Large sets of generating tuples for Lie groups

TL;DR

This paper investigates the properties of generating pairs and subgroups in connected, semisimple linear Lie groups, showing their generative sets are dense and well-behaved in various topologies.

Contribution

It establishes that spaces of generating pairs and conjugates of circle subgroups are Zariski-open, Euclidean-open, and dense, providing a comprehensive understanding of generative structures in Lie groups.

Findings

Generating pairs form Zariski-open sets in the compact case.

Generating pairs are Euclidean-open in general.

Spaces of conjugates of generic circle subgroups are dense and Zariski-open.

Abstract

We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficiently generic circle subgroups $H_i$, $i=1,2$ of $G$, the space of conjugates of $H_i$ that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra $Lie(G)$.