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Sang-hyun Kim

Publications and source records attributed to Sang-hyun Kim.

At least 37 records · Page 2Linked to original sources
arXiv2020

Non-freeness of groups generated by two parabolic elements with small rational parameters

Let $q\in\mathbb{C}$, let \[a=\begin{pmatrix} 1&0\\1&1\end{pmatrix},\quad b_q=\begin{pmatrix} 1&q\\0&1\end{pmatrix},\] and let $G_q<\mathrm{SL}_2(\mathbb{C})$ be the group generated by $a$ and $b_q$. In this paper, we study the problem of determining when the group $G_q$ is not free for $|q|<4$ rational. We give a robust computational criterion which allows us to prove that if $q=s/r$ for $|s|\leq 27$ then $G_q$ is non-free, with the possible exception of $s=24$. In this latter case, we prove that the set of denominators $r\in\mathbb{N}$ for which $G_{24/r}$ is non-free has natural density $1$. For a general numerator $s>27$, we prove that the lower density of denominators $r\in \mathbb{N}$ for which $G_{s/r}$ is non-free has a lower bound \[ 1- \left(1-\frac{11}{s}\right) \prod_{n=1}^\infty \left(1-\frac{4}{s^{2^n-1}}\right). \] Finally, we show that for a fixed $s$, there are arbitrarily long sequences of consecutive denominators $r$ such that $G_{s/r}$ is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.

math.GR↗
arXiv2020

Small $C^1$ actions of semidirect products on compact manifolds

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $ψ$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $ψ^*$ on $H^1(S,\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\mathcal U$ of the trivial action in the space of $C^1$ actions of $π_1(T)$ on $M$ such that any action in $\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\mathbb{R})\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds.

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arXiv2020

Diffeomorphism groups of critical regularity

Let $M$ be the circle or a compact interval, and let $α=k+τ\ge1$ be a real number such that $k=\lfloor α\rfloor$. We write $\mathrm{Diff}_+^α(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are Hölder continuous with exponent $τ$. If $α\ge1$, we prove that there exists a finitely generated subgroup $G_α\le\mathrm{Diff}_+^α(M)$ with the property that $G_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^β(M)$ for all $β>α$. If $α>1$, we also show the dual result: there exists a finitely generated group $H_α\le\bigcap_{β<α}\mathrm{Diff}_+^β(M)$ with the property that $H_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^α(M)$. We can further require that the same properties are inherited by all finite index subgroups, and also by the commutator subgroups, of $G_α$ and $H_α$. The commutator groups of $G_α$ and of $H_α$ are countable simple groups. As a consequence, whenever $1\leα<β$ we have a continuum of isomorphism types of finitely generated subgroups of $\mathrm{Diff}_+^α(M)$ whose images under arbitrary homomorphisms to $\mathrm{Diff}_+^β(M)$ are abelian. We give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms. For example, we show that if $k\neq 2$ is an integer and if $k<β$ then there is no nontrivial homomorphism $\mathrm{Diff}_+^k(S^1)\to\mathrm{Diff}_+^β(S^1)$.

math.GR↗
arXiv2019

Some groups with planar boundaries

In this expository note, we illustrate phenomena and conjectures about boundaries of hyperbolic groups by considering the special cases of certain amalgams of hyperbolic groups. While doing so, we describe fundamental results on hyperbolic groups and their boundaries by Bowditch and Haissinsky, along with special treatments for the boundaries of free groups by Otal and Cashen.

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arXiv2019

Algebraic Structure of Diffeomorphism Groups of One-Manifolds

It is a celebrated result of Mather that the group of $C^k$--diffeomorphisms of an $n$--manifold is simple, provided that a mild isotopy condition is satisfied, with the possible exception of $k=n+1$. The purpose of this article is mostly expository, and in it we give a detailed account of Mather's proof in the case when $n=1$.

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arXiv2018

Flexibility of group actions on the circle

In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if $L$ is a nontrivial limit group then the nonlinear representation variety $\mathrm{Hom}(L,\mathrm{Homeo}_+(S^1))$ contains uncountably many semi-conjugacy classes of faithful actions on $S^1$ with pairwise disjoint rotation spectra (except for $0$) such that each representation lifts to $\mathbb{R}$. For the case of most Fuchsian groups $L$, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on $S^1$ that is never semi-conjugate to any action that factors through a finite--dimensional connected Lie subgroup in $\mathrm{Homeo}_+(S^1)$. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on $S^1$. In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of $\mathrm{PSL}_2(\mathbb{R})$ which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of $\mathbb{Z}$--valued subadditive defect--one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.

