Homogeneous space with non virtually abelian discontinuous groups but without any proper SL(2,R)-action

2015/03/07 by Takayuki Okuda, Okuda, Takayuki
Mathematics · #22E40 #22F30 #53C30 #53C35 (Secondary) #57S30 (Primary) #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Finite Group Theory Research #Mathematical Analysis and Transform Methods #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1503.02186

openalex publication_date 2015/03/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of "continuous analogue" gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space G/H admits a discontinuous group which is not virtually abelian if and only if G/H admits a proper SL(2,R)-action (T. Okuda [J. Differential Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces G/H of reductive type. We give a counterexample in the case G = SL(5,R).

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