Compact Clifford-Klein forms of homogeneous spaces of reductive and nonreductive types

2018 Fiscal Year Final Research Report

• Project/Area Number: 17H06784
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Morita Yosuke 京都大学, 理学研究科, 助教 (70804318)
• Project Period (FY): 2017-08-25 – 2019-03-31
• Keywords: 幾何学 / Lie群 / 等質空間 / 固有な作用 / Clifford-Klein群

Outline of Final Research Achievements

I studied proper actions on homogeneous spaces. When G/H is a semisimple symmetric space satisfying some condition, I constructed a homogeneous space G/H_θ of nonreductive type such that a discrete subgroup of G acts properly and cocompactly on G/H_θ if and only if it acts properly and cocompactly on G/H. One can prove that G/H_θ does not admit a compact Clifford-Klein form (i.e. does not admit a proper cocompact action), which implies that G/H cannot have a compact Clifford-Klein form. I also found that H_θ can be applied to the nonexistence of compact Clifford-Klein forms in some nonsymmetric cases.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

1. コンパクト Clifford-Klein 形の存在問題は、1980年代後半から様々な手法によって研究されてきたが、非簡約型の等質空間を利用して簡約型等質空間に関する結果を導くという手法は新しい。
2. 簡約型等質空間上の固有な作用と、その「極限」（geometric transition）に現れる非簡約型等質空間上の固有な作用の間に関係がつけられる、という現象がいくつかの場合で観察されており（小林-吉野、Danciger-Gueritaud-Kassel）、今回の研究で得られた結果もその一例とみなせるかもしれない。