Generic subgroups of Lie Groups

Jörg Winkelmann

Let G be a real Lie group. We define D_k(G) as the subset of G^k containing all (g₁,..,g_k) such that the g_i together generate a discrete subgroup of G. We discuss measure-theoretic properties of this set. In particular, we prove that this set and its complement have both positive infinite measure if k>1 and G is not amenable. In contrast, for an amenable Lie group G there is a number n such that D_k(G) is of measure zero for k>n and has a complement of measure zero otherwise.
(A real Lie group G is amenable if the quotient of G by its maximal normal solvable Lie subgroup is compact.)

Topology, 41, 163--181 (2002)

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