TITLE: Fixed point theorems and equiangular lines
QUESTION [12 upvotes]: I've been thinking about the equiangular lines (or SIC-POVM) conjecture, 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.

REPLY [7 votes]: 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.