Location: Online (Zoom)

Title: Connected components of spaces of type-preserving representations

Abstract: We investigate the spaces of representations of surface groups into PSL(2, R). For a closed surface, by the classic result of Goldman, the Euler class together with the Milnor-Wood inequality provide a complete classification of the connected components of the spaces of the representations. However, describing the connected components becomes more subtle when considering the space of type-preserving representations for punctured surfaces. In this talk, I will present a recent joint work with Tian Yang that addresses this problem.

Location: Online (Zoom)

Title: Geometry, topology, and combinatorics of fine curve graph variants

Abstract: The fine curve graph of a surface S is a graph whose vertices are essential simple closed curves in S and whose edges connect curves that are disjoint. This contrasts past work on the curve graph, which is similar to the above but considers everything up to isotopy. In this talk, we will explore existing and new results on both classical and fine curve graphs and their variants. We will reach into the fields of geometry, topology, and combinatorics to gain new perspectives on these graphs and the surfaces used to construct them. We will further prove a sampling of results about finitary curve graphs, whose vertices are essential simple closed curves in S and edges connect curves that intersect at finitely many points.

Location: Online (Zoom)

Title: Train tracks for end periodic graph maps

Abstract: Relative train track representations for mapping classes of finite graphs encodes important topological and dynamical properties of these maps and serve as a kind of “normal form” of them. With Yan Mary He we developed an analogous construction for some proper mapping classes of infinite graphs, namely the end periodic maps. I will give an overview to our construction and also discuss some applications.

Location: KIAS 1423

Title: On the kernel of group actions on asymptotic cones

Abstract: The concept of an asymptotic cone was first suggested by Gromov and he used it to establish Gromov's polynomial growth theorem. An asymptotic cone of a group reflects many properties of the group. For example, a group is virtually nilpotent if and only if all of its asymptotic cones are locally compact (equivalently, proper). Also, a finitely generated group is hyperbolic if and only if all of its asymptotic cones are real trees.

In this talk, we characterize the natural kernel of the action of a group G on its asymptotic cone. Our main theorem states that if G is acylindrically hyperbolic, then the kernel of G-action on an asymptotic cone of G is the same as K(G), the unique maximal finite normal subgroup. Then it turns out that the kernel also coincides with many algebraically defined subgroups. Moreover, this result does not depend on the choice of ultrafilter and sequence that we need to define asymptotic cones so it implies that the kernel is invariant under the choice of these. It is known that a group may have distinct (actually, non-homeomorphic) asymptotic cones, and indeed some acylindrically hyperbolic groups have various asymptotic cones.

As an application, we relate this kernel to other kernels of group actions on other spaces at "infinity", for instance, the limit set of convergence group action, Floyd boundary, and many boundaries of CAT(0) spaces with some conditions. In addition, we will introduce another action of G on an asymptotic cone, called Paulin's construction, and describe the kernel of Paulin's construction. If time permits, we will prove that the kernel can determine whether a given action is non-elementary, under specific conditions, and give the kernel of several (not acylindrically hyperbolic) groups. This work is joint with my advisor, Hyungryul Baik.

Location: KIAS 8406 (4th floor of Building #8)

Title: Hyperbolic-like actions of hyperbolic groups

Abstract: The actions of Fuchsian groups in the circle at infinity can be generalized in terms of hyperbolic-like actions. In general, hyperbolic-like actions are more flexible than the Fuchsian group actions. In this talk, I will give an overview of the previous researches about the flexibility and discuss related conjectures. This talk is based on ongoing projects and discussions with Michele Triestino and with Shuhei Maruyama.

Location: KIAS 1423

Title: Outer automorphism groups of right-angled Artin groups and maximal SIL-pair systems

Abstract: The existence of separating intersections of links (or SIL-pairs, for short) serves as a criterion for classifying the defining graph of right-angled Artin groups. Regarding this notion, we will define a maximal SIL-pair system to combine the parts of graphs generating SIL-pairs, so that one can decompose defining graphs to understand the group structures of the outer automorphism groups of right-angled Artin groups. We will also see how maximal SIL-pair systems help us to detect acylindrical hyperbolicity of a subgroup of the outer automorphism group of a right-angled Artin group, called the pure symmetric outer automorphism group.

