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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!

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    $\begingroup$ Very few of these will be finite. Here is a list which contains some finite ones: mathoverflow.net/questions/56363/… . $\endgroup$
    – Thomas Rot
    Commented Nov 2, 2016 at 20:46
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    $\begingroup$ 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. $\endgroup$
    – Oscar Randal-Williams
    Commented Nov 2, 2016 at 20:48
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    $\begingroup$ You should check group cohomology. $\endgroup$
    – Alex Degtyarev
    Commented Nov 2, 2016 at 21:06
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    $\begingroup$ 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.) $\endgroup$
    – HJRW
    Commented Nov 3, 2016 at 8:02
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    $\begingroup$ The action of $G$ on $EG$ must be free; omitted from your definition. $\endgroup$
    – Peter May
    Commented Nov 3, 2016 at 14:52

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