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Math Help - A continuum of homomorphic images

  1. #1
    MHF Contributor Swlabr's Avatar
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    A continuum of homomorphic images

    Let G be a finitely-generated (and thus countable) group and let H_i be a homomorphic image of G such that H_i \not\cong H_j for i \neq j with i, j \in I where I is some index set with cardinality equal to that of the reals.

    Such groups do exists (for example, non-elementary hyperbolic groups), and I initially found this quite surprising (although there are uncountably many maps from the natural numbers to itself, so it does make sense). My question is...is there an *easy* proof of the existence of such groups? I can't seem to conjure up one.
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  2. #2
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    Quote Originally Posted by Swlabr View Post
    Let G be a finitely-generated (and thus countable) group and let H_i be a homomorphic image of G such that H_i \not\cong H_j for i \neq j with i, j \in I where I is some index set with cardinality equal to that of the reals.

    Such groups do exists (for example, non-elementary hyperbolic groups), and I initially found this quite surprising (although there are uncountably many maps from the natural numbers to itself, so it does make sense). My question is...is there an *easy* proof of the existence of such groups? I can't seem to conjure up one.

    How did you come to know that non-elementary hyperbolic groups are like the above? To me it looks pretty non-intuitive since every homomorphic image of a f.g. group its completely determined by the image of a generator set, and it doesn't look easy to create a non-countable set of non-isomorphic homomorphic images out of such a f.g. group...

    Tonio
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    How did you come to know that non-elementary hyperbolic groups are like the above? To me it looks pretty non-intuitive since every homomorphic image of a f.g. group its completely determined by the image of a generator set, and it doesn't look easy to create a non-countable set of non-isomorphic homomorphic images out of such a f.g. group...

    Tonio
    I was reading this paper (it is in Groups, Geometry and Dynamics, vol. 3 (2009), issue 3, pp.423-452, but the link is to ArXiV) and started to wonder...
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