• Ser Peow Tan

Title : Identities on hyperbolic surfaces old and new 

Abstract : We will survey some older results on identities on hyperbolic surfaces due to Basmajian, McShane, Bridgeman and Luo-Tan, and talk about some newer results on generalizations of the Basmajian and Bridgeman identities to surfaces with only cusps and/or cone points, and the main ideas behind the proofs. The newer results are based on joint work with Ara Basmajian, Hugo Parlier and Nhat Minh Doan.

• Ara Basmajian

Title : Curves, optimal metrics, and the inf spectrum of moduli space 

Abstract : The focus of this talk will be on finding metrics tailored to curves. That is, given an essential closed curve on a closed surface of genus at least 2, find, among all hyperbolic metrics, a metric that minimizes the length of the curve. We call this the optimal (designer) metric tailored to the curve, and the length of the curve with respect to this optimal metric is called the inf invariant of the curve. It is not difficult to see that the inf invariant remains unchanged in the mapping class group orbit of a curve.

By considering the inf invariant over each mapping class group orbit of a curve we are naturally led to a discrete spectrum of positive real numbers we call the inf spectrum of the topological surface. This spectrum is independent of metric and is associated to the moduli space of hyperbolic structures. In this talk, we outline a construction leading to coarse growth bounds for this spectrum. This is joint work with Sayantika Mondal and Hugo Parlier.

• Federica Fanoni

Title : Brouwer classes: tempered, well tempered and flows

Abstract : An orientation-preserving homeomorphism of the plane is a Brouwer homeomorphism if it has no fixed points. By looking at the mapping class of such a homeomorphism relative to finitely many orbits, one obtains a mapping class of an infinite-type surface. For these surfaces an interesting family of mapping classes are the tempered ones: loosely speaking, those without any pseudo-Anosov behavior. I will discuss work with Juliette Bavard, Frédéric Le Roux and Nelson Schuback where we relate the fact that a Brouwer class is tempered to a dynamical property of the underlying homeomorphism.

• Sanghoon Kwak 

Title : Thurston's theorem: entropy in dimension one 

Abstract : In his paper named "Entropy in dimension one," Thurston showed that a positive real number h occurs as the topological entropy for an ergodic train track representative of an outer automorphism of a free group if and only if its expansion constant exp(h) is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. In this talk, I will present a streamlined account of Thurston's proof, following joint expository work with Ryan Dickmann, George Domat, Thomas Hill, Carlos Ospina, Priyam Patel, and Rebecca Rechkin. 

• Youngju Kim

Title : Tubular neighborhood theorem in complex hyperbolic n-manifolds

Abstract : In a real hyperbolic $2$-manifold, the collar lemma says that a closed geodesic has an embedded tubular neighborhood whose width depends only on the length of the geodesic. In fact, a codimension $1$ totally geodesic embedded submanifold in a real hyperbolic $n$-manifold also has such a tubular neighborhood that depends only on its $(n-1)$-volume. The width of the tubular neighborhood does not depend on the geometry of underlying manifold. On the other hand, a totally geodesic surface with codimension bigger than $1$ in a hyperbolic manifold can be arbitrarily close to itself.

Here, we will discuss a tubular neighborhood theorem for an embedded complex codimension $1$ submanifold in a complex hyperbolic n-manifold. We note that complex codimension $1$ is real codimension $2$. We provide an explicit estimate for this width that depends only on the codimension $1$ volume of the embedded complex submanifold. We present two applications of this tubular neighborhood theorem: the first is a lower volume bound for such manifolds, and the second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold. This is joint work with Ara Basmajian. 

• Jan Kim

Title : Maximal free abelian subgroups of generalized cactus groups

Abstract :  Cactus groups are finitely presented groups acting properly and cocompactly on median graphs, and therefore can be viewed as nonpositively curved groups. This naturally leads to the question of whether they are hyperbolic. For cactus groups, their hyperbolicity can be characterized by the absence of subgroups isomorphic to \mathbb{Z}^2. Regarding this, Genevois studied the algebraic dimension of cactus groups and proposed a conjecture on the maximal ranks of their free abelian subgroups. Motivated by Genevois's work, we study free abelian subgroups of generalized cactus groups, which form a natural generalization of cactus groups. In this talk, we discuss upper bounds on the ranks of free abelian subgroups of generalized cactus groups, and construct explicit free abelian subgroups realizing these bounds in several cases. This is joint work with Junseok Kim.

• Bram Petri

Title : Apollonian random manifolds and their bass notes

Abstract : I will speak about joint work with Will Hide, Anna Roig Sanchis and Joe Thomas. We use a model of random hyperbolic 3-manifolds to investigate the set of all possible spectral gaps of hyperbolic 3-manifolds of finite volume.

• Hyeran Cho

Title : Random branched covers of groups

Abstract : We construct a random model for an n-fold branched cover of a finite acceptable 2-complex X. This includes presentation 2-complexes for finitely presented groups satisfying some mild conditions. For any k > 0, we show that as n goes to infinity, a random branched cover asymptotically almost surely is homotopy equivalent to a 2-complex satisfying geometric small cancellation C'(k). As a consequence the fundamental group of a random branched cover is asymptotically almost surely Gromov hyperbolic, coherent and has small cohomological dimension. This is joint work with Jean-Francois Lafont and Rachel Skipper.

• Inhyeok Choi

Title : Counting closed curves for subgroups of mapping class groups

Abstract : Let X be a closed negatively curved surface and let C be a closed curve on X. How many closed curves on X are there with the same topological type as C and with length less than L? This is a counting problem for the mapping class group orbit. Mirzakhani’s celebrated theorems assert that the count grows polynomially, with precise asymptotics. This was later generalized by Rafi and Souto, who counted filling curves with respect to a positive length function. 

In this talk, I will present an analogous counting problem for non-elementary subgroups of mapping class groups. Key ingredients include the north-south dynamics of pseudo-Anosov mapping classes and the non-arithmeticity of the length spectrum. Joint work with Dongryul M. Kim.

• Sebastian Hensel

Title : Rotation sets as seen from the fine curve graph

Abstract : The rotation set is a conjugacy invariant for torus homeomorphisms, which generalizes the rotation number in the case of the circle. We will present joint work with Frédéric Le Roux in which we show that the shape of the rotation set can (in many cases) be determined from the action of the homeomorphism on the fine curve graph. (No familiarity with rotation sets will be assumed!)