arXiv subjects

Hiroshige Shiga

Publications and source records attributed to Hiroshige Shiga.

9 recordsLinked to original sources

arXiv2026

Thompson's groups and Teichmüller modular groups of generalized Cantor sets

Thompson's groups, which are denoted by $F, T$ and $V$, were introduced by R. Thompson. It is known that they are related to various fields in mathematics. In this paper, we establish that Thompson's groups are regarded as subgroups of Teichmüller modular groups of Teichmüller spaces of generalized Cantor sets. Moreover, Thompson's groups $F$ and $T$ act properly discontinuously on such Teichmüller spaces but Thompson's group $V$ does not. In some sense, those results are improvements of the results by E. de Faria, F. P. Gardiner and W. J. Harvey on Thomnpson's group $F$ and asymptotic Teichmüller spaces. We also show that Thompson's groups act infinitely many Teichmüller spaces of generalized Cantor sets.

math.CV↗

arXiv2026

Structures of moduli spaces of generalized Cantor sets

For each $ω\in (0, 1)^{\mathbb N}$, we may construct a Cantor set $E(ω)\subset [0, 1]$ called a generalized Cantor set for $ω$. We study the moduli space of $ω$ denoted by $\mathcal M(ω)\subset (0, 1)^{\mathbb N}$. It is the set of $ω'$ so that $E(ω')$ is quasiconformally equivalent to $E(ω)$. In this paper, we show that the set $\mathcal M(ω)$ is measurable in $(0, 1)^{\mathbb N}$ and we give a necessary condition for $ω'$ to belong to $\mathcal M(ω)$. By using this condition, we show that there are uncountably many moduli spaces in $(0, 1)^{\mathbb N}$. We also show that except for at most one moduli space, the volume of the moduli space with respect to the standard product measure of $(0, 1)^{\mathbb N}$ vanishes.

math.CV↗

arXiv2025

On the spectrum of the number of geodesics and tight geodesics in the curve complex

Let $S$ be an oriented surface of type $(g, n)$. We are interested in geodesics in the curve complex $\mathcal C(S)$ of $S$. In general, two $0$-simplexes in $\mathcal C(S)$ have infinitely many geodesics connecting the two simplexes while another geodesics called tight geodesics are always finitely many. On the other hand, we may find two $0$-simplexes in $\mathcal C(S)$ so that they have only finitely many geodesics between them. In this paper, we consider the spectrum of the number of geodesics with length $d (\geq 2)$ in $\mathcal C(S)$ and tight geodesics, which is denoted by $\mathfrak{Sp}_d(S)$ and $\mathfrak{Sp}_d^T(S)$, respectively. In our main theorem, it is shown that $\mathfrak{Sp}_d(S) \subset \mathfrak{Sp}_d^T(S)$ in general, but $\mathfrak{Sp}_2(S)= \mathfrak{Sp}_2^T(S)$. Moreover, we show that $\mathfrak{Sp}_2(S)$ and $\mathfrak{Sp}_2^T(g, n)$ are completely determined in terms of $(g, n)$.

math.GT↗

arXiv2024

Uniform domains and moduli spaces of generalized Cantor sets

We consider a generalized Cantor set $E(ω)$ for an infinite sequence $ω=(q_n)_{n=1}^{\infty}\in (0, 1)^{\mathbb N}$, and consider the moduli space $M(ω)$ for $ω$ which are the set of $ω'$ for which $E(ω')$ is conformally equivalent to $E(ω)$. In this paper, we may give a necessary and sufficient condition for $D(ω):=\mathbb C\setminus E(ω)$ to be a uniform domain. As a byproduct, we give a condition for $E(ω)$ to belong to $M(ω_0)$, the moduli space of the standard middle one-third Cantor set. We also show that the volume of the moduli space $M(ω)$ with respect to the standard product measure on $(0, 1)^{\mathbb N}$ vanishes under a certain condition for $ω$.

math.CV↗

arXiv2023

On the moduli space of the standard Cantor set

We consider a generalized Cantor set $E(ω)$ for an infinite sequence $ω=(q_n)_{n=1}^{\infty}$ of positive numbers with $0<q_n<1$, and examine the quasiconformal equivalence to the standard middle one-third Cantor set $E(ω_0)$. We may give a necessary and sufficient condition for $E(ω)$ to be quasiconformally equivalent to $E(ω_0)$ in terms of $ω$.

math.CV↗

arXiv2019

The quasiconformal equivalence of Riemann surfaces and the universal Schottky space

In the theory of Teichmüller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In this paper, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to obtain the universal Schottky space which contains all Schottky spaces, the deformation spaces of Schottky groups as the universal Teichmüller space contains all Teichmüller spaces.

math.CV↗

arXiv2019

On the quasiconformal equivalence of dynamical Cantor sets

The complement of a Cantor set in the complex plane is itself regarded as a Riemann surface of infinite type. The problem is the quasiconformal equivalence of such Riemann surfaces. Particularly, we are interested in Riemann surfaces given by Cantor sets which are created through dynamical methods. We discuss the quasiconformal equivalence for the complements of Cantor Julia sets of rational functions and random Cantor sets.

math.CV↗

arXiv2017

Extending holomorphic motions and monodromy

Let $E$ be a closed set in the Riemann sphere $\widehat{\mathbb{C}}$. We consider a holomorphic motion $ϕ$ of $E$ over a complex manifold $M$, that is, a holomorphic family of injections on $E$ parametrized by $M$. It is known that if $M$ is the unit disk $Δ$ in the complex plane, then any holomorphic motion of $E$ over $Δ$ can be extended to a holomorphic motion of the Riemann sphere over $Δ$. In this paper, we consider conditions under which a holomorphic motion of $E$ over a non-simply connected Riemann surface $X$ can be extended to a holomorphic motion of $\widehat{\mathbb{C}}$ over $X$. Our main result shows that a topological condition, the triviality of the monodromy, gives a necessary and sufficient condition for a holomorphic motion of $E$ over $X$ to be extended to a holomorphic motion of $\widehat{\mathbb{C}}$ over $X$. We give topological and geometric conditions for a holomorphic motion over a Riemann surface to be extended. We also apply our result to a lifting problem for holomorphic maps to Teichmüller spaces.

math.CV↗

arXiv1995

Projective structures with discrete holonomy representations

Let $K(X)$ denote the set of projective structures on a compact Riemann surface $X$ whose holonomy representations are discrete. We will show that each component of the interior of $K(X)$ is holomorphically equivalent to a complex submanifold of the product of Teichmüller spaces and the holonomy representation of every projective structure in the interior of $K(X)$ is a quasifuchsian group.

math.DG↗