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Hyungryul Baik

Publications and source records attributed to Hyungryul Baik.

50 records · Page 3Linked to original sources
arXiv2018

Minimal asymptotic translation lengths of Torelli groups and pure braid groups on the curve graph

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 $Mod(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group $PB_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group $B_n$, which behaves like $1/n^2$.

math.GT↗
arXiv2018

The Smallest Positive Eigenvalue Of Fibered Hyperbolic 3-Manifolds

We study the smallest positive eigenvalue $λ_1(M)$ of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold $M$ which fibers over the circle, with fiber a closed surface of genus $g\geq 2$. We show the existence of a constant $C>0$ only depending on $g$ so that $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2, C\log {\rm vol}(M)/{\rm vol}(M)^{2^{2g-2}/(2^{2g-2}-1)}]$ and that this estimate is essentially sharp. We show that if $M$ is typical or random, then we have $λ_1(M)\in [C^{-1}/{\rm vol}(M)^2,C/{\rm vol}(M)^2]$. This rests on a result of independent interest about reccurence properties of axes of random pseudo-Anosov elements.

math.GT↗
arXiv2018

Subgroup growth of right-angled Artin and Coxeter groups

We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting.

math.GR↗
arXiv2018

An algorithm to compute the Teichmueller polynomial from matrices

In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmueller polynomial corresponding to those surface homeomorphisms by first constructing an invariant track whose first homology group can be naturally identified with the first homology group of the surface, and computing its Alexander polynomial.

math.GT↗
arXiv2016

Is a typical bi-Perron number a pseudo-Anosov dilatation?

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 unit 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 genus cases.

math.GT↗
arXiv2016

Unsmoothable group actions on compact one-manifolds

We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by $C^{1+\mathrm{bv}}$ diffeomorphisms on the circle, which generalizes a result of Farb-Franks, and which parallels a result of Ghys and Burger-Monod concerning differentiable actions of higher rank lattices on the circle. This answers a question of Farb, which has its roots in the work of Nielsen. We prove this result by showing that if a right-angled Artin group acts faithfully by $C^{1+\mathrm{bv}}$ diffeomorphisms on a compact one-manifold, then its defining graph has no subpath of length three. As a corollary, we also show that no finite index subgroup of $\textrm{Aut}(F_n)$ and $\textrm{Out}(F_n)$ for $n\geq 3$, the Torelli group for genus at least $3$, and of each term of the Johnson filtration for genus at least $5$, can act faithfully by $C^{1+\mathrm{bv}}$ diffeomorphisms on a compact one-manifold.

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arXiv2016

Spaces of invariant circular orders of groups

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 action of Aut(G) on CO(G) and relate it to results of Koberda regarding the action on LO(G). We then study two families of circularly orderable groups: finitely generated abelian groups, and free products of circularly orderable groups. For finitely generated abelian groups A, we use a classification of elements of CO(A) to describe the homeomorphism type of the space CO(A), and to show that Aut(A) acts faithfully on the subspace of circular orders which are not linear. We define and characterize Archimedean circular orders, in analogy with linear Archimedean orders. We describe explicit examples of circular orders on free products of circularly orderable groups, and prove a result about the abundance of orders on free products. Whenever possible, we prove and interpret our results from a dynamical perspective.

math.GR↗
arXiv2015

Constructing pseudo-Anosov maps with given dilatations

In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.

math.GT↗
arXiv2014

Fuchsian Groups, Circularly Ordered Groups, and Dense Invariant Laminations on the Circle

We propose a program to study groups acting faithfully on S^1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on S^1 is conjugate to a Fuchsian group if and only if it admits three very-full laminations with a variation of the transversality condition. Some partial results toward a similar characterization of hyperbolic 3-manifold groups which fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated 3-manifolds developed by Thurston and Calegari-Dunfield.

math.GT↗
arXiv2012

The Space of Geometric Limits of One-generator Closed Subgroups of PSL2(R)

We give a complete description of the closure of the space of one-generator closed subgroups of PSL2(R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) = PSL2(R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps transform convergent sequences of closed subsets of the domain into convergent sequences of closed subsets of the range. In particular, this allows us to compute certain geometric limits of PSL2(R) only by looking at the Hausdorff limit of some closed subsets of C.

math.GT↗