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I heard the phrase from many mathematician using in the colloquials. I heard algebraic geometer using it. I was never bother about it until one of my professor responded to one my question as follows:

The graph product is by definition the fundamental group of the space K obtained from the underlying graph by blowing up.

what this phrase means in that context.

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Madeel, the phrase "graph product" has a number of meanings. I suspect your professor is referring to viewing a graph of groups as the fundamental group of a graph of spaces. There's a blog post about that here, which might help: 392c.wordpress.com/2009/03/08/… . –  HJRW Feb 16 '10 at 23:15
    
Adeel, I now realise that you have seen the blog entry in question already! :) –  HJRW Feb 17 '10 at 4:46
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Here the graph product is defined as in the Hatcher.

Anyway I am posting a response by my professor which has answered my question:

In algebraic geometry, blowing up is a process of replacing a point inan algebraic variety by a projective space. In graph products, it is the process of replacing each vertex v of the graph by K(G_v,1), and each edge by the appropriate mapping cylinder (as described in Hatcher).

See section 1B on K(G,1) spaces for further detail.

\Adeel

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