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Results tagged with big-list
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user 1463
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
47
votes
Most interesting mathematics mistake?
Not just a great mistake, but also a great documentation of a mistake: Stallings's How not to prove the Poincaré Conjecture.1 (I think this paper is my answer to every community-wiki question.)
1Here …
43
votes
What are some of the big open problems in 3-manifold theory?
ADDED (29 May, 2013)
As has been pointed out in the comments, there has been great progress since this answer was first written, and the conjectures below have now been proved, thanks to ground-breaki …
37
votes
What are your favorite instructional counterexamples?
The Baumslag--Solitar groups have presentations
$BS(p,q)=\langle a,b\mid a^p=b^{-1}a^q b\rangle$.
They have the following nice properties:
they're two generator, one relator groups;
they can be wr …
17
votes
A single paper everyone should read?
Stallings's How Not To Prove the Poincare Conjecture (cached at Citeseer) is the funniest paper I've ever read.
17
votes
Applications of the notion of of Gromov-Hausdorff distance
Gromov's Theorem was, as far as I'm aware, the first but very far from the last application of Gromov–Hausdorff distance to group theory. One particularly fruitful line of reasoning starts with a seq …
17
votes
How helpful is non-standard analysis?
The asymptotic cone of a metric space (and hence of a finitely generated group endowed with the word metric) is constructed using non-standard analysis, and has been used to prove many nice theorems. …
- 25.2k
15
votes
Accepted
Judging whether a finitely presented group is a 3-manifold group?
Apologies for the shameless self-promotion, but as you ask for necessary conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'.
Aschenbrenner, …
11
votes
Examples of residually-finite groups
It's perhaps worth mentioning a powerful construction of finitely presented residually finite groups due to Bridson--Grunewald (though the residual finiteness comes from work of Wise).
Wise produced a …
10
votes
Research-level mathematical bookstores
I haven't been there in a while, but Foyles (in London) used to have an excellent selection of mathematics books.
10
votes
Interesting applications of the pigeonhole principle
The easiest proof I know of the Morse Property for word-hyperbolic groups (which says that quasigeodesics are uniformly close to geodesics) uses the pigeon-hole principle several times.
10
votes
Results from abstract algebra which look wrong (but are true)
Let $F$ be a non-abelian free group and let $G=\prod_\omega F$ be the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharla …
6
votes
Most important mathematical results in last 30 years
Perelman's proof of the Geometrization conjecture (see here, here and here) was the crowning achievement of decades of work. It was the most important of Thurston's conjectures about the topology of …
6
votes
What are some good group theory references?
For infinite discrete groups:
Lyndon & Schupp is authoritative for classical, combinatorial methods.
Bridson & Haefliger has a lot of material for more geometric classes, like hyperbolic and CAT(0) …
5
votes
Most important mathematical results in last 30 years
Agol's proof of the Virtual Haken conjecture was a wonderful application of the tools developed by Wise and his coauthors in geometric group theory to 3-manifold topology. The Virtual Haken conjectur …
5
votes
Accepted
Fundamental theorems
In his book Topics in Geometric Group Theory, Pierre de la Harpe calls the following result the Fundamental Observation of Geometric Group Theory (though he also calls it a theorem!). It is also ofte …