On relatively free subsets of Lie groups

Abstract

In an arbitrary neighborhood U of the identity e of a connected Lie group there is a subset S of cardinality c and relatively free , i.e., the only nontrivial equations xf'x|2 • • • xjfe, 8, « ± 1, satisfied by substitution for distinct symbols among the xi distinct elements of S are equations that are identities throughout G. 0. In (2) the existence of the free topological group F(X) associated with a completely regular space A is shown by a construction involving quaternions. In brief (and corrected) form, the argument proceeds as follows: the (algebraic) free group F0(A) generated by A is embedded isomorphically in the group Hloo(A) = PfHXp the Cartesian product of the multiplicative group Hj of quaternions of norm 1 (each Ht/ = H,) where the index/ranges over C(A, H(), the set of continuous maps /: A—>H,. In this embedding, A preserves its topology and FQ(X) is endowed with the topology of a topologi- cal group. Standard results show that ?Tmax = sup{?T: F0(A) is a topological group in the topology 9 and A inherits its topology from 5} = sup{TGF0(A)} produces F(A). The role played by the compact group Hloo(A) is that of insuring that the set TGF0(X) is nonempty. The argument hinges on the existence of an infinite free subset of H,. Since H, is a connected Lie group, a question related to this aspect of H, is explored below and is answered as follows: In every connected Lie group G there is a subset S, of cardinality c = card(R) and as free as any subset of G can be: if a word evaluated on 5 yields e (the identity of G), then the word is identically e on G (see below for details). In particular, if no (word) identities hold universally in G, then S is free. 1. The proof depends on two lemmas, the second of which emerged in the form given below as a result of illuminating conversations with Professor S. Schanuel of SUNY/Buffalo. The first lemma is given in (1), (4). The proof below is somewhat more elementary than that in (4) and is therefore included here.