The Milnor-Wolf theorem says that any solvable group has either polynomial or exponential growth. I wonder about the existence of alternative proofs of this fact. I have an impression that the original proof is more brutal and less ideological. If you have a citation or a suggestion of an alternative proof, please post.
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1$\begingroup$ Would you say something about what the original proof is? the only proof I have in mind consists in proving 1) if $G$ is f.g. virtually nilpotent, then it has polynomial growth 2) if $G$ is f.g. solvable and not virtually nilpotent, then it has a free subsemigroup. There are probably several proofs of (2). $\endgroup$– YCorCommented Oct 11, 2014 at 16:57
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$\begingroup$ I could check Milnor's paper but not Wolf's. Milnor (J. Diff Geom. 2, p447-449, 1968) checks that if $G$ is f.g. solvable but not polycyclic then it has exponential growth. His proof actually shows it contains a free subsemigroup (see the reasoning by contradiction in his Lemma 1), although he doesn't say it. $\endgroup$– YCorCommented Oct 11, 2014 at 17:30
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$\begingroup$ Yes, Yves, this is exactly how it goes. What are the alternative proofs of 2? $\endgroup$– Kate JuschenkoCommented Oct 11, 2014 at 19:10
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$\begingroup$ There's one proof of 2 using the fact, due to Groves (1978), that a non-vn f.g. solvable group has a homomorphism into $K^*\ltimes K$ with non-vn image for some non-discrete locally compact field $K$, and using this we can play ping-pong on the tree/hyperbolic 2/3-space, without distinction between the polycyclic and non-polycyclic case. $\endgroup$– YCorCommented Oct 11, 2014 at 19:42
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$\begingroup$ But the original proof has the advantage to generalize to elementary amenable groups (Chou, 1980). $\endgroup$– YCorCommented Oct 11, 2014 at 19:45
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There seems to be a different proof of a sharper result by our own T. Tao: http://terrytao.wordpress.com/tag/milnor-wolf-theorem/
answered Oct 12, 2014 at 3:53