math.GT↗
arXiv2018

Free products and the algebraic structure of diffeomorphism groups

Let $M$ be a compact one--manifold, and let $\mathrm{Diff}^{1+\mathrm{bv}}(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation. We prove that if $G$ is a group which is not virtually metabelian, then $(G\times\mathbb{Z})*\mathbb{Z}$ is not realized as a subgroup of $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. This gives the first examples of finitely generated groups $G,H\le \mathrm{Diff}_+^\infty(M)$ such that $G\ast H$ does not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. By contrast, for all countable groups $G,H\le\mathrm{Homeo}^+(M)$ there exists an embedding $G\ast H\to \mathrm{Homeo}^+(M)$. We deduce that many common groups of homeomorphisms do not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$, for example the free product of $\mathbb{Z}$ with Thompson's group $F$. We also complete the classification of right-angled Artin groups which can act smoothly on $M$ and in particular, recover the main result of a joint work of the authors with Baik. Namely, a right-angled Artin group $A(Γ)$ either admits a faithful $C^{\infty}$ action on $M$, or $A(Γ)$ admits no faithful $C^{1+\mathrm{bv}}$ action on $M$. In the former case, $A(Γ)\cong\prod_i G_i$ where $G_i$ is a free product of free abelian groups. Finally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on $S^1$.

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arXiv2017

Chain groups of homeomorphisms of the interval

We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson's group $F$. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a canonical quotient which is also a chain group, and which has a simple commutator subgroup. On the other hand, every finitely generated subgroup of $\operatorname{Homeo}^+(I)$ can be realized as a subgroup of a chain group. As a corollary, we show that there are uncountably many isomorphism types of chain groups, as well as uncountably many isomorphism types of countable simple subgroups of $\operatorname{Homeo}^+(I)$. We consider the restrictions on chain groups imposed by actions of various regularities, and show that there are uncountably many isomorphism types of $3$--chain groups which cannot be realized by $C^2$ diffeomorphisms, as well as uncountably many isomorphism types of $6$--chain groups which cannot be realized by $C^1$ diffeomorphisms. As a corollary, we obtain uncountably many isomorphism types of simple subgroups of $\operatorname{Homeo}^+(I)$ which admit no nontrivial $C^1$ actions on the interval. Finally, we show that if a chain group acts minimally on the interval, then it does so uniquely up to topological conjugacy.

math.GT↗
arXiv2017

Finite subgraphs of an extension graph

Let $Γ$ be a finite graph and let $Γ^{\mathrm{e}}$ be its extension graph. We inductively define a sequence $\{Γ_i\}$ of finite induced subgraphs of $Γ^{\mathrm{e}}$ through successive applications of an operation called "doubling along a star". Then we show that every finite induced subgraph of $Γ^{\mathrm{e}}$ is isomorphic to an induced subgraph of some $Γ_i$.

math.GR↗
arXiv2016

Exotic mapping class group actions on the circle

It has been known since the time of Nielsen that the mapping class group $\text{Mod}_{g,1}$ of a surface of genus $g$ and one puncture acts faithfully by homeomorphisms on the circle. In this note, we show that this standard representation of the mapping class group is not rigid, precisely, if $G<\text{Mod}_{g,1}$ is a finite index subgroup then there exist infinitely many non--conjugate faithful representations $G\to \text{Homeo}^+(S^1)$. We thus answer a question of B. Farb.

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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arXiv2015

Coxeter groups, hyperbolic cubes, and acute triangulations

Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called "flag no-square". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.

math.GT↗
arXiv2015

Anti-trees and right-angled Artin subgroups of braid groups

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk and the 2--sphere, answering questions due to Crisp--Wiest and M. Kapovich. Another corollary is that a pure braid group contains a closed hyperbolic manifold group as a quasi-isometrically embedded subgroup up to dimension eight. Finally, we show that the isomorphism problem, conjugacy problem, and membership problems are unsolvable in the class of finitely presented subgroups of braid groups.

math.GR↗
arXiv2014

Right-angled Artin groups and finite subgraphs of curve graphs

We show that for a sufficiently simple surface $S$, a right-angled Artin group $A(Γ)$ embeds into $\Mod(S)$ if and only if $Γ$ embeds into the curve graph $\mC(S)$ as an induced subgraph. When $S$ is sufficiently complicated, there exists an embedding $A(Γ)\to\Mod(S)$ for some $Γ$ not contained in $\mC(S)$.

math.GT↗
arXiv2014

The geometry of the curve graph of a right-angled Artin group

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we are able to develop a Nielsen--Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.

math.GR↗
arXiv2014

Lagrangian Floer potential of orbifold spheres

For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov-Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.

math.SG↗
arXiv2013

Hyperbolic surface subgroups of one-ended doubles of free groups

Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank two, or (2) every generator is used the same number of times in the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).

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