Hyungryul Baik

Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set CO (G) of all left-invariant circular orders on a fixed countable and discrete group G . CO (G) contains as a closed subspace LO (G) , the space of all left-invariant linear orders of G , as first topologized by Sikora. We use the compactness of these spaces to show the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable. We describe the act

Abstract In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which

Fuchsian groups, circularly ordered groups and dense invariant laminations on the circle

We describe the topology of the space of all geometric limits of closed abelian subgroups of PSL2C. Main tools and ideas come from the previous paper [BC12].

Let $M$ be a hyperbolic fibered 3-manifold. We study properties of sequences $(S_{\alpha_n}, \psi_{\alpha_n})$ of fibers and monodromies for primitive integral classes in the fibered cone of $M$. The main tool is the asymptotic translation length $\ell_{\mathcal{C}} (\psi_{\alpha_n})$ of the pseudo-Anosov monodromy $ \psi_{\alpha_n}$ on the curve complex. We first show that there exists a constant $C>0$ depending only on the fibered cone such that for any primitive integral class $(S, \psi)$ in

Abstract We prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>→</m:mo> <m:mi>S</m:mi> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S , converging to the universal cover . Th

We study the asymptotic behavior of the asymptotic translation lengths on the curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered hyperbolic 3-manifold $M$ with $b_1(M) \geq 2$. For a sequence $(Σ_n, ψ_n)$ of fibers and monodromies in the fibered cone, we show that the asymptotic translation length on the curve complex is bounded above by $1/χ(Σ_n)^{1+1/r}$ as long as their projections to the fibered face converge to a point in the interior, where $r$ is the dimension of

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic units whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$ . As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area-one abelian differentials for low-

We study the asymptotic behavior of the asymptotic translation lengths on the curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered hyperbolic 3-manifold M with b1(M ) ≥ 2. For a sequence (Σn, ψn) of fibers and monodromies in the fibered cone, we show that the asymptotic translation length on the curve complex is bounded above by 1/|χ(Σn)| 1+1/r as long as their projections to the fibered face converge to a point in the interior, where r is the dimension of the ψn-invariant

Let $M$ be a hyperbolic fibered 3-manifold. We study properties of sequences $(S_{\alpha_n}, \psi_{\alpha_n})$ of fibers and monodromies for primitive integral classes in the fibered cone of $M$. The main tool is the asymptotic translation length $\ell_{\mathcal{C}} (\psi_{\alpha_n})$ of the pseudo-Anosov monodromy $ \psi_{\alpha_n}$ on the curve complex. We first show that there exists a constant $C>0$ depending only on the fibered cone such that for any primitive integral class $(S, \psi)$ in

Abstract We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov, and almost every random walk on $\mathrm {Out}(F_n)$ eventually becomes fully irreducible. If the underlying measu