arXiv subjects

Kazuki Kannaka

Publications and source records attributed to Kazuki Kannaka.

6 recordsLinked to original sources

arXiv2026

Hurwitz-Radon numbers and proper actions of semisimple Lie groups

We study proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces. Motivated by Okuda's classification of semisimple symmetric spaces admitting proper $SL(2,\mathbb{R})$-actions [J. Differential Geom., 2013], we focus on symmetric spaces lying on the boundary of the existence of proper $SL(2,\mathbb{R})$-actions. As a rigidity result, we show that any connected non-compact semisimple Lie group acting properly on these symmetric spaces must be globally isomorphic to $Spin(n,1)$ up to compact factors. Moreover, the Hurwitz-Radon number arises as the largest value of $n$ for the existence of $Spin(n,1)$-proper actions. Our symmetric spaces include the pseudo-Riemannian hyperbolic space $\mathbf{H}_{+}^{N,N-1}$ of signature $(N,N-1)$.

math.DG↗

arXiv2025

Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces

Let $X=G/H$ be a homogeneous space of a Lie group $G$. When the isotropy subgroup $H$ is non-compact, a discrete subgroup $Γ$ may fail to act properly discontinuously on $X$. In this article, we address the following question: in the setting where $G$ and $H$ are reductive Lie groups and $Γ\backslash X$ is a standard quotient, to what extent can one deform the discrete subgroup $Γ$ while preserving the proper discontinuity of the action on $X$? We provide several classification results, including conditions under which local rigidity holds for compact standard quotients $Γ\backslash X$, when a standard quotient can be deformed into a non-standard quotient, a characterization of the largest Zariski-closure of discontinuous groups under small deformations, and conditions under which Zariski-dense deformations occur.

math.DG↗

arXiv2025

Deformations of Standard Locally Homogeneous Spaces

Let $X=G/H$ be a homogeneous space, where $G \supset H$ are reductive Lie groups. We ask: in the setting where $Γ\backslash G/H$ is a standard quotient, to what extent can the discrete subgroup $Γ$ be deformed while preserving the proper discontinuity of the $Γ$-action on $X$? We provide several classification results, including: conditions under which local rigidity holds for compact standard quotients $Γ\backslash X$; criteria for when a standard quotient can be deformed into a nonstandard one; a characterization of the maximal Zariski-closure of discontinuous groups under small deformations; and conditions under which Zariski-dense deformations occur. Proofs of the results stated in this paper are provided in detail in arXiv:2507.03476.

math.DG↗

arXiv2025

Zariski dense discontinuous surface groups for reductive symmetric spaces

Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The study of deformations of discontinuous groups for $G/H$ was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group $Γ$ admits a non-standard small deformation as a discontinuous group for $G/H$ if $Γ$ is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we also prove that if $G/H$ is a symmetric space and admits some non virtually abelian discontinuous groups, then $G$ contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on $G/H$. As a key part of our proofs, we show that for a discrete surface subgroup $Γ$ of high genus contained in a reductive group $G$, if the Zariski closure of $Γ$ is locally isomorphic to $SL(2,\mathbb{R})$, then $Γ$ admits a small deformation in $G$ whose Zariski closure is a reductive subgroup of the same real rank as $G$.

math.DG↗

arXiv2023

Counting orbits of certain infinitely generated non-sharp discontinuous groups for the anti-de Sitter space

Inspired by an example of Gueritaud-Kassel [Geom. Topol. 2017], we construct a family of infinitely generated discontinuous groups $Γ$ for the 3-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$. These groups are not necessarily sharp (a kind of "strong" properly discontinuous condition introduced by Kassel and Kobayashi [Adv. Math. 2016]), and we give its criterion. Moreover, we find upper and lower bounds of the counting $N_Γ(R)$ of a $Γ$-orbit contained in a pseudo-ball $B(R)$ as the radius $R$ tends to infinity. We then find a non-sharp discontinuous group $Γ$ for which there exist infinitely many $L^2$-eigenvalues of the Laplacian on the noncompact anti-de Sitter manifold $Γ\backslash\mathrm{AdS}^{3}$, by applying the method established by Kassel-Kobayashi. We also prove that for any increasing function $f$, there exists a discontinuous group $Γ$ for $\mathrm{AdS}^{3}$ such that the counting $N_Γ(R)$ of a $Γ$-orbit is larger than $f(R)$ for sufficiently large $R$.

math.GR↗

arXiv2021

Linear Independence of Generalized Poincaré Series for Anti-de Sitter 3-Manifolds

Let $Γ$ be a discrete group acting properly discontinuously and isometrically on the three-dimensional anti-de Sitter space $\mathrm{AdS}^{3}$, and $\square$ the Laplacian which is a second-order hyperbolic differential operator. We study linear independence of a family of generalized Poincaré series introduced by Kassel-Kobayashi [Adv. Math. 287 (2016), 123-236, arXiv:1209.4075], which are defined by the $Γ$-average of certain eigenfunctions on $\mathrm{AdS}^{3}$. We prove that the multiplicities of $L^{2}$-eigenvalues of the hyperbolic Laplacian $\square$ on $Γ\backslash\mathrm{AdS}^{3}$ are unbounded when $Γ$ is finitely generated. Moreover, we prove that the multiplicities of stable $L^{2}$-eigenvalues for compact anti-de Sitter 3-manifolds are unbounded.

math.SP↗