arXiv subjects

Hiroaki Nagaya

Publications and source records attributed to Hiroaki Nagaya.

3 recordsLinked to original sources

arXiv2026

Proper Actions on Homogeneous Spaces of the Upper Triangular Matrix Group via Coarse Geometry

We study proper actions on homogeneous spaces of the group $\mathrm{B}(n)$ of invertible upper triangular matrices from a coarse-geometric viewpoint. We obtain sufficient conditions for the properness of the natural $L$-action on $\mathrm{B}(n)/H$, where $L,H$ are closed subgroups of $\mathrm{B}(n)$. We also give an explicit sufficient condition for properness of the $L$-action on a homogeneous space naturally identified with $\mathrm{B}(n)/H\cong \mathbb{R}^{n-1}$. In addition, we discuss extensions to the case where the ambient group $G$ is a subgroup of $\mathrm{B}(n)$ and $L,H$ are closed subgroups of $G$, with particular attention to the unipotent subgroup $\mathrm{N}(n)$.

math.DG↗

arXiv2026

Some categorical remarks on coarse subspaces of coarse spaces

In this paper, we provide a categorical framework for understanding coarse subspaces of coarse spaces. First, we introduce the notion of a controlled total relation between coarse spaces and show that the category whose morphisms are closeness classes of controlled total relations is isomorphic to the conventional category of coarse spaces defined using closeness classes of controlled maps. Next, we show that the assignment associating to each coarse space the finite-join partially ordered set of its coarse subspaces is functorial, and prove that this partially ordered set is naturally isomorphic to the poset of subobjects in the category of coarse spaces. Furthermore, we formulate asymptotic disjointness between coarse subspaces and show that mono-morphisms preserve this relation. These results provide a categorical interpretation of the framework of coarse subspaces introduced by Leitner--Vigolo [Lecture Notes in Math.~(2023)] and characterize coarse subspaces as objects intrinsic to the category of coarse spaces. They also provide a foundation for a coarse-geometric interpretation of the properness criterion established by Kobayashi [Math.~Ann.~(1989); J.~Lie Theory (1996)] and Benoist [Ann.~of Math.~(1996)] (cf.~Nagaya--Ogawa--Okuda [Proc.~Japan Acad.~Ser.~A (2025)]).

math.CT↗

arXiv2025

A characterization of proper actions with bornology and coarse geometry

In 1961, Palais showed that every smooth proper Lie group action on a smooth manifold admits a compatible Riemannian metric on the manifold such that the action becomes isometric. In 2006, Yoshino studied a continuous proper action of a locally compact Hausdorff group on a locally compact Hausdorff space, and showed that the space carries a compatible uniform structure making the action equi continuous in an appropriate setting. In this paper, we focus on bornological proper actions on bornological spaces and prove that the space admits a compatible coarse structure such that the action becomes equi controlled.

math.DG↗