2025/01/24 by Bochenski, Maciej, Morita, Yosuke
#20H10 #22E40 #22F30 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Primary 57S30 #Representation Theory (math.RT) #Secondary 17B08
paper · doi:10.48550/arxiv.2501.14274
We construct a series of homogeneous spaces G/H of reductive type which admit proper actions of discrete subgroups of G isomorphic to cocompact lattices of O(n,1) (n=2,3,4) but do not admit proper actions of non-compact semisimple subgroups of G. The existence of such homogeneous spaces was previously not known even for n=2. Our construction of proper actions of discrete subgroups is based on Guéritaud-Kassel's work on convex cocompact subgroups of O(n,1) and Danciger-Guéritaud-Kassel's work on right-angled Coxeter groups. On the other hand, the non-existence of proper actions of non-compact semisimple subgroups is proved by the theory of nilpotent orbits and elementary combinatorics.