Limiting distribution of dense orbits in a moduli space of rank $m$ discrete subgroups in $(m+1)$-space

TL;DR

This paper investigates the distribution of dense orbits of certain lattice subgroups acting on a moduli space of rank m-discrete subgroups in (m+1)-space, highlighting the complexity introduced by groups with infinitely many connected components.

Contribution

It extends the understanding of orbit distributions to cases where the acting groups have infinitely many connected components, specifically analyzing the space of rank m-discrete subgroups.

Findings

Describes the limiting distribution of dense orbits in the moduli space.

Identifies the space as a quotient with infinitely many connected components.

Builds on previous work related to random walks on these spaces.

Abstract

We study the limiting distribution of dense orbits of a lattice subgroup $\Gamma\le \text{SL}(m+1,\mathbb{R})$ acting on $H\backslash\text{SL}(m+1,\mathbb{R})$, 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 $H\backslash G$ identifies with $X_{m,m+1}$, a moduli space of rank $m$-discrete subgroups in $\mathbb{R}^{m+1}$. This study is motivated by the work of Shapira-Sargent who studied random walks on $X_{2,3}$.