arXiv subjects
Publications and source records attributed to Takuya Katayama.
This paper focuses on using the theory of bicorn curves in the context of closed surfaces to understand hyperbolic phenomena of the curve graphs of those surfaces. We prove that the curve graph of any closed surface is 15-hyperbolic with one exception. Furthermore, we provide significantly tighter bounds for the bounded geodesic image theorem, originally proven by Masur--Minsky.
In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.
Let $N$ be a closed nonorientable surface with or without marked points. In this paper we prove that, for every finite full subgraph $Γ$ of $\mathcal{C}^{\mathrm{two}}(N)$, the right-angled Artin group on $Γ$ can be embedded in the mapping class group of $N$. Here, $\mathcal{C}^{\mathrm{two}}(N)$ is the subgraph, induced by essential two-sided simple closed curves in $N$, of the ordinal curve graph $\mathcal{C}(N)$. In addition, we show that there exists a finite graph $Γ$ which is not a full subgraph of $\mathcal{C}^{\mathrm{two}}(N)$ for some $N$, but the right-angled Artin group on $Γ$ can be embedded in the mapping class group of $N$.
Let $N$ be a connected nonorientable surface with or without boundary and punctures, and $j\colon S\rightarrow N$ be the orientation double covering. It has previously been proved that the orientation double covering $j$ induces an embedding $ι\colon\mathrm{Mod}(N)$ $\hookrightarrow$ $\mathrm{Mod}(S)$ with one exception. In this paper, we prove that this injective homomorphism $ι$ is a quasi-isometric embedding. The proof is based on the semihyperbolicity of $\mathrm{Mod}(S)$, which has already been established. We also prove that the embedding $\mathrm{Mod}(F') \hookrightarrow \mathrm{Mod}(F)$ induced by an inclusion of a pair of possibly nonorientable surfaces $F' \subset F$ is a quasi-isometric embedding.
In this article, we determine the function $\ell(S_{g, p})$ such that the right-angled Artin group $G(P_{m})$ is embedded in the mapping class group $\mathrm{Mod}(S_{g, p})$ if and only if $m$ is not more than $\ell(S_{g, p})$. Using this function and Birman--Hilden theory, we prove that $\mathrm{Mod}(S_{0, p})$ is virtually embedded in $\mathrm{Mod}(S_{g, 0})$ if and only if $p \leq 2g+2$.
In this article, we prove that embeddings of right-angled Artin group $A_1$ on the complement of a linear forest into another right-angled Artin group $A_2$ can be reduced to full embeddings of the defining graph of $A_1$ into the extension graph of the defining graph of $A_2$.
For a finite graph $Γ$, let $G(Γ)$ be the right-angled Artin group defined by the complement graph of $Γ$. We show that, for any linear forest $Λ$ and any finite graph $Γ$, $G(Λ)$ can be embedded into $G(Γ)$ if and only if $Λ$ can be realised as a full subgraph of $Γ$. We also prove that if we drop the assumption that $Λ$ is a linear forest, then the above assertion does not hold, namely, for any finite graph $Λ$, which is not a linear forest, there exists a finite graph $Γ$ such that $G(Λ)$ can be embedded into $G(Γ)$, though $Λ$ cannot be embedded into $Γ$ as a full subgraph.