arXiv subjects

Hyunshik Shin

Publications and source records attributed to Hyunshik Shin.

10 recordsLinked to original sources

arXiv2021

On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs

We study the right-angled Artin group action on the extension graph. We show that this action satisfies a certain finiteness property, which is a variation of a condition introduced by Delzant and Bowditch. As an application we show that the asymptotic translation lengths of elements of a given right-angled Artin group are always rational and once the defining graph has girth at least 6, they have a common denominator. We construct explicit examples which show the denominator of the asymptotic translation length of such an action can be arbitrary. We also observe that if either an element has a small syllable length or the defining graph for the right-angled Artin group is a tree then the asymptotic translation lengths are integers.

math.GR↗

arXiv2021

Asymptotic translation lengths and normal generation for pseudo-Anosov monodromies of fibered 3-manifolds

Let $M$ be a hyperbolic fibered 3-manifold. We study properties of sequences $(S_{α_n}, ψ_{α_n})$ of fibers and monodromies for primitive integral classes in the fibered cone of $M$. The main tool is the asymptotic translation length $\ell_{\mathcal{C}} (ψ_{α_n})$ of the pseudo-Anosov monodromy $ ψ_{α_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 $(S, ψ)$ in the fibered cone, $\ell_{\mathcal{C}} (ψ)$ is bounded from above by $C/|χ(S)|$. We also obtain a moral connection between $\ell_{\mathcal{C}} (ψ)$ and the normal generating property of $ψ$ in the mapping class group on $S$. We show that for all but finitely many primitive integral classes $(S, ψ)$ in an arbitrary 2-dimensional slice of the fibered cone, $ψ$ normally generates the mapping class group on $S$. In the second half of the paper, we study if it is possible to obtain a continuous extension of normalized asymptotic translation lengths on the curve complex as a function on the fibered face. An analogous question for normalized entropy has been answered affirmatively by Fried and the question for normalized asymptotic translation length on the arc complex in the fully punctured case has been answered negatively by Strenner. We show that such an extension in the case of the curve complex does not exist in general by explicit computation for sequences in the fibered cone of the magic manifold.

math.GT↗

arXiv2020

Topological and dynamical properties of Torelli groups of partitioned surfaces

Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, first we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned surfaces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the first and second authors on Torelli groups for closed surfaces.

math.GT↗

arXiv2020

An upper bound on the asymptotic translation lengths on the curve graph and fibered faces

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 the $ψ_n$-invariant homology of $Σ_n$ (which is independent of $n$). As a corollary, if $b_1(M) = 2$, the asymptotic translation length on the curve complex of such a sequence of primitive elements behaves like $1/χ(Σ_n)^{2}$. Furthermore, together with a work of E. Hironaka, our theorem can be used to determine the asymptotic behavior of the minimal translation lengths of handlebody mapping class groups and the set of mapping classes with homological dilatation one.

math.GT↗

arXiv2019

On Translation Lengths of Anosov Maps on Curve Graph of Torus

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any given Anosov map. The application of our result is threefold: (a) to determine which word realizes the minimal translation length on the curve graph within a specific class of words, (b) to establish the effective bound on the ratio of translation lengths of an Anosov map on the curve graph to that on Teichmüller space, and (c) to estimate the overall growth of the number of Anosov maps which have a sufficient number of Anosov maps with the same translation length.

math.GT↗

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

Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex

Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $ψ$. We show that there exists a sequence $(R_n, ψ_n)$ of fibers and monodromies contained in the fibered cone of $(S,ψ)$ such that the asymptotic translation length of $ψ_n$ on the curve complex $\mathcal{C}(R_n)$ behaves asymptotically like $1/|χ(R_n)|^2$. As applications, we can reprove the previous result by Gadre--Tsai that the minimal asymptotic translation length of a closed surface of genus $g$ asymptotically behaves like $1/g^2$. We also show that this also holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.

math.GT↗

arXiv2017

Spectral radius of a star with one long arm

A tree is said to be starlike if exactly one vertex has degree greater than two. In this paper, we will study the spectral properties of $S(n,k \cdot 1)$, that is, the starlike tree with $k$ branches of length 1 and one branch of length $n$. The largest eigenvalue $λ_1$ of $S(n,k \cdot 1)$ satisfies $\sqrt{k+1} \leq λ_1 < k/\sqrt{k-1}$. Moreover, the largest eigenvalue of $S(n,k \cdot 1)$ is equal to the largest eigenvalue of $S(k \cdot (n+1) )$, which is the starlike tree that has $k$ branches of length $n-1$. Using the spectral radii of $S(n,k \cdot 1)$ we can show

math.CO↗

arXiv2015

Algebraic degrees of stretch factors in mapping class groups

We explicitly construct pseudo-Anosov maps on the closed surface of genus $g$ with orientable foliations whose stretch factor $λ$ is a Salem number with algebraic degree $2g$. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree $d$, for each positive even integer $d$ such that $d \leq g$.

math.GT↗

arXiv2015

Pseudo-Anosov mapping classes not arising from Penner's construction

We show that Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner's construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner's construction. This resolves a conjecture of Penner.

math.GT↗