Snippets

Tits Alternative for abstract Lie groups …

… with finitely many connected components follows from the Tits alternative for linear groups. The reduction appears to be standard (and is used e.g. in the proof of Gromov’s polynomial growth theorem) but not written down anywhere; I am recording it here for future reference.

Let be a Lie group with finitely many connected components. Ad(G) is a subgroup of a linear group; by the Tits Alternative for linear groups, it is either virtually solvable or contains a free group on two generators. In the former case, G, being (a finite extension of) an extension of Ad(G) by Z(G), is virtually solvable as well. In the latter case, the two generators of the free group have pre-images which are generators of a free group in G.

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