Abstracts

Mini-course

One-relator groups and the coherence property

A group is said to be coherent if all of its finitely generated subgroups are finitely presented. An old conjecture of Baumslag's predicted that all one-relator groups are coherent. The aim of this minicourse will be to explain the recent developments in the theory of one-relator groups that have led to the recent resolution of this conjecture. In the first talk I will cover the classical theory of one-relator groups with a modern spin. In the second talk, I will cover the work of Wise and Louder--Wilton on the non-positive and negative immersions properties that led to a deeper understanding of the subgroup structure of one-relator groups. Finally, in the third talk I will cover homological aspects of one-relator groups which allowed Andrei Jaikin-Zapirain and myself to confirm Baumslag's conjecture.



Research Talks

Residually finiteness in lacunary hyperbolic groups

Gromov discovered that the interrelation between the geometric, topological, and algebraic properties of finitely generated groups can be explored through their asymptotic cones. For example, if all asymptotic cones of a finitely generated group G are simply connected, then G is finitely presented, its Dehn function is bounded by a polynomial, and its isodiametric function is linear. A finitely generated group G is hyperbolic if every asymptotic cone of G is an ℝ-tree. If at least one asymptotic cone of G is an ℝ-tree, G is called lacunary hyperbolic. In this talk, we will discuss lacunary hyperbolic groups and related problems.



Moment formulas and the Logarithm law for a unipotent flow

In 1999, Kleinbock and Margulis established the Logarithm law for a geodesic flow, saying that the growth of generic geodesics in homogeneous spaces are logarithmic, which is a generalization of the work of Sullivan (1982). Their theorem was proven using the exponential mixing property of the geodesic flow.

In a different context, the logarithmic behavior of orbits under unipotent flows on the homogeneous space SL(d,R)/SL(d,Z) can be analyzed through moment formulas for the Siegel transform, as shown by Athreya and Margulis (2009). By adopting their strategy, we can derive the Logarithm law for one-parameter unipotent subgroups on the S-arithmetic space SL(d,Q_S)/SL(d,Z_S). This is a joint work with Samantha Fairchild.



On generating mapping class groups by pseudo-Anosov elements

Wajnryb proved that the mapping class group of a closed oriented surface is generated by two elements. We proved that the mapping class group is generated by two pseudo-Anosov elements. In particular, if the genus is greater than or equal to nine, we can take the generators to two conjugate pseudo-Anosov elements with arbitrarily large dilatations. This is a joint work with Naoyuki Monden (Okayama University, Japan.)



Approaching Gromov's question through the lens of the Hopf property

Gromov's question of whether every hyperbolic group is residually finite remains a longstanding open problem in geometric group theory.  Mal'cev showed that every finitely generated residually finite group is Hopfian, enabling an approach from the perspective of the less restrictive property, the Hopf property.  Recently, Reinfeldt-Weidmann and Fujiwara-Sela independently proved that every hyperbolic group is Hopfian.  In this context, we present our recent findings on relatively hyperbolic groups and discuss observations from ongoing research that expand upon this perspective.



A categorified eigenring action on Jacobi diagrams in handlebodies

For an algebra $R$ and a multiplicative submodule $J$, the eigenring, denoted by $E_{R}(J)$, is a canonical subquotient algebra of $R$ by $J$. In the case where $J$ is a two-sided ideal, $E_{R}(J)$ coincides with the quotient algebra $R/J$. The formal resemblance of linear categories to algebras allows us to extend this construction to that of a subquotient linear category. This talk presents an example that arises in the context of quantum topology, demonstrating this general framework. To be precise, we consider a linear category $A$ generated by Jacobi diagrams in handlebodies. We then relate the Lie operad to a “categorified” eigenring of $A$. The category $A$ was introduced by Habiro and Massuyeau in order to extend the Kontsevich integral.



Tiling Conjecture and polygonality of words

For a word w in a free group F, one defines the Baumslag double of F along w as the amalgamated free product D(F;w) of two copies of F glued along the copies of the word w. The group D(F;w) is one-ended and word-hyperbolic, when the amalgamating word w is minimal (that is, shortest in its automorphic orbit) and diskbusting (that is, it doesn't belong to any proper free factor). The group D(F;w) may be regarded as one of the most ancient constructions of one-ended word-hyperbolic groups, and have played the role of testbeds for more general conjectures. For instance, people were interested in answering a question of Gromov: does a generic hyperoblic group contain a surface subgroup? Each D(F;w) has a very natural K(pi,1), which is a locally CAT(0) square complex X(F;w). Henry Wilton and I conjectured that X(F;w) virtually contains a closed hyperbolic π1-injective surface. We still do not know whether or not this conjecture (Tiling Conjecture) is true, although Henry Wilton proved that there exists a hyperbolic surface subgroup (group theoretically) inside D(F;w), and Oum and I proved the existence of such a surface when the rank is two. I will present this conjecture, along with a purely group/graph theoretic reformulation ("every minimal diskbusting word is polygonal") and along with a recent progress toward this reformulation discovered by my REU students, Le Xuan Hoang (VNU-HUS) and Tran Nguyen Nam Hung (VNU-HCM).



