arXiv subjects

Dounnu Sasaki

Publications and source records attributed to Dounnu Sasaki.

4 recordsLinked to original sources

arXiv2025

Counting subgroups via Mirzakhani's curve counting

Given a hyperbolic surface $Σ$ of genus $g$ with $r$ cusps, Mirzakhani proved that the number of closed geodesics of length at most $L$ and of a given type is asymptotic to $cL^{6g-6+2r}$ for some $c>0$. Since a closed geodesic corresponds to a conjugacy class of the fundamental group $π_1(Σ)$, we extend this to the counting problem of conjugacy classes of finitely generated subgroups of $π_1(Σ)$. Using `half the sum of the lengths of the boundaries of the convex core of a subgroup' instead of the length of a closed geodesic, we prove that the number of such conjugacy classes is similarly asymptotic to $cL^{6g-6+2r}$ for some $c>0$. As a special case, these conjugacy classes can be interpreted as subsurfaces of $Σ$ via their convex cores, and the result can be viewed as counting subsurfaces of a given type. Furthermore, we see that the above length measurement for subgroups is `natural' within the framework of subset currents, which serve as a completion of weighted conjugacy classes of finitely generated subgroups of $π_1(Σ)$.

math.GT↗

arXiv2022

Currents on cusped hyperbolic surfaces and denseness property

The space $\mathrm{GC} (Σ)$ of geodesic currents on a hyperbolic surface $Σ$ can be considered as a completion of the set of weighted closed geodesics on $Σ$ when $Σ$ is compact, since the set of rational geodesic currents on $Σ$, which correspond to weighted closed geodesics, is a dense subset of $\mathrm{GC}(Σ)$. We prove that even when $Σ$ is a cusped hyperbolic surface with finite area, $\mathrm{GC}(Σ)$ has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on $Σ$ but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to $\mathrm{GC}(Σ)$. To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.

math.GT↗

arXiv2019

Subset currents on surfaces

Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group $π_1 (Σ)$ of a compact hyperbolic surface $Σ$. Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on $π_1(Σ)$, which we call subset currents on $Σ$. We prove that the space $\mathrm{SC}(Σ)$ of subset currents on $Σ$ is a measure-theoretic completion of the set of conjugacy classes of non-trivial finitely generated subgroups of $π_1 (Σ)$, each of which geometrically corresponds to a convex core of a covering space of $Σ$. This result was proved by Kapovich-Nagnibeda in the case of free groups, and is also a generalization of Bonahon's result on geodesic currents on hyperbolic groups. We will also generalize several other results of them. Especially, we extend the (geometric) intersection number of two closed geodesics on $Σ$ to the intersection number of two convex cores on $Σ$ and, in addition, to a continuous $\mathbb{R}_{\geq 0}$-bilinear functional on $\mathrm{SC}(Σ)$.

math.GT↗

arXiv2015

An intersection functional on the space of subset currents on a free group

Kapovich and Nagnibeda introduced the space $\mathcal{S} {\rm Curr}(F_N)$ of subset currents on a free group $F_N$ of rank $N\geq 2$, which can be thought of as a measure-theoretic completion of the set of all conjugacy classes of finitely generated subgroups of $F_N$. We define a product $\mathcal{N} (H,K)$ of two finitely generated subgroups $H$ and $K$ of $F_N$ by the sum of the reduced rank $\overline{\rm rk}(H\cap gKg^{-1})$ over all double cosets $HgK\ (g\in F_N)$, and extend the product $\mathcal{N}$ to a continuous symmetric $\mathbb{R}_{\geq 0}$-bilinear functional $\mathcal{N} \colon \mathcal{S} {\rm Curr} (F_N)\times \mathcal{S} {\rm Curr} (F_N)\to \mathbb {R}_{\geq 0}$. We also give an answer to a question presented by Kapovich and Nagnibeda. The definition of $\mathcal{N}$ originates in the Strengthened Hanna Neumann Conjecture, which has been proven by Mineyev and can be stated as follows: $\mathcal{N} (H,K)\leq \overline{\rm rk} (H) \overline{\rm rk} (K)$ holds for any finitely generated subgroups $H$ and $K$ of $F_N$. As a corollary to our theorem, this inequality is generalized to the inequality for subset currents.

math.GR↗