2023/09/15 by Kazuki Kannaka, Takayuki Okuda, Kannaka, Kazuki +3
Mathematics · #22F30 #30F35 #30F60 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Primary 57S30 #Representation Theory (math.RT) #Secondary 22E40
paper · pdf · doi:10.48550/arxiv.2309.08331
openalex publication_date 2023/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,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,ℝ), then Γ admits a small deformation in G whose Zariski closure is a reductive subgroup of the same real rank as G.