(2,R), equipped with the Chabauty topology. The quotient of the hyperbolic plane by the action of a discrete subgroup of G is a hyperbolic 2-orbifold, and we will mostly focus on sets of subgroups where the quotient orbifold has a fixed finite topological type, and their closures in Sub(G).

In her thesis, Maryam Mirzakhani counted the number of simple closed curves on a (real) hyperbolic surface of bounded length. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll survey these connections as well as discuss answers to some qualitative strengthenings: what do long curves look like on a hyperbolic surface? And what do hyperbolic surfaces with long curves look like? Answers to these questions represent joint work with Francisco Arana-Herrera and James Farre, respectively.

Left and right lifting diagrams, like the dual left and right (Kan) extensions, can be defined in any 2-category. In this talk, we'll make the case that absolute lifting diagrams are particularly expedient for formal category theory, using them to provide a streamlined version of the classical proof that left adjoints preserve colimits and right adjoints preserve limits. This is joint work with Dominic Verity.

Pleated surfaces are an important tool introduced by Thurston to study hyperbolic 3-manifolds and can be described as piecewise totally geodesic surfaces immersed in the 3-manifold and bent along a geodesic lamination. Bonahon generalized this notion to representations of surface groups in PSL_2(C) and described a holomorphic parametrization of the resulting open chart of the character variety in term of shear-bend cocycles. In this talk I will discuss joint work with Martone, Mazzoli and Zhang, where we generalize this theory to surface group representations in PSL_d(C). In particular, I will discuss the notion of d-pleated surfaces, and their holomorphic parametrization.

I will talk about joint work with Andrew Lobb related to Toeplitz's square peg problem, which asks whether every (continuous) Jordan curve in the Euclidean plane contains the vertices of a square. Specifically, we show that every smooth Jordan curve contains the vertices of a cyclic quadrilateral of any similarity class. I will describe the context for the result and its proof, which involves symplectic geometry in a surprising way.

Relatively hyperbolic groups generalize geometrically finite Kleinian groups acting on real hyperbolic space H^3. The boundaries of relatively hyperbolic groups generalize the limit sets of Kleinian groups. Since the boundary of H^3 is S^2, the limit set of every Kleinian group is planar. Can every relatively hyperbolic group with planar boundary be realized a Kleinian group? The answer is no, and we will give illustrative examples to show the many ways this can fail. However, we prove that if G is one-ended, (G,P) is relatively hyperbolic, and the boundary of (G,P) is planar and without cut points, then all the peripheral subgroups are surface groups. This is consistent with G being a Kleinian group. We also formulate a conjecture about the general situation (which is an extension of the Cannon Conjecture). This is joint work with Chris Hruska.

Topological Hochschild homology (THH) is an invariant of associative ring spectra, closely related to algebraic K-theory. Bhatt–Morrow–Scholze defined a "motivic filtration" on the THH of ordinary commutative rings, after completion at a prime number p, relating it to invariants (new and old) in p-adic Hodge theory. These new structures have led to many advancements in the study of algebraic K-theory and in arithmetic geometry. In this talk, I will try to give an accessible overview of this landscape of invariants, and then discuss a new construction of the motivic filtration on THH, with the advantages that it is quite simple to state and applies not only to ordinary commutative rings but a broad class of commutative ring spectra; the new work is joint with Jeremy Hahn and Dylan Wilson.

Given a pair of finite degree (not necessarily regular) covers (p,X),(q,Y) of a finite type surface S, we show that the covers are equivalent if and only if the following holds: for any closed curve gamma on S, some power of gamma admits an embedded lift to X if and only if some power of gamma admits an embedded lift to Y. We apply this to study the well-known construction of Sunada which yields pairs of hyperbolic surfaces (X,Y) that are not isometric but that have the same unmarked length spectrum. In particular we show that the length-preserving bijection from closed geodesics on X to those on Y arising from the Sunada construction never sends simple closed geodesics to simple closed geodesics. We also show that length-isospectral surfaces arising from several of the most well-known manifestations of the construction are not simple length isospectral. Even more, we construct length-isospectral hyperbolic surfaces so that for each finite n, the set of lengths corresponding to closed geodesics with at most n self intersections disagree. This represents joint work with Maxie Lahn, Marissa Loving, and Nicholas Miller.