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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.
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 …
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 …
3
votes
Most important mathematical results in last 30 years
Kahn--Markovic's proofs of the Surface Subgroup conjecture and the Ehrenpreis conjecture.
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 …
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 …
3
votes
Geometric or topological results from group theory
I'm still a little uncertain about this question, but I'll try to say something about the Virtual Haken conjecture (discussed above) and in the process explain why I think it's a good example.
The Vi …
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.
5
votes
Fixed point theorems
The main theorem of Smith theory asserts that if a $p$-group $G$ acts on a mod-$p$-acyclic space $X$ (which must also be 'finitistic', a fairly weak condition), then the fixed point set $X^G$ is also …
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, …
1
vote
Examples of results first proved using geometrical methods?
I gave an example here of a topological proof that a product of two commutators in a free group is not itself always a commutator. In answer to the same question, Arturo Magidin indicated how to give …
2
votes
Examples of results first proved using geometrical methods?
The proof of Stallings's Ends Theorem is topological. Note that the set of ends of a group $\Gamma$ can be identified with $H^1(\Gamma,\mathbb{Z}_2\Gamma)$, so you don't have to define ends geometric …
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 …
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 …
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 …
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) …