# Thread: A continuum of homomorphic images

1. ## A continuum of homomorphic images

Let $\displaystyle G$ be a finitely-generated (and thus countable) group and let $\displaystyle H_i$ be a homomorphic image of $\displaystyle G$ such that $\displaystyle H_i \not\cong H_j$ for $\displaystyle i \neq j$ with $\displaystyle i, j \in I$ where $\displaystyle I$ 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.

2. Originally Posted by Swlabr
Let $\displaystyle G$ be a finitely-generated (and thus countable) group and let $\displaystyle H_i$ be a homomorphic image of $\displaystyle G$ such that $\displaystyle H_i \not\cong H_j$ for $\displaystyle i \neq j$ with $\displaystyle i, j \in I$ where $\displaystyle I$ 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

3. Originally Posted by tonio
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...