I've been thinking about the [equiangular lines (or SIC-POVM) conjecture](http://en.wikipedia.org/wiki/SIC-POVM), and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking specifically of geometric fixed point theorems, like Brouwer's. So my (rather vague) questions are:


1) is there some good survey article or classification for fixed point theorems?  

2) are there fixed-point theorems which are related to actions of groups on geometric spaces?  

3) has anybody tried this idea?


*Added:* In response to Joe's comment below, let me note that while the motivation is from quantum information theory, the equiangular lines conjecture is a purely classical geometry problem (see my comment below). The conjecture is really intriguing: numerical constructions of sets of equiangular lines have been found up to dimension 67, at which point the computer time required exceeded the patience of the investigators. However, only a handful of these numerical solutions have been shown to be rigorously correct by finding corresponding algebraic numbers. See [this recent paper](http://arxiv.org/abs/0910.5784).



Fixed point theorems and equiangular lines
The book "Fixed point theory" by Dugundji and Granas is a nice reference. The headers of the sections in the book give some kind of classification of fixed point theorems.


* results based on compactness
* order theoretic results
* results based on convexity
* Borsuk theorem and topological transitivity
* homology and fixed points
* Leray-Shauder degree and fixed point index


Part VI of the bibliography is really extensive and contains a finer classification of fixed point theorems.


