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Note: this question was edited after a comment below

I'm reading into classifying spaces for the moment and I have some questions about these things. I'm using the following definition:

Given a group $G$, define the total space $EG$ for this group to be a $G$-CW-complex that is contractible. Next we define $BG=EG/G$ to be the classifying space of $G$.

These spaces seem to exist and be unique up to $G$-homotopy for any group. I was wondering if there are criteria such that if these are satisfied by $G$, then the CW-complex $BG$ is finite (or, more general, of finite type)?

I do not immediatly find an obvious answer. For example to ask that $BG$ is finite already fails when taking $G$ to be finite. i.e. $\mathbb{Z}_2$ has $B\mathbb{Z}_2$ the infinite projective space $\mathbb{R}P^{\infty}$, which has one cell in each dimension.

If someone could recommend additional literature on classifying spaces, please do. This will be highly appreciated!

Tom Ultramelonman
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    Very few of these will be finite. Here is a list which contains some finite ones: http://mathoverflow.net/questions/56363/list-of-classifying-spaces-and-covers?rq=1 . – Thomas Rot Nov 02 '16 at 20:46
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    What people usually mean by "classifying space" is the quotient $BG = (EG)/G$. What you have called $EG$ is never finite (unless $G$ is trivial): if $G$ has an element of finite order then $EG$ must be infinite-dimensional; if $G$ is infinite then $EG$ must have infinitely-many vertices. – Oscar Randal-Williams Nov 02 '16 at 20:48
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    You should check group cohomology. – Alex Degtyarev Nov 02 '16 at 21:06
  • @OscarRandal-Williams Yes, I might have made an error when composing my question. When I say finite, I mean the number of orbits by $G$ are finite. As such the better question would be, when is $BG$ finite or of finite type. I'll change the question! – Tom Ultramelonman Nov 03 '16 at 06:29
  • Suggested references: Brown, Cohomology of groups. Geoghegan, Topological methods in group theory. – YCor Nov 03 '16 at 06:53
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    A discrete group with finite BG is said to be of type F. Many, many groups satisfy this property: all torsion-free hyperbolic groups, for instance. (But notice that they have to be finitely presentable.) – HJRW Nov 03 '16 at 08:02
  • I once wrote some expository notes on classifying spaces which also conatin a list of references; maybe one day when I have a webpage I will also put them there... if you are interested in reading them, please contact me via jreinh AT stanford DOT edu – Jens Reinhold Nov 03 '16 at 09:15
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    The action of $G$ on $EG$ must be free; omitted from your definition. – Peter May Nov 03 '16 at 14:52
  • Have you looked at the Sullivan conjecture? – Sean Tilson Nov 04 '16 at 13:43

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