Limiting Distribution of Dense Orbits in a Moduli Space of Rank m Discrete Subgroups in (m+1)-Space

We study the limiting distribution of dense orbits of a lattice subgroup Γ ≤ SL(m+1,R) acting on H SL(m+1,R), with respect to a filtration of growing norm balls. The novelty of our work is that the groups H we consider have infinitely many non-trivial connected components. For a specific such H, the homogeneous space H G identifies with Xm,m+1, a moduli space of rank m-discrete subgroups in Rᵐ⁺¹. This study is motivated by the work of Shapira-Sargent who studied random walks on X2,3.