Remarks on discrete subgroups with full limit sets in higher rank Lie groups

TL;DR

This paper investigates the existence and properties of discrete subgroups with full limit sets in higher rank Lie groups, providing criteria and examples related to their structure and boundary behavior.

Contribution

It demonstrates the existence of infinitely generated discrete subgroups with full limit sets and establishes criteria for such properties in specific groups like SL(3,R).

Findings

Existence of infinitely generated discrete subgroups with full limit sets.

Criteria for full limit sets in SL(3,R).

Construction of Zariski-dense subgroups with specific Jordan projections.

Abstract

We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of $G = \operatorname{SL}(3,\mathbb{R})$ must have a full limit set in the Furstenberg boundary of $G$. In the appendix, we show the the existence of Zariski-dense discrete subgroups $\Gamma$ of $\operatorname{SL}(n,\mathbb{R})$, where $n\ge 3$, such that the Jordan projection of some loxodromic element $\gamma \in\Gamma$ lies on the boundary of the limit cone of $\Gamma$.