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I'll be delighted to get some help from the experts around here, regarding the following questions:

1) If we take the collection of all rank 2 groups. Does this collection countable? Unctountable? Does someone know about any work concerning putting a topology on the collction of all groups or maybe on the collection of all rank 2 groups?

2) Does someone know about works concerning estimations on the number of metabelian groups, or on their density in the collection of groups?

3) Does someone know about any works about estimating the density of groups satisfying some property? i.e.- given a family of groups defined by a property $X$ , what is the limit of $\frac{no. X}{no. grps}$ when the order becomes infinite?

Thanks in advance !

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What do you mean by 'rank'? I assume you mean the size of a minimal generating set. – HJRW May 26 '12 at 14:51
Also, you should retag your question – HJRW May 26 '12 at 14:55
This is indeed what I mean Thanks – jason mfash May 26 '12 at 16:32

In answer to your first question, you might want to look up the Gromov--Grigorchuk topology on the space of marked groups, which topologizes the set of 'marked' groups of rank $r$, where a marking is a choice of generating set of size $r$. References can be found in this paper of Champetier and Guirardel.

It's very well known that there are uncountably many groups of rank two. To see this, one constructs a family of groups of rank two with uncountably many different isomorphism types of $H_2$.

Yves de Cornulier, who is active on MO, is an expert on the subspace of metabelian groups in the space of marked groups.

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Thanks ! I'll be glad to receive further suggestions ! Thanks again ! – jason mfash May 26 '12 at 16:13

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