On a type of maximal abelian torsion free subgroups of connected Lie groups

For an arbitrary real connected Lie group G we define \(\mathrm {p}(G)\) as the maximal integer p such that \(\mathbb {Z}^p\) is isomorphic to a discrete subgroup of G and \(\mathrm {q}(G)\) is the maximal integer q such that \(\mathbb {R}^q\) is isomorphic to a closed subgroup of G. The aim of this paper is to investigate properties of these two invariants. We will show that if G is a noncompact connected Lie group, then \(1\le \mathrm {q}(G)\le \mathrm {p}(G)\le \dim (G/K)\) where K is a maximal compact subgroup of G. In the cases when G is an exponential Lie group or G is a connected nilpotent Lie group, we give explicit relationships between these two invariants and a well known Lie algebra invariant \(\mathcal M(\mathfrak {g})\), i.e. the maximum among the dimensions of abelian subalgebras of the Lie algebra \(\mathfrak {g}:=\mathrm {Lie}(G)\).