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Erika Kuno

Publications and source records attributed to Erika Kuno.

13 recordsLinked to original sources
arXiv2026

Large-scale geometry of graphs interpolating between curve graphs and pants graphs

We study two types of graphs interpolating between the curve graph and the pants graph from the viewpoint of large-scale geometry. One was introduced by Erlandsson and Fanoni, and the other by Mahan Mj. These graphs were developed independently in different contexts. In this paper, we provide explicit formulae for computing their quasi-flat ranks. These formulae depend on the genus and the number of boundary components of the underlying surface, as well as the interpolation parameter. We also classify geometries of the interpolating graphs into the hyperbolic, relatively hyperbolic, and thick cases. Our approach relies on the theory of twist-free graphs of multicurves, which is developed by Vokes and Russel.

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arXiv2025

The bicorn curves on closed surfaces

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.

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arXiv2024

Uniform hyperbolicity of nonseparating curve graphs of nonorientable surfaces

Let $N$ be a connected finite type nonorientable surface with or without boundary components and punctures. We prove that the graph of nonseparating curves of $N$ is connected and Gromov hyperbolic with a constant which does not depend on the topological type of the surface by using the bicorn curves introduced by Przytycki and Sisto. The proof is based on the argument by Rasmussen on the uniform hyperbolicity of graphs of nonseparating curves for finite type orientable surfaces.

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arXiv2023

Automorphisms of fine curve graphs for nonorientable surfaces

The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of essential simple closed curves on the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the fine curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. In this paper, based on their argument, we prove that the automorphism group of the fine curve graph of a closed nonorientable surface $N$ of genus $g \geq 4$ is isomorphic to the homeomorphism group of $N$.

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arXiv2021

Quasimorphisms on nonorientable surface diffeomorphism groups

Bowden, Hensel, and Webb constructed infinitely many quasimorphisms on the diffeomorphism groups of orientable surfaces. In this paper, we extend their result to nonorientable surfaces. Namely, we prove that the space of nontrivial quasimorphisms $\widetilde{QH}(\mathrm{Diff}_0(N_g))$ on the identity component of the diffeomorphism group $\mathrm{Diff}_0(N_g)$ on a closed nonorientable surface $N_g$ of genus $g\geq 3$ is infinite-dimensional. As a corollary, we obtain the unboundedness of the commutator length and the fragmentation length on $\mathrm{Diff}_0(N_g)$.

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arXiv2021

Right-angled Artin groups and curve graphs of nonorientable surfaces

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 $N$.

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arXiv2021

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

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.

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arXiv2018

The RAAGs on the complement graphs of path graphs in mapping class groups

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 $p \leq 2g+2$.

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arXiv2016

Abelian subgroups of the mapping class groups for non-orientable surfaces

Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free subgroup of the mapping class groups.

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arXiv2015

Right-angled Artin groups on finite subgraphs of disk graphs

Koberda proved that if a graph $Γ$ is a full subgraph of a curve graph $\mathcal{C}(S)$ of an orientable surface $S$, then the right-angled Artin group $A(Γ)$ on $Γ$ is a subgroup of the mapping class group ${\rm Mod}(S)$ of $S$. On the other hand, for a sufficiently complicated surface $S$, Kim-Koberda gave a graph $Γ$ which is not contained in $\mathcal{C}(S)$, but $A(Γ)$ is a subgroup of ${\rm Mod}(S)$. In this paper, we prove that if $Γ$ is a full subgraph of a disk graph $\mathcal{D}(H)$ of a handlebody $H$, then $A(Γ)$ is a subgroup of the handlebody group ${\rm Mod}(H)$ of $H$. Further, we show that there is a graph $Γ$ which is not contained in some disk graphs, but $A(Γ)$ is a subgroup of the corresponding handlebody groups.

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arXiv2015

Uniform hyperbolicity for curve graphs of non-orientable surfaces

Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs of (non-)orientable surfaces are 9-hyperbolic.

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