Meandering hyperbolic group actions

We introduced the stable notion of meandering hyperbolic group action and proved that the actions of cocompact lattices in higher rank on flag varieties are meandering hyperbolic and hence they are structurally stable.
In addition, I will give interesting examples of meandering hyperbolic group actions. This is a joint work with Misha Kapovich and Jaejeong Lee. 



Equidistribution of divergent geodesics via entropy methods

Duke’s Theorem (1988) is concerned with the equidistribution of closed geodesics on the modular surface, and Einsiedler, Lindenstrauss, Michel and Venkatesh (2012) gave an ergodic theoretic proof of Duke's Theorem using maximal entropy methods. Along these lines, David and Shapira (2018) established an equidistribution result of certain divergent geodesics on the modular surface under non-escape of mass assumption. In this talk, we will relax this non-escape of mass assumption and give its applications to continued fractions of rationals. This is joint work with Ofir David, Ron Mor, and Uri Shapira.



Invariant quasimorphisms and mixed scl

Quasimorphisms and stable commutator length (scl) are closely related through the Bavard duality. Recently, we considered their variants, invariant quasimorphisms and mixed scl, and proved the Bavard duality between them. In this talk, I will discuss the extension problem of invariant quasimorphisms, the comparison problem between scl and mixed scl, and the relationship between these problems. 

This talk is based on joint works with Morimichi Kawasaki, Shuhei Maruyama, Takahiro Matsushita, and Masato Mimura.



Automorphisms of fine curve graphs for nonorientale surfaces

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of the actual essential simple closed curves on the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface of genus g ≥ 2 is isomorphic to the homeomorphism group of the surface. We generalized their result to closed nonorientable surfaces of genus g ≥ 4. This is a joint work with Mitsuaki Kimura.



Mapping class groups of Infinite graphs -- Big Out(Fn)

Surfaces and graphs are closely related; there are many parallels between the mapping class groups of finite-type surfaces and finite graphs, where the mapping class group of a finite graph is the outer automorphism group of a free group of (finite) rank. A recent surge of interest in infinite-type surfaces and their mapping class groups begs a natural question: What is the mapping class group of an “infinite” graph? In this talk, I will explain the answer given by Algom-Kfir and Bestvina, and present recent work, joint with George Domat and Hannah Hoganson, on the coarse geometry of such groups.



On the faithfulness of the Burau representation of B₃ modulo p

The Burau representation is one of the most well-studied representations of the braid group. While the question of its faithfulness has a long history, the case of B₃ was relatively easily solved in the mid-20th century by using the fact that the quotient of B₃ by its center is isomorphic to the modular group. In this talk, I extend this result to the Burau representation of B₃ modulo a prime p, and present an algorithm to determine the faithfulness of the representation for a given prime p.



A Higher-Dimensional Generalization of the Pseudo-Anosov Mapping Class

In this talk, I will present an example of applying concepts from Geometric Group Theory and surface mapping class group theory to symplectic topology. I will begin with a brief introduction to symplectic topology and then illustrate a specific application: a higher-dimensional generalization of the pseudo-Anosov mapping class in the symplectic setting.



Diophantine approximation on Apollonian circle packing

Apollonian circle packing is a fractal set which is the union of recursively determined tangent circles starting from three tangent circles. On this fractal set, one can consider the problem of approximating limit points by tangency points of circles in the packing. In this talk, we will discuss the Hausdorff dimension of badly approximable points in this context. This is a joint work in progress with Kangrae Park and Yongquan Zhang.



Two-bridge knot groups and problems arising from their study

Two-bridge knot groups are particularly elegant examples of one-relator groups.  In a series of joint works with Donghi Lee conducted between 2011 and 2016,  we fully resolved several problems related to two-bridge knot groups,  which had been on my mind since around 1997 and formally proposed in 1999. These solutions found applications in McShane’s identity,  Bowditch-Tan-Wong-Zhang’s end invariants, and epimorphisms between two-bridge knot groups. These studies further inspired Yuya Koda and me to introduce the concept of homotopy motion groups  of surfaces in 3-manifolds. In this talk, I will briefly review the joint work with Donghi Lee on two-bridge knot groups 

and discuss how this work led to the development of homotopy motion groups.  I will then present some open questions for further exploration.



Decision problems in braid and Artin groups

The 3-strand braid group is one of the oldest one-relator groups studied. Artin groups generalize braid groups, and are a natural class of groups arising from conditions on the presentation itself. Particular classes of Artin groups arise in many contexts, including knot theory and recently in the study of quasiconvex hierarchies of 3-manifold groups. Nevertheless, for the class of all Artin groups decidability of the word problem remains an open problem. In this talk, I will describe recent progress on decision problems in braid and Artin groups groups, and give, among other results, a full classification of when the submonoid membership problem is decidable in braid and Artin groups, in terms of forbidden subgraphs of the underlying graph. This is joint work with R. D. Gray (East Anglia).



The Pants Graph of a Rose

In this talk, we introduce the concept of a pants decomposition for a rose, defined as a pants decomposition of a marked surface. Elementary moves between pants decompositions of a rose generate a simplicial graph known as the pants graph of a rose. We will explore fundamental properties of the pants graph, with a focus on constructing coarsely Lipschitz maps.