Timeline for Milnor-Wolf result on growth of solvable groups
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2014 at 17:41 | comment | added | YCor | Osin actually improved Chou's result, see arxiv.org/abs/math/0404075, and indeed had to go into the technicalities. He proves uniform exponential growth. Actually there are a few other improvements that are known for non-vn f.g. solvable groups, but not known for elementary amenable groups, such as: - the existence of a QI-embedded free subsemigroup on 2 generators - exponential conjugacy growth. | |
| Oct 12, 2014 at 16:31 | comment | added | Kate Juschenko | I am also interested in generalization of Chou to the class of elementary amenable groups. It would wonderful if this can be explained in a better form. All books that discuss amenability deliberately avoid these proofs, since it is too technical. | |
| Oct 12, 2014 at 3:53 | answer | added | Igor Rivin | timeline score: 3 | |
| Oct 11, 2014 at 23:36 | comment | added | Ian Agol | Certain special classes of solvable groups are covered by the simple argument in the appendix to Gromov's paper on groups of polynomial growth by Tits. For example, the argument applies to a lattice in a 3-dimensional solvable Lie group. The point is that a finitely generated abelian-by-cyclic group has exponential growth if the cyclic group acts on the abelian subgroup with an eigenvalue $>1$ in absolute value. Maybe one can show that a non-vn solvable group contains such a subgroup or else contains a solvable Baumslag-Solitar subgroup? | |
| Oct 11, 2014 at 19:45 | comment | added | YCor | But the original proof has the advantage to generalize to elementary amenable groups (Chou, 1980). | |
| Oct 11, 2014 at 19:42 | comment | added | YCor | 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. | |
| Oct 11, 2014 at 19:10 | comment | added | Kate Juschenko | Yes, Yves, this is exactly how it goes. What are the alternative proofs of 2? | |
| Oct 11, 2014 at 17:30 | comment | added | YCor | 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. | |
| Oct 11, 2014 at 17:29 | history | edited | Alain Valette | CC BY-SA 3.0 |
Corrected 2 typos
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| Oct 11, 2014 at 16:57 | comment | added | YCor | 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). | |
| Oct 11, 2014 at 16:49 | history | asked | Kate Juschenko | CC BY-SA 3.0 |