Location: KIAS 8101 (Building #8, First floor - next to the tea room)

Title: Comparing Two Lengths in the Mapping Class Group of a Handlebody

Abstract: The mapping class group of a 3-dimensional 1-handlebody, often referred to as the handlebody group, is a finitely presented group that plays a significant role in 3-manifold theory and geometric group theory. This group exhibits two distinct natural isometric actions on Teichmüller space and Outer Space, respectively. In this talk, we analyze these actions through the lens of stable translation length. This research is conducted in collaboration with KyeongRo Kim.

Location: KIAS 8406  (4th floor of Building #8)

Title: Topological normal generation of big mapping class groups 

Abstract: For closed surfaces, it is well-known that the mapping class group is normally generated by one element. By Lanier and Margalit, any pseudo-Anosov map with stretch factor is less than √2 normally generates the mapping class group. Also, for closed surfaces of genus more than 2, any torsion element except hyperelliptic involution is a normal generator. We ask the same question that when the big mapping class group is normally generated, namely the mapping class group of infinite type surfaces.

In this talk, I will first introduce the the topology of big mapping class groups. After that I will answer when the big mapping class group is topologically normally generated by one element, and give an upper bound of how many generators are needed to topologically normally generate the group.

Location: KIAS 8101  (Building #8, First floor - next to the tea room)

Title: Embeddability of RAAGs into combinatorial HHG

Abstract: Hierarchically Hyperbolic Space(HHS) is a large-scale geometric structure which utilize geometry of both mapping class groups of surfaces and CAT(0) cube complex. One main interest about HHS theory is to extend geometric, algebraic results of hyperbolic groups and MCG to the groups acting on HHS. In this talk, we focus on groups acting on a combinatorial HHS and its slice stabilizer property, and we will show that given a finite collection of unbounded domains of CHHS, there exists a collection of axial elements which generates a right-angled Artin subgroup of CHHG.

Location: KIAS 1423 (4th floor, Building #1)

Title: On divergent trajectories on homogeneous spaces.

Abstract: In the theory of dynamical systems, the asymptotic behaviorof typical trajectories has traditionally been of primary interest. Recently, there has been active research on special trajectories,particularly divergent trajectories. In this talk, we will especially consider divergent trajectories on homogeneous spacesand explore recent developments in this area.

Location: Online (Zoom)

Title: Classification of Stable Surfaces with respect to Automatic Continuity

Abstract: We provide a complete classification of when the homeomorphism group of a stable surface, Σ, has the automatic continuity property: Any homomorphism from Homeo(Σ) to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of Σ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable second countable Stone space has the automatic continuity property. Under the presence of stability this answers two questions of Mann. This is joint work with George Domat and Kasra Rafi.

Location: Online (Zoom)

Title: Graphical models for topological groups

Abstract: By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley-Abels graph of a totally disconnected, locally compact group, we introduce countable connected graphs associated to Polish groups that we term Cayley-Abels-Rosendal graphs. A group admitting a Cayley-Abels-Rosendal graph acts on it continuously, coarsely metrically properly and cocompactly by isometries of the path metric. By an expansion of the Milnor-Schwarz lemma, it follows that the group is generated by a coarsely bounded set and is quasi-isometric to the graph. In other words, groups admitting Cayley-Abels-Rosendal graphs are topological analogues of the finitely generated groups. We will see these concepts in action by considering homeomorphism groups of countable Stone spaces (i.e. homeomorphism groups of countable end spaces/countable ordinals). We completely characterize when these homeomorphism groups are coarsely bounded, when they are locally bounded (all of them are), and when they admit a Cayley-Abels-Rosendal graph, and if so produce a coarsely bounded generating set. This is joint work with Beth Branman, Hannah Hoganson, and Robbie Lyman.

Location: Online (Zoom)

Title: Structural stability of meandering hyperbolic group action

Abstract: We proved that the actions of cocompact lattices in higher rank semisimple Lie groups on flag varieties are structurally stable via using a new stable notion of meandering hyperbolic group action. This is a generalization of convex cocompact groups in hyperbolic geometry and Anosov groups in semisimple Lie groups. This is a joint work with Misha Kapovich and Jaejeong Lee.

Location: KIAS 1424

Title: Coding of geodesic flow and continued fraction

Abstract: Measure-preserving transformations and flows are the main objects in studying dynamical systems. The "coding" generated by an induced map of the flows is a powerful tool to analyze the behavior of the flows. The connection between the continued fractions and the geodesic flows on hyperbolic surfaces has been developed over many decades. In this talk, we will see that the continued fraction is interpreted as coding of geodesics on a hyperbolic surface. This interpretation can be applied to the measure-theoretical problems of geodesic flow.