Timeline for Examples of results first proved using geometrical methods?
Current License: CC BY-SA 3.0
5 events
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| Sep 5, 2011 at 20:56 | comment | added | Benjamin Steinberg | @Agol, I'm pretty sure that people have used length to describe the number of letters in a word since the time of Post, Thue and Dyck, if not earlier. Length of a word is just a combinatorial notion. But I agree nobody considered growth before Milnor and Svarc, who were geometrically motivated. But people could have, and the notion is considered for finitely generated algebras. | |
| Sep 5, 2011 at 15:20 | comment | added | Ian Agol | @Ben: I agree it's combinatorial, but since you used the term "length", you (and others) are clearly thinking of it in a geometric fashion. In fact, I think Milnor originally introduced the notion of growth of groups with geometric applications in mind. A choice of generating set (and therefore Cayley graph) is in some sense a geometric choice, not intrinsic to the group (like the choice of a Riemannian metric on a manifold). The answer "virtually nilpotent" is free of notions of length, which makes the theorem interesting. | |
| Sep 5, 2011 at 0:33 | comment | added | Benjamin Steinberg | @Agol, although one can, and should, think of growth in terms of balls in the Cayley graph the definition of growth is purely combinatorial. You are counting the number of elements that have a shortest length representative of size at most n. People consider growth of other algebraic structures like semigroups and algebras where one doesn't have the corresponding geometry. | |
| Sep 4, 2011 at 17:33 | comment | added | Ian Agol | This is a good example, but since the definition of "polynomial growth" is geometric, maybe it's not surprising that geometry is an ingredient in the solution. | |
| Sep 4, 2011 at 17:18 | history | answered | Benjamin Steinberg | CC BY-SA 3.0 |