# Thread: Question about Asymptotic and Geometric Group theory

1. ## 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).

2. Originally Posted by Turloughmack
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

3. Originally Posted by tonio
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...

4. Originally Posted by Swlabr
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

5. 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...