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Math Help - Question about Asymptotic and Geometric Group theory

  1. #1
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    Question about Asymptotic and Geometric Group theory

    1.Show that a free product of finite group is hyperbolic.

    2. Let G, H be finitely generated groups with growth series g(z) and h(z) and generating sets X and Y. Then show that the growth series for [G x H] with respect to the disjoint union of X and Y is g(z).h(z).
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  2. #2
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    Quote Originally Posted by Turloughmack View Post
    1.Show that a free product of finite group is hyperbolic.


    Here it must be a free finite product of finite groups, otherwise we end with a non-fin. gen. group which can't then

    be hyperbolic.

    Now we can use the fact that finite groups are (trivially) hyperbolic groups and thus a finite product, either free or cartesian, of

    them is hyperbolic, too.


    2. Let G, H be finitely generated groups with growth series g(z) and h(z) and generating sets X and Y. Then show that the growth series for [G x H] with respect to the disjoint union of X and Y is g(z).h(z).
    This seems to me to follow immediately from the definitions, but I'm really not that sure now. Check this and I will, too.

    Tonio
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    This seems to me to follow immediately from the definitions, but I'm really not that sure now. Check this and I will, too.

    Tonio
    How do you differ between a free product and a free finite product? I have come across this distinction before...
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    Quote Originally Posted by Swlabr View Post
    How do you differ between a free product and a free finite product? I have come across this distinction before...

    I'm not sure I completely understand your question, but what I meant to convey is that

    a free finite product has a finite number of factors, and a free infinite product has

    an infinite number of factors.

    Tonio
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Oh right - I have always read `a free product of groups' as `a free product of two groups'. I have never contemplated an infinite free product!

    But then - I very rarely contemplate non-finitely generated groups